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EXTERNAL APPROXIMATIONS BY FINITE ELEMENTS

Victor Apanovitch, SIMSOLID Corporation

Proprietary & Restricted Rights Notice

© 2016 SIMSOLID Corporation All Rights Reserved. This document is an unpublished draft of book “External Approximations by Finite Elements” by Victor Apanovitch. It is intended to provide background information on the mathematical foundations used in the commericial software application, SIMSOLID. Unauthorized use, distribution or duplication is prohibited.

Table of Contents Preface Introduction 1. Partitions, spaces, and boundary operators 1.1. Basic definitions 1.2. Spaces associated with a partition 1.3. Generalized Green’s formula 1.4. Properties of the boundary operators

2. The hallmarks of external approximations by finite elements 2.1. Approximations on fixed and variable partitions

2.2. More spaces associated with a partition 2.3. Some auxiliary assertions 2.4. The hallmarks of external approximations

3. Types of finite elements for external approximations and their construction 3.1. Finite elements conformed by a subspace 3.1.1. Element definition 3.1.2. Construction of basic functions of an element 3.1.3. A sufficient condition for existence of an element basis 3.1.4. Virtual finite element 3.2. Homogeneous and composite finite elements 3.3. Spaces of finite elements

4. Error estimates and convergence of external approximations of Sobolev spaces 4.1. Operators of approximations 4.2. Convergence of approximations on a fixed partition 4.3. Convergence of approximations on a family of partitions 4.3.1. Some definitions 4.3.2. Error estimates for piecewise polynomial approximations 4.3.3. Convergence of approximations

5. Approximations of boundary spaces 5.1. Definitions 5.2. Operators of boundary approximation 5.3. Approximations on a fixed partition 5.4. Approximations on a family of partitions

6. External approximations of variational equations 6.1. Construction and total error estimate 6.2. External error estimates and convergence of approximations 6.3. Convergence of approximations in case when bilinear form is not V -elliptic

7. Examples of finite elements 7.1. Justification of some classic non-conformed finite elements 7.1.1. Wilson's element

7.1.2. Family of Crouzeix-Raviart ‘s elements 7.1.3. Morley's element 7.1.4. Fraeijs de Veubeke’s element 7.2. Elements for external approximations 7.2.1. 2D elements for boundary value problems of second and fourth orders 7.2.2. 3D elements 7.2.3. Elements for approximations of spaces of vector functions 7.2.4. Axisymmetrical elements 7.2.5. Finite elements for stress concentration

8. Numerical experiments 8.1. Torsion 8.1.1. Formulation 8.1.2. Model problem 8.1.3. Stress concentration at a semi-round notch 8.2. Thin plates bending 8.2.1. Formulation 8.2.2. Bending of a cantilever plate 8.2.3. Bending of a square plate 8.3. 2D elasticity problems 8.3.1. Formulation 8.3.2. Extension of a square plate 8.3.3. Bending of a circular bar

8.3.4. Extension of a plate with a round hole 8.4. Special elements in stress concentration 8.4.1. Extension of a plate with an elliptic hole 8.4.2. Extension of a strip weakened by a square hole with filleted corners 8.4.3. Extension of a perforated plate 8.4.4. Strip with a round hole 8.4.5. Extension of a strip with a cutout 8.5. 2D problems of linear fracture mechanics 8.5.1. Plate with an inclined crack 8.5.2. Strip with a crack 8.6. Axis-symmetrical elasticity problems 8.6.1. Formulation 8.6.2. A hollow sphere under internal pressure 8.6.3. Stress concentration around a spherical cavity 8.6.4. Round plate under pressure 8.6.5. Spherical dome under pressure 8.6.6. Sine-shaped shell of infinite length under pressure 8.6.7. Shell compensator Reference list

Preface The Finite Element Method proved to be a powerful analysis tool in many fields of engineering. Its wide use and still great potential stimulate research and development of new effective finite element based technologies. This book presents an innovative approach to the finite element approximations. The approach possesses some new unique features and at the same time it inherits the computational advantages of the conventional finite element analysis. For example: finite elements of arbitrary shape with approximating functions of arbitrary type can be easily formulated; the types of the approximating functions do not depend on the element’s geometry; the element is not mapped onto a canonical shape; the matrix of the equation system is sparse, symmetric, and positive definite; and others. I believe that the suggested concept will lead to new analysis technologies created on its basis.

Generality and constructiveness are the major features of the theory proposed. All the results are obtained in abstract form under minimal restrictions on finite element geometry and contents of approximating spaces. It provides wide applicability of the suggested theory to different areas of analysis and simulation. It is described in detail how the methods of construction of the finite elements and approximations follow from the basic theorems, and how they are implemented in numerical algorithms. Numerous benchmark problems illustrate the applicability and accuracy of the method.

Section 1 introduces the definitions of spaces and operators associated with a partition of the domain into finite elements. Section 2 presents the necessary and sufficient sign, as well as two sufficient signs of the external finite element approximations of Sobolev space H m () ( m  0 is

arbitrary integer) under minimal limitations on the shape of finite elements (it is assumed that the boundary of the finite element K must be sectionally smooth only) and the contents of spaces of approximating functions (it is assumed that the functions just belong to the space H m ( K ) ). Herewith two types of the external approximations are considered: one is associated with a fixed partition of the domain  into finite elements, the other is associated with a family of partitions. Section 3 addresses the formulations of different classes of finite elements for external approximations. Section 4 introduces the operators of approximations and studies their convergence properties on fixed and variable partitions of the domain. Section 5 develops the approximations of spaces of functions determined on the boundaries of the partition. These spaces are used to construct external approximations by necessary and sufficient sign. Section 6 studies the external approximations of variational equations. Section 7 justifies some classic nonconformed finite elements as well as presents examples of finite elements for external approximations. Section 8 discusses results of numerical experiments on numerous benchmark problems.

This book should appeal to numerical analysts and software developers in different fields of engineering such as structural and dynamic analysis, applied mechanics, multiphysics, and computational mathematics.

The theory presented in the book was developed in 1981-1991 when I worked as a professor of Department of Theoretical Mechanics at Belarus State Polytechnic Academy. I take this opportunity to thank my former colleges for the support.

Victor Apanovitch

Introduction Elliptic boundary value problems can be stated in two equivalent forms: differential formulation or weak formulation. The equivalency means that under the assumption of sufficient smoothness of the input data (for example, the smoothness of the coefficients of the differential operator of the problem and the smoothness of the boundary of the domain) both formulations can be transformed one to another and their solutions coincide. The idea of a weak solution of a boundary value problem generalizes the idea of its classic solution.

In the weak form a boundary value problem is stated as follows: find element u  V that satisfies an abstract variational equation

v  V

a  u, v   f  v  ,

(1)

where: 

V is an enclosed subspace of Sobolev space (or some product of such subspaces)



a  u, v  is a continuous on V  V bilinear form



f  v  is some linear form on V

In computations, the infinite-dimensional space V is approximated by a finite-dimensional space X h . Herewith the continuous variational problem (1) is approximated by the discrete one as

follows: find element uh  X h that satisfies the discrete variational equation

vh  X h

a  uh , vh   f  vh  .

(2)

In result, the equation (1) is reduced to a system of linear algebraic equations from which the approximant uh of the exact solution u of the variational problem (1) is found.

In Finite Element Method (FEM) the space X h is constructed by partitioning the domain  into non-intersecting sub-domains (finite elements (FE)), and by defining on each FE smooth approximating functions (shape functions) that satisfy specific compatibility conditions on interelement boundaries. The fulfillment of the compatibility conditions is the major problem for the construction of FE approximations of the space V . These conditions define the smoothness requirements imposed onto the approximating functions from the space X h of finite elements. In conventional FEM, the space X h is constructed in such a way that the following inclusion holds Xh  V

(3)

If the spaces X h meet the condition (3) then the corresponding approximations of the space V and equation (1) are called internal approximations. For the internal approximations the closeness of the approximate and the exact solutions is evaluated by the norm of the space V . Herewith the unique solvability of the continuous variational problem (1) guarantees the unique solvability of the discreet variational problem (2), and the density of the family X h in the space

V guarantees the convergence of the approximate solutions to the exact one. Finite elements that generate internal approximations are referred to as conformed FE.

The construction of the conformed FE of high accuracy to approximate the Sobolev space

H m    is not a simple task. When m  1 then the functions from the space X h must be continuous, but their first derivatives can be discontinuous on inter-element boundaries. When

m  2 , the approximating functions must be continuous with their first derivatives. It necessitates the use the complex structures of degrees of freedom even for finite elements of the simplest form (triangle or quadrilateral for 2D problems). The problem becomes more complicated in case of curved boundaries. Such boundaries have to be simulated with elements that have non-planar faces. The curved elements are constructed through the mapping of some initial FE of simple form. However, such mapping worsens the approximating properties of the FE, particularly, if the FE is significantly distorted.

It is much simpler to construct FE which do not meet the condition (3). Such FE are called non-conformed FE. However, the justification of non-conformed FE is a complex mathematical problem. To solve it, one has to accomplish the following steps: 

introduce suitable spaces to compare the approximate uh and the exact u solutions (the space V can not be used for that because X h  V )



prove the existence and the uniqueness of the solution of the discreet variational problem (2) (these features are not provided automatically like in the case of internal approximations)



estimate the non-conformity errors and prove the convergence of the approximate solutions to the exact one

Due to the mathematical difficulties, the justification of non-conformed FE often is not conducted, and the accuracy and robustness of the approximations are verified through numerical experiments on benchmark problems only. Nevertheless, the efficiency of the non-conformed schemes caused their wide use in computations.

Problems of the mathematical justification of non-conformed FE were considered in works [1 28]. There had been developed several justification techniques. In 1965, in work [1] the patch test was introduced as the criterion of convergence of non-conformed FE. The patch test has been used to justify many non-conformed approximations applied to the specific problems [2 8]. In work [9] a variational treatment of the patch test was developed and a general estimate of the non-conformity error of the approximations of the variational equations was derived. The non-conformity error accounts the discontinuity of derivatives up to

 m  1 -order

of the

approximating functions from the space of FE at the inter-element boundaries. The proof of convergence of a non-conformed FE usually consists in obtaining the estimates of the non-conformity errors [5, 13, 14, 18 – 20, 29, 30]. In works [9, 12, 31] there was demonstrated that for a particular type of problems, the success of patch testing is a sufficient condition for the convergence of some non-conformed FE. Some of the non-conformed schemes were analyzed in work [17]. The results of [17] were generalized in work [24] by establishing some sufficient conditions of convergence. The iterative method of solution for schemes with non-conformed elements was suggested in work [27].

In 1980, in work [23], there was developed an approximation that successfully passed the patch test, but did not converge. Thus, it was demonstrated that, in general, the patch test is not a sufficient condition for the convergence of non-conformed FE. The generalization of the patch test was given in work [22], in which have been established the necessary and sufficient conditions of convergence of the non-conformed approximations. The generalized patch test did not depend on the type of the boundary value problem, but it was not constructive because in fact

it just stated that the error of non-conformity of the approximation must tend to zero when the mesh of FE is refined. So, the justification of the specific non-conformed FE was performed in work [22] for the model boundary value problems using the conventional technique for obtaining the estimates of the non-conformity errors.

In the above-mentioned works, the justification of the specific non-conformed FE was performed for specific types of boundary value problems, and the properties of the corresponding equations were extensively utilized. In such justifications, the proof technique usually assumed that a subspace of conformed FE was present in the space of non-conformed FE. This assumption reduced the applicability of the results obtained and prevented the development of general approaches to the construction of non-conformed approximations. In fact, the suggested criteria only helped to find out what kind of non-conformed functions can be added to the basic functions of conformed FE in order to improve accuracy on rough partitions of the domain.

Another approach to treat non-conformed approximations is based on the concept of the external approximations introduced in work [32]. The general theory of the external approximations of Sobolev spaces and variational equations has been developed in work [33] where were also given the examples of its application to the finite difference approximations and the approximations by parts. In terms of the theory of external approximations some non-conformed FE have been studied in works [11, 28, 34].

Let us note that the problem of justification of non-conformed FE is closely related to the problem of combination of approximations of different types on sub-domains, and with the

problem of approximate fulfillment of the essential boundary conditions. Some methods of the combinations have been studied in works [35, 36]. In work [37] it has been demonstrated that the non-conformed basic functions can lead to a significant error in the solution. The combined approximations are important when solving the boundary value problems in presence of singularity in the solution, and for unbounded domains. For example, in works [38, 39] the non-conformed combined approximation has been successfully applied for the solution of a boundary value problem for a domain with a cut. Herewith in the circle that contained the tip of the cut, the asymptotic representation of the solution for an infinite domain with a cut has been used. On the rest of the domain the solution has instead been approximated using linear triangular FE. In work [40], in terms of the theory of external approximations, it has been studied the approximate fulfillment of the essential boundary conditions in a thin plate bending problem. In work [41] it was shown that the methods of arbitrary accuracy could be developed through the replacement of the interpolation of the boundary conditions with the orthogonalization of the boundary error to some system of boundary functions.

The external approximations of Sobolev spaces and variational equations by finite elements have been systematically studied by the author in works [96-100]. This book generalizes the results of these works.

1. Partitions, spaces, and boundary operators Basic definitions Let us  be an open bounded domain in R n . For any integer m  0 , the Sobolev space H m    consists of functions v  L2    for which all generalized derivatives D v,   m (where   1 ,..., n  is a multi-integer, and   a1  ...   n ) belong to the space L2    of all functions

integrable with square, i.e.  ,   m

  D   

D v  L2   

 D v dx   1  vD  dx 









where dx  dx1...dxn ; D    is the space of all infinitely differentiable functions which have compact supports in  .

In the space H m    , norm and semi-norm are defined as follows:  m  2  D v dx    0    

1

  2 v k,m   D v dx    k     1 k  m

1

v

m, 





2

2

The Sobolev space H 0m    is defined as a closure of the space D    by the norm  m, .

A bounded domain  in the Euclidean space R n is said to belong to the class C k , where k is a non-negative integer, if there exists the finite number of local coordinate systems and local mappings fr , 1  r  R , as well as real numbers   0,   0 such that the following conditions are satisfied (see Fig. 1.1): 

functions fr , 1  r  R are k -times continuously differentiable in a closed (n-1)-cube

 xˆ

r

 Rn 1 : xnr  



Here





xˆ r  x1r , ..., xnr 1 , xˆ r  

means that xir   , 1  i  n  1



the boundary of the domain  is 

R

r

r 1



 xˆ , x  ; r n

 

xnr  xˆ r , xˆ r  



in any local coordinate system 1  r  R

 xˆ , x  ; f  xˆ   x  f  xˆ    ,  xˆ , x  ; f  xˆ     x  f  xˆ  , r

r

r n

r

r

r n

r

r

r n

r

r

r n

r

r

    

xˆ r     xˆ r

It is said that any domain of class C0 has a regular (or Lipshitz) boundary. The boundary  (or its part S) is called smooth if fr , 1  r  R belong to the class C  (i.e. if the boundary  is a (n-1)-dimensional infinitely differentiable manifold in R n ). The boundary  of the domain  is called sectionally smooth if it can be divided into finite number of pieces i , of class C  , so that

N



i .

i 1

Spaces associated with a partition Let us  be a bounded domain in R n with a sectionally smooth boundary  . Let us  denote a partition of  into a finite number of sub-domains K (finite elements). Let us the partition  fulfills the following conditions: 



K K 



for any K   , domain K is closed and the set of its inner points K is not empty (i.e. K  0 )



for any pair of elements K1 , K 2   : K1  K 2  0



for any K   the boundary K is sectionally smooth and regular

Partitioning the domain  into FE, let us define the space V 

 H K , m

K

K  .

(1.1)

Mutually orthogonal subspaces

 0, 0, ..., H

m

 K  , ..., 0  are contained in the space

V . These

subspaces can be identified with the spaces H m  K  . Thus, the space V is a direct sum of its mutually orthogonal subspaces V 

 H  K ,

K  ,

m

K

and any element v  V can be represented as follows v 

v

K

,

K  ,

(1.2)

K

where element  0, 0, ..., v K , ..., 0  is identified with an element v K  H m  K  .

The inner product and the norm in the space V are defined as follows:

 u , v V

  

u

V







 u



  K  m   K  K D u ,D v  0, K K  0  m     D u K   D v K  dx    0   K  K    1  2   2  K u     K  m, K    1  2 2 m     D u K dx    K  0       1 2 2   m    K     D u  dx     0   K      K

, vK

m, K





 







 

One can see that the norm  m,  is the restriction of the norm  V on the space H m    .

(1.3)

To approximate the space H m    let us define a finite-dimensional space, also referred to as a space of finite elements Xh 

P

K h

 V,

K 

K

where PhK are some finite-dimensional spaces of functions determined on the sub-domains

K  .

For any FE K   the space PhK approximates the space H m  K  ; the finite element space X h approximates the space V . Approximating functions from the space X h are sufficiently smooth on each FE of the partition ( PhK  H m ( K ) ) but they are discontinuous across the inter-element boundaries. Thus the following inclusions take place X h  V  L2    ,

and approximating functions from the space of FE do not belong to the space H m    . The functions are “external” with respect to the space H m    and the concept of external approximations must be used to justify them.

The formal definition of external approximations is as follows (see [33]). Assuming that: 

h is a family parameter



Vh is a finite-dimensional space with some norm



ph is a continuous linear mapping from Vh onto V (an extension operator)



rh is a mapping from V onto Vh (a restriction operator)

approximations  Vh , ph , rh  of a space V are called the external approximations of the closed subspace V from V , if the family  Vh , ph , rh  possesses the following property: if phuh  u weakly in V then u  V . Approximations  Vh , ph , rh  of the space V are called converging in V if

u V

u  ph rh u V  0 .

Let us assume that V  H m    , and the space V , associated with the partition  , is defined by (1.1). Then V  V is a closed subspace.

Let us also define a finite-dimensional space of vectors N h Vh  R   .

Extension operators ph : uh  Vh  phuh  X h

(1.4)

are defined by the inclusion X h  V .

The primal goal is to establish conditions that must be met by the approximating functions ph uh  X h  L2 () in order to restore the smoothness properties of the functions in limit, when

the weak convergence phuh  u in the space V implies the inclusion u  H m    .

Generalized Green’s formula

The validity of generalized Green’s formula is proved in [33] under the following assumptions. Let us there be given Hilbert spaces V and H , and the operator   L  V , T  , such that: 

 maps V onto T



H contains V , and V has a stronger topology

 the kernel V0 of the operator  is dense in H

Further, let us there be given a continuous on V  V bilinear form a  u , v  and a linear on V0 functional u defined by the relation

 u, v   a  u, v 

v  V0 .

The formal operator  is a continuous linear operator from V into V0 . The domain of determination of the operator  is a Hilbert space V     u  V

u  H 

with the norm u V  



u

2 V

 u

2 H



1

2

.

Under the given assumptions there exists a unique operator  such that the following Green’s formula is valid (see [33]) u  V    , v  V

a  u, v    u, v    u ,  v ,

(1.5)

where  ,   is the inner product defined on H  H ; ,  is the duality pairing defined on T '  T (here T  is a conjugate space)

Various Green’s formulas can be established from the general formula (1.5) by choosing of bilinear forms and spaces. Such formulas are widely used in the theory of elliptic boundary value problems and in particular when proving the equivalence of weak and differential formulations of the problems. To determine the hallmarks of the external FE approximations of Sobolev spaces let us introduce specific spaces, operators, and generalized Green’s formula associated with the partition of the domain.

Let us partition the domain  into FE, then let us determine the space V by the formula (1.1), and let us introduce the continuous on V  V bilinear form a  u , v  

 a  u K

K

, vK



K

where



 u

aK u K , v K 

K

D v K dx

K

u

K

u

K

,

v

K

vK,

 m

The formal operator  , associated with the form a   u , v  , is defined by the expression  u 

  u K

K

,

K 

K

where K u K   1 D u K , 

 m

The domain of determination of the operator  is the space V  

 H K   V . m

K

Let us define a face K r of the FE K as a (n-1)-dimensional sub-domain of the boundary K , which either divides the adjacent elements or coincides with a patch of the boundary  of the domain  (Fig. 1.2). Let us introduce the following notations: 

M  K r  is the set of all faces of the FE K



M  Kr ,l  is the set of all smooth face sections K r , l of the face K r (in brackets a typical

element of the set is indicated) In these notations symbols r and l identify the face and the smooth face section respectively (they are not indices).

Now the following spaces are introduced:

H 

 L  K   L  2

2

K

V0 

 H K  m 0

K

T 

m 1

 H K

m  j 1 2

K r K r , l j  0

         K r , l   

where K   , Kr  M  Kr  , Kr , l  M  Kr , l  .

The space T contains mutually orthogonal subspaces

 0, 0, ..., H

m  j 1 2

 Kr ,l  , ..., 0 

that can be identified with spaces H m  j 1 2  Kr , l  . Then T 

m 1

   H K Kr Kr , l j  0

m  j 1 2

 Kr ,l 

(1.6)

and any element g  T can be represented as follows g 

m 1

   g

K r ,l , j

.

K K r K r , l j  0

The trace operator associated with the partition is defined as follows m 1

    rK, l , j , K

(1.7)

Kr Kr , l j  0

where  rK,l , j is an operator of j-order differentiation along the outward normal to the smooth section K r , l of the face K r of FE K   . When j  0 , the function  rK, l , 0 u is a restriction of the function u on K r , l .

According to the trace theorem from [33] each component  rK,l , j of the operator 

is a

continuous linear mapping from the space H m  K  onto the space H m  j 1 2  Kr , l  . Hence the operator  maps the space V onto the space T . Since the space V is enclosed into the space H with stronger topology, and the space V0 is a kernel of the operator  that is dense in the space H , then the assumptions of the Theorem 6.2.1 from [33] are met. So, for any  ,   m there

exists a unique operator   such that for any u , v  V the following generalized Green’s formula is valid





a   u , v    u , v    u ,  v ,

where:

(1.8)

 

m

  K



K , r,l, j

 rK, l,,j

K r K r , l j  2 m  1



 L H m  K  , H m  j  1 2  K r , l 



m  j  2m  1

        

Taking into account the representation (1.2), the formula (1.8) can be rewritten as: u , v  V

 u  

  K

 D v K  dx   1 



  

  D u

K

K

   vdx   u ,  u , 

(1.9)

where: uK  u

K

; vK  v

K

.

The operators  and   are referred to as boundary operators.

Properties of the boundary operators The Sobolev space H m    consists of functions v  L2    such that all their generalized derivatives D v,   m belong to the space L2    , i.e. for any multi-integer  ,   m there exists such a function D v  L2    that   D   

D





v dx   1



 vD  dx  0 , 

(1.10)



where D    is the space of all infinitely differentiable functions that have a compact support in .

Since the space D    is dense in the space H 0m    , it is possible to extend the expression (1.10) onto the space H 0m    , i.e. it holds for any function   H 0m    .

Let us vˆ denotes the function that is equal to the function v  V in  and that equals zero in a compliment of  in R n . It is known that if vˆ  H m  R n  then v  H 0m    . Thus, if for any function   D  R n  from the space of infinitely differentiable functions, the expression below holds 

    D

Rn

  vˆ dx   1 

 K

K

ˆ  0,  D  vdx 

(1.11)

Rn

then vˆ  H m  R n  and v  H 0m    . Here vˆ K  vˆ K .

Because of the density of the space D  R n  in the space H m  R n  , the expression (1.11) can be extended by continuity onto the space H m  R n  . Integrals in (1.11) are evaluated only on the domain  because vˆ  0 outside  . It must be also noted that, since by assumption the domain  has a regular boundary, the space H m    coincides with the space of restrictions on  of all

functions u from the space H m  R n  .

Now let us consider the space V defined by (1.1). Spaces H m    and H 0m    are the closed subspaces of V . The conditions to determine if the function v  V belongs to the space H m    or H 0m    readily follow from the formula (1.9). In fact, if the following inclusion holds

u

u

K

 H m 

K

then D u 

 D u

K

.

K

Thus, using (1.9)-(1.11), the following lemma can be formulated.

Lemma 1.1 A function v  V belongs to the space H m    -and respectively to H 0m    - if, and only if, for any u  H 0m    -and respectively for any u  H m    - and for all  ,   m , one of the two equivalent conditions holds:



 u   D



  v  dx   1 

 K

K

 D uvdx  0 





 u,  v  0 

Now, some properties of the boundary operators  and   that act in the spaces H m    and H 0m    can be established. Let us introduce the following designations:



r is the common face of adjacent FE K 1 and K 2



M  r  is the set of common faces of all elements of the partition



M   r ,l  is the set of common smooth face sections of all elements of the partition

For any r ,l  M   r ,l  there exist the face sections Kr1,l and Kr2,l of adjacent FE K 1 and K 2 so that r ,l  Kr1,l  Kr2,l . The j-order derivative of a function u K  H m  K  along the outward normal to the face section K r ,l is defined as follows

 rK, l , j u K 

 j!

   D u  !    j

K

nr, l

where 

  1 , ..., n  is a multi-integer



nr ,l   n1 , ..., nn  is an outward unit normal (see Fig. 1.2.b)



 !  1 !2 !...n !;

vr, l  v11 v22 ...vnn

The outward normals to the common face section of the adjacent elements K 1 and K 2 are opposite. Hence for any function v  H m    the following relation holds

 r , l  M   r , l 

 r1, l , j v1   1  r2, l , j v 2 , j

0  j  m 1

where

v1  v K1 ,

v2  v K 2 .

Here and below superscripts 1 and 2 denote the functions and the operators associated with the adjacent elements K 1 and K 2 .

Therefore, the operator  maps the space H m    onto the closed subspace T of the space T characterized as follows

  Tˆ   g   

m 1

   g

K r,l, j

T

K r K r , l j  0

K

g 1r , l , j   1 g r2, l , j , j

 r , l  M   r , l 

(1.12)

0  j  m  1

Identically, considering the space H 0m    it is apparent that the operator  maps it onto the closed subspace



Tˆ0  g  Tˆ



g  0 .

(1.13)

Now some properties of the boundary operator   can be established.

Lemma 1.2 

Operator   ,   m maps the space H 0m    onto the orthogonal complement Tˆ  of the space Tˆ in the space T '



Operator   ,   m maps the space H m    onto the orthogonal complement Tˆ0 of the space Tˆ0 in the space T 

◊ Proof – The operator  maps the space H m    onto T . According to Lemma 1.1:  ,   m u  H 0m    , v  H m   

  u,  v  0

u  H m    , v  H 0m   

  u,  v  0 

Now the characterizations of the spaces Tˆ  and Tˆ0 can be established.It has been demonstrated that:

f  Tˆ0 , g  Tˆ0

f , g  0;

(1.14)

f  Tˆ  , g  Tˆ

f, g  0.

(1.15)

Taking into account the definition (1.13), the expression (1.14) can be rewritten as follows

f,g 

m 1

   K



K r K r , l j  0

 m 1

r ,l j  0



m 1

 r ,l j  0

f rK, l , j , g rK, l , j 

f r1, l , j , g 1r , l , j  f r2, l , j , g r2, l , j



(1.16)

f r1, l , j   1 f r2, l , j , g 1r , l , j  0 j

where K   , K r  M  K r  , K r , l  M  K r , l  ,  r , l  M   r , l  .

Since (1.16) must hold for any g  Tˆ , then the space Tˆ0 is defined by the characterization

Tˆ0 

f

 T ' |  r , l  M   r , l 

f r1, l , j    1 f r2, l , j , f

0  j  m  1 .

Identically from (1.15) one can obtain



Tˆ   f  Tˆ0 | K r, l  M  K r, l 

f r, l , j  0,

0  j  m  1

where M  K r, l  is a set of smooth face sections of FE that coincide with the boundary  of the domain  .

Figure legends for Section 1:

Fig. 1.1 To the definition of the domain boundary

Fig. 1.2 Adjacent finite elements: a) linked along the common boundary b) separated

2. The hallmarks of external approximations by finite elements Approximations on fixed and variable partitions Two types of the approximations are under consideration: 

The first type is associated with a fixed partition  of the domain  . This type is an analogy of p-version of FEM



The second type is associated with a family  h of partitions. This type is an analogy of hversion of FEM

Let us define a fixed partition  of the domain  and let us associate with it a family X h of FE spaces Xh 

P

K h

,

K 

K

where PhK  H m  K  are finite-dimensional spaces.

The absence of the parameter h in the partition designation indicates that each space X h is associated with the same partition  . The dependence between h and approximations must be such that: when h  0

 

dim PhK   .

For instance, if PhK is a space generated by N basic functions, then h  1 N .

Second type of the approximations is associated with a family  h of partitions. In this case the parameter h has the meaning of maximum diameter of FE: hK  diam  K  .

h  max hK , K

It is assumed that the family  h is regular so: 

there exists such positive constant  that K   h

hK  K  

(2.1)

where:

hK  diam  K  ,

 K  sup diam  S  , S is a sphere from K 



diameters hK approximate zero

The family  h is constructed through the partition refinement. Spaces PK of the approximating functions are fixed; the number of FE - card  h  - tends to infinity when h tends to zero. The FE space is defined as follows Xh 

P

K

,

K  h .

K

For the approximations associated with the fixed partition  , definitions (1.1) and (1.6) of spaces V , V0 , H , T contain finite products of spaces. For approximations associated with the family

 h of partitions, the finite products in expressions (1.1) and (1.6) are substituted by numerable products, because when h  0 then card  h    . In this case, for instance, the space V consists of infinite vectors u   u1 , u 2 , ..., u K , ...  ,

uK  H m  K 

such that u

2 V





uK

K

2 m, K

 .

More spaces associated with a partition

In order to establish the signs of the external FE approximations of Sobolev spaces, the duality pairing

, 

on T '  T in formula (1.8) is to be substituted by the inner product in the

corresponding space. Let us introduce the space

HT 

m 1

  L  K  (2.2) 2

r ,l

K

Kr Kr , l j  0

with the norm

g 

m 1

 g K

K r K r , l j  0

K r,l, j

 HT

g

HT

     K Kr Kr ,l 

1

2

m 1



j 0

g rK, l , j 0, K r , l

 2  .  

Space T , defined by (1.6), is continuously and densely enclosed into the space HT . According to the Theorem 2.1.5 from [33], T and HT can be identified with subspaces which are dense in the conjugate space T ' : T  H T  T ' , and these inclusions are continuous and dense. Then the duality pairing , 

on T '  T can be identified with the unique extension of the inner

product  ,   H in HT . Further in the text the closures of the spaces Tˆ , Tˆ0 in the space HT are T

designated as Hˆ and Hˆ 0 , and the orthogonal complements of these closures in HT as Hˆ  and

Hˆ 0 correspondingly. Thus: Hˆ 0 

f

 H T |  r , l  M   r , l 



Hˆ   f  Hˆ 0 |

f r1, l , j    1 f r2, l , j , 0  j  m  1 (2.3) j

f



 0

(2.4)

Functions that are determined on the face K r can be represented through their components determined on the face sections as follows

f rK, j 



K r , l

f rK, l , j  L2  K r , l  ,

0  j  m  1.

Then the characterization (2.3) can be rewritten as follows Hˆ 0 

f

 HT |  r  M   r 

f r1, j    1 f r2, j , 0  j  m  1 j

(2.5)

Characterizations of the spaces Hˆ and Hˆ 0 can be rewritten in the same manner.

It must be mentioned that, for the fixed partition of the domain, the space HT is a fixed separable Hilbert space. Therefore, there exists a family of closed subspaces that is dense in HT . When the family  h of partitions is under consideration, then the family of spaces H Th , defined in accordance with (2.2), take place. When h  0 the product of finite number of spaces in (2.2) becomes a product of numerable spaces. In this case a family of spaces Rh is called dense in space HT if, for any sequence xh  H Th , there exists such a sequence yh  Rh that

lim xh  yh h0

HTh

 0.

Some auxiliary assertions Now let us consider two auxiliary lemmas.

Lemma 2.1 Let us M be a Hilbert space and N  M is a closed subspace. A bounded sequence  h  M satisfies the relationship

  N

lim  ,  h   0 h0

(2.6)

if, and only if, there exists the family of closed subspaces Gh  N which is dense in the space

N and, for any h , it possesses the following property

gh  Gh

 gh , h   0

(2.7)

 Necessity - The space M can be represented as a direct sum of the subspaces M  N  N  , where N  is an orthogonal complement of the subspace N in the space M . Then any element of the sequence  h  can be expressed uniquely as follows

 h   h1   h2

 h1  N ,

 h2  N  .

For any h , let us Fh1 be a one-dimensional subspace generated by the element  h1 , i.e. f  Fh1

f   h1 ,

  R.

Because the space Fh1 is one-dimensional, therefore the orthogonal complement Gh of the space Fh1 in the space N does exist and has a co-dimension equal to one.

The space N can be represented as a direct sum of its mutually orthogonal subspaces, i.e. N  Gh  Fh1 . Hence any element   N can be uniquely represented as follows

   h  h h1

h  R,  h  Gh .

Then lim  ,  h   lim  h  h h1 ,  h1   h2   lim h  h1 h0

h0

h0

2

0

and h h1  0 strongly in M . That implies

  lim  h  hh1   lim  h . h 0

h 0

Thus Gh is the family of the subspaces that is dense in N and possesses the property (2.8).  Sufficiency - Let us the relationship (2.7) holds. Then for any element   N

lim  , h   lim   , h    gh , h    lim   gh , h   lim    h , h   0 . h 0 h 0 h 0 h 0 Here  h is a projection of  onto the closed subspace Gh  N .

Lemma 2.2 Let us M , N and  h  fulfill the assumptions of Lemma 2.1. If there exists a sequence g h  M such that

  N

 , g h   0 ,

lim  h  g h  0 , h 0

then (2.6) holds.

◊ Proof – It follows from (2.8) that

  N

lim  ,  h   lim  ,  h  g h  . h0

h0

(2.8) (2.9)

Then (2.6) follows from (2.9) and the inequality

 ,  h

 g h     h  g h .‫ٱ‬

The hallmarks of external approximations The necessary and sufficient sign of external FE approximations of a Sobolev space is defined by the following theorem.

Theorem 2.1 The FE approximations Vh , ph , rh  are external approximations of the space H m    -and space H 0m    respectively- if, and only if, the following condition is met g h  Gh , ph uh  X h

 gh ,  ph uh H

 0,

(2.10)

T

where Gh is a dense family of closed subspaces of the spaces Hˆ  -and Hˆ 0 respectively.

 Proof – Let us consider the approximations of the space H m    . Let us assume that a sequence ph uh weakly converges in the space V . It means that, for any function   H m    and for any multi-integer  ,   m , the following holds:    u  D   dx   u  D   dx  h



where uh  phuh 

u K

k h

.

0

    K   D u  dx  u  dx  h     K     

(2.11)

According to Lemma 1.1, in order that u0  H m    , the fulfillment of the following condition is necessary and sufficient

  H 0m    ,  ,   m   lim     D uhK h 0  K

    dx   1 

  u dx   1 





   u



K h

K

    D  dx    

 u D  dx  0 

0



This condition is equivalent to the condition

 ,   m,

  H0m   

lim    ,  ph uh  h 0

HT

 0.

(2.12)

According to Lemma 1.2, the operator   maps H 0m    onto the space Tˆ  . Thus the necessary result follows from (2.12) and from Lemma 2.1, in which it is assumed:

M  HT , N  Hˆ  , h   ph uh .

Let us consider the approximations of the space H 0m    . Let us the sequence ph uh weakly converges in space V , i.e. relationship (2.11) holds. According to Lemma 1.1, in order that

u0  H 0m    , the following condition is necessary and sufficient  ,   m,

  H m   

lim    ,  ph uh  h 0

HT

 0.

(2.13)

Since the operator   maps H m    onto Tˆ0 (see Lemma 1.2), the necessary result follows from (2.13) and from Lemma 2.1 where it is assumed that:

M  HT , N  Hˆ 0 , h   ph uh . ‫ٱ‬

The following theorems determine two sufficient signs of the external FE approximations of Sobolev space.

Theorem 2.2 The FE approximations  Vh , ph , rh  are external approximations of the space H m    -and space

H0m    respectively- when, in some family of spaces Yh  H m    -and Yh  H 0m    respectively-, there exists a sequence ph vh  Yh such that ph uh  X h

lim  ph uh   ph vh

h0

HT

0 .

(2.14)

 Proof - According to Lemma 1.1, since Yh  H m    -and Yh  H 0m    respectively-, for any   H 0m    -and   H m    respectively- it holds the following  ,   m, ph vh  Yh





 ,  ph vh



HT

 0.

(2.15)

Let us consider the approximations of the space H m    . Let us phuh be a sequence that converges weakly in the space V . Then, the sufficiency of the condition (2.14) follows from (2.12), (2.15) and from Lemma 2.2 in which it is assumed that: M  HT , h   ph uh , gh   ph vh .

When approximations of the space H 0m    are considered, the sufficiency of condition (2.14) can be proven in a similar way. ‫ٱ‬

The trace operator  , defined by (1.7), can be written as a direct sum of j-order trace operators  

m 1



j

j 0

where



m 1

 j       ij  rK, l , i  K K r K r , l i  0

     

0  j  m 1  1, i  j  0, i  j

 ij  

(2.16)

The boundary operators   in (1.8) can be presented in a similar way m



 

 j ,

 m

j  2 m 1

where  j 

m

  K

K r K r , l i  2 m  1

m  j  2m  1  1, i  j  0, i  j

 ij  

The space from (2.2) can be represented as follows HT 

m 1

H j 0

where

T, j

,



 ij  rK, l,,i        

(2.17)

HT , j 

m 1

    K

2 ij L

K r K r , l i  0

 Kr , l  .

 1, i  j  ij    0, i  j

Subspaces -see characterizations (2.3) and (2.4)- can be represented as follows Hˆ  

m 1

 Hˆ

 j

j 0

Hˆ 0 

m 1

 Hˆ

 0, j

j 0

where



Hˆ 0, j  f  HT , j |  r , l  M   r , l 



Hˆ j  f  Hˆ 0, j |

f





0



f r1, l , j    1 f r2, l , j (2.18) j

(2.19)

Theorem 2.3 Let us  Vh , ph , rh  be a family of FE approximations of the space H m    . Let us represent the set I of values of the index j , 0  j  m  1 as a sum of its not intersecting and not empty subsets I  I1  I 2 .

Approximations

Vh , ph , rh  are external approximations of the space H m    -and the space

H0m    respectively- if the following holds: 

For any j  I1 , there exist families of spaces Yhj  H j 1    - and Yhj  H 0j 1    respectively- and sequences ph vhj  Yhj such that

lim  j ph uh   j ph vhj

ph uh  X h

h 0

HT

 0,

(2.20)

 0,

(2.21)

where  j is the trace operator (2.16). 

For any j  I 2

g

ghj  Ghj , ph uh  X h

j h,

 j ph uh



HT

where Ghj is a dense family of closed subspaces of the spaces Hˆ j -and Hˆ 0, j respectively-. Here spaces Hˆ j and Hˆ 0, j are defined by the characterizations (2.19) and (2.18).

 Proof – Let us consider approximations of the space H m    . Let us the sequence phuh converges weakly in the space V , i.e. the relationship (2.11) holds. According to Lemma 1.1, in order that u0  H m    , it is necessary and sufficient that the following condition is fulfilled  ,   m,   H 0m   



lim   ,  ph uh

h 0



HT

 0.

(2.22)

If the representations (2.16) and (2.17) are taken into account, the inner product in (2.22) can be rewritten as follows



  ,  ph uh



HT



m 1

  

 ,  j ph uh

2 m  j 1

j 0



HT

.

(2.23)

The relationship (2.22) will be fulfilled if, for any j  I and  ,   m



lim  2m  j 1 ,  j phuh

h 0



HT

 0.

(2.24)

Let us j  I1 . Because of the inclusion Yhj  H j 1    , the following relationship holds  ,

  m,



 ,  j ph uh



2 m  j 1

  H 0m    ,



HT

ph vhj  Yhj

0

(2.25)

Thus, when j  I1 , then (2.24) follows from (2.20), (2.25) and Lemma 2.2 in which it is assumed that M  HT , j , h   j ph uh , gh   j ph vhj . When j  I 2 , the validity of (2.24) follows from (2.21) and Lemma 2.1 in which it is assumed that M  HT , j , H  Hˆ j , h   j ph uh .

When approximations of space H 0m    are considered, the result can be proven in a similar way. ‫ٱ‬

The Theorems 2.1-2.3 can be applied not only to justify well-known non-conformed finite elements (in Section 7.1 it is done for the elements of Wilson, Crouzeix-Raviart, Morly, etc.), but also to develop novel highly efficient finite elements and approximations. Some methods of construction of such finite elements are elaborated in the next Section.

3. Types of finite elements for external approximations and their construction Finite elements conformed by a subspace Element definition Let us consider how to construct FE that meet the necessary and sufficient sign established in Theorem 2.1. There are two types of functions in the formulation of the theorem: 

Functions g h from the family of enclosed subspaces Gh  Hˆ  (or Gh  Hˆ 0 ). The domains of determination of the functions are boundaries of the elements. These functions are referred to as boundary functions



Functions ph uh from the family X h of FE spaces. The restrictions of the functions ph uh onto the FE K belong to the spaces H m ( K ) but the functions itself belong to the space L2 () only

Let us Gh be a family of finite-dimensional spaces. These spaces will be referred to as the spaces of boundary functions. They are elaborated in Section 5. Here, to satisfy the requirements of the Theorem 1.1, the properties of the approximating functions from the spaces PhK  H m  K  are established. The types of finite elements that meet the conditions of the theorem are defined, as well as the method of their construction is developed.

Let us consider the approximations of the space H m    . Using the designations introduced in the Section 1, (2.10) can be rewritten as follows

g h  Gh  Hˆ  , ph uh  X n

 g h ,  ph uh H

m 1

   

 T

K



K r K r , l j  0 K r , l

m 1

  g r ,l j  0 r ,l

1 r,l, j

g rK, l , j  rK, l , j p K d  

(3.1)

 r1, l , j p1  g r2, l , j  r2, l , j p 2 d   0

where

K   , K r  M  K r  , K r , l  M  K r , l  ,  r , l  M   r , l  p1  P1  H m  K 1  ,

p2  P2  H m  K 2  ,

and where 

K 1 and K 2 are the adjacent FE



g rK, l , j is a restriction of a function from the space Gh onto the smooth face section K r , l of the face K r

Because of the inclusion Gh  Hˆ  and of the characterization (2.4), restrictions of the functions from the space Gh onto the boundary  of the domain  equal zero, and so are the corresponding integrals in the expression (3.1).

Let us assume that basic functions of the space Gh  H  have local supports - they are the subdomains K r , j of the common face  r  K r1  K r2 of the adjacent FE K 1 and K 2 , with not empty multitude of internal points K r , j  0 (it is also possible that K r , j   r ). Further it will

be shown that such choice of the basis in the space Gh leads to such basic functions of the finite element space X h of that also have local supports in the domain  .

Let us GrK, j designates a finite-dimensional space generated by the basic functions grK,l ,t whose supports belong to the boundary K of the element K . Because Gh  Hˆ  , then, according to the characterization (2.4), every basic function g r , j , t is a pair g r , j , t   g 1r , j , t , g r2, j , t  whose components satisfy the relation g1r , j , t    1 g r2, j , t , j

0  j  m  1.

(3.2)

When the approximations of the space H 0m    are under consideration, then the inclusion

Gh  Hˆ 0 takes place. In this case, due to the characterization (2.5), the boundary basic functions with supports on the boundary  of the domain  must be added to the functions whose components satisfy the relation (3.2) (see Fig. 3.1).

From (3.1) it follows that

pi  Pi  H m  K i  , g ri , j , t  Gri , j , i  1, 2;

 g

1 r, j, t

 r1, j p1  g r2, j , t  r2, j p 2 d   0

r

where

 rK, j 



K r , l

K r ,l , j

.

0  j  m 1 (3.3)

Let us consider FE K 1   and adjacent elements K 2 , ..., K N (see Fig. 3.1). Let us G K designates a space of boundary functions of the element

GK 

K r  M  K r  .

0  j  m  1,

GrK, j , j , Kr

From (3.3) follows:

pi  Pi  H m  K i  ,

g ri , j , t  G i ,

1  i  N , 0  j  m  1,

 g

1 r, j, t

2k  N

(3.4)

 r1, j p1  g rk, j , t  rk, j p k d   0

K 1

For (3.4) to hold there must exist such functions in the spaces Pi ,1  i  N that meet the following conditions for some arbitrary numbers  r , j , t



g ri , j , t ri , j pi d    r , j , t .

(3.5)

K 1

For i  1 , the integrals in the left hand side of (3.5) are continuous linear forms (functionals) on the space H m  K 1  (further, the designation K is used instead of K 1 ). Let us denote these integrals through i  p  and let us rewrite (3.5) as follows

i  p   i ,

1 i  M,

where M  card  K  , and where  K is a set of the functionals (3.5).

The structure of the FE basis is determined by the following theorem.

(3.6)

Theorem 3.1 Let us the functionals  i ,1  i  M from (3.5) be linearly independent on the space PK . Then: 

PK can be represented as a direct sum of two subspaces P K  P  PZ ,

(3.7)

where PZ   p  P K |

i  p   0, 1  i  M 

(3.8)

and P is a complement to PZ , isomorphic to the factor-space P K PZ . 

In the space P there exists a basis

 p , 1  i  M  j

that satisfies the following

condition 1, i  k (3.9)  0, i  k

 i  pk    

Any element p  P K can be uniquely expressed as follows p

M

 i  p  pi  i 1

N M

   p p k 1

Z k

k

where k  p  , 1  k  N  M are some coefficients;

 p , 1  k  N  M

of the space PZ .

 Proof - The mapping P K  RM , defined by the formula F  p    1  p  , ...,  M  p  

is surjective. If the opposite is assumed, then there is a vector

  1, ..., M   RM ,   0 such that

,

Z k

(3.10)

is a basis

M

   p   0 i

i 1

i

for any p  P K , i.e. the functionals  i are linearly dependent, which contradicts the initial assumption.

Canonical basis in the space R M consists of vectors f k , 1  k  M that have a k-component equal to one and the other components equal to zero. Due to the surjectivity of the mapping F , for any f k there can be found such an element pk  P K that F  pk   f k .

Then there exists a set of linearly independent elements p1 , ..., pM  P K that satisfy the condition (3.9).

Let us P designates a space generated by the elements pi , 1  i  M . Then the subspace PZ defined by (3.8) is a closed complement of the subspace P . Any element p  P K can be expressed as follows

p  p1  p2 , where

p1 

M

   p p i 1

 i

i

 P

M

p2  p   i  p  pi  PZ i 1

The representation p  p1  p2 is unique because, if 0  p1  p2 , where:

p1  P ,

p2  PZ

then

p2 

M

c p i 1

i

 k  p2  

 i

 c  p  M

i 1

i

k

 i

From latter follows that

ck  k  p2   0,

1 k  M

i.e. p2  0, p1  0 . Therefore, (3.7) and (3.10) are exact. ‫ٱ‬

The functionals i , 1  i  M associated with FE boundary are referred to as boundary degrees of freedom; their multitude is denoted by  K . The coefficients k , 1  k   N  M  are referred to as internal degrees of freedom; their multitude is denoted by AK .

The requirement of linear independence of the set of functionals i  p  on the space PK is equivalent to the requirement of PK -multi-solvency of the multitude  K of boundary degrees of freedom that can be formulated as follows: in the space PK there exists at least one function that satisfies the conditions (3.6) for any set of numbers  i . When dim  P K   card   K  , the definition of

PK -multi-solvency is transformed into the classic definition of

PK

-unique-solvency of the multitude  K (see [10, 27]). When dim  P K  >card   K  the multitude

 K is PK -poly-solvent and P  -unique-solvent. In this case there can exist several subspaces P  PK that provide P  -unique-solvency of the multitude  K .

Now a general definition of a conformed by subspace FE can be formulated. A conformed by subspace FE is a quartet  K , G K , P K , P  , where: 

K is a closed domain in R n with not empty multitude of internal points and sectionally

smooth regular boundary 

PK is a finite-dimensional space of functions determined on the domain K



G K is a finite-dimensional space of functions determined on the boundary K of the domain K



P is a subspace of space PK on which the multitude  K of boundary degrees of freedom is PK -unique-solvent

Construction of basic functions of an element Let us consider how to construct the basic functions of spaces P and PZ . Let us P  H m  H  be some finite-dimensional space, that below will be referred to as initial, and let us

 pk  , 1  k

 N be its basis, i.e.: p  P

p

N

b p k 1

where bk are some coefficients.

k

k

,

(3.11)

The following relations take place

i  p  

N

 b   p , k 1

k

i

k

1 i  M

and they can be rewritten in a matrix form RbT   T ,

(3.12)

where: 

R is a matrix with dimensions M  N



b is a vector of coefficients b   bk k 1



 is a vector of boundary degrees of freedom of FE    i i 1

N

M

Then (3.12) can be rewritten as follows RbT  RZ bZT   T ,

(3.13)

where: 

R is a block with dimensions M  M



RZ is a block with dimensions M   N  M 

Let us the block R be non-singular. Such a block does exist if the rank of the matrix R equals to M , i.e. if the multitude of boundary degrees of freedom of FE is P-multi-solvent. Multiplying (3.13) from the left by the matrix R1 , the expression for the vector b through the vectors bZ and  is obtained bT  R1 T  R1RZ bZT .

(3.14)

Now the representation (3.11) can be rewritten as follows p  1bT   2bZT ,

where

1   pk k 1 ; M

(3.15)

 2   pk k  N  M  1 . N

Substituting (3.14) into (3.15), one obtains

p  P K

p  1 R1 T    2  1 R1 RZ  bZT 



M

M



N M



j 1

  k  p    pi rik    i 1

k 1



 b Zj  p j  M  

M

 l 1

M  pl  rlk rkjZ  k 1 

(3.16)

where: 

rik , 1  i, k  M , are the elements of the matrix R1



rkjZ , 1  k  M , 1  j  N  M are the elements of the matrix RZ

One can see that the element p  P  P K has been represented by the elements of the new basis. When (3.10) and (3.16) are compared, it is apparent that pk 

M

 pr i 1

 i ik

,

1 k  M

are the basic functions of the space P , while M

M

l 1

k 1

p Zj  p j  M   pl  rlk rkjZ ,

are the basic functions of the space PZ .

1  j N  M

Thus, to find the basic functions

p   k

M

k 1

,

p  Z j

N M j 1

one has to choose a basis

 pi i 1 N

of the

initial space P and then to linearly transform it. Herewith, the initial space P and the space PK coincide.

A sufficient condition for existence of an element basis The Theorem 3.1, that determines the structure of FE basis, is based on the assumption of linear independence, or PK -multi-solvency, of the multitude  K of boundary degrees of freedom of FE on the space P K  H m  K  of approximating functions of FE. This assumption implies the existence of the representation (3.10). The question remained is whether it is always possible to determine the space PK for a given multitude  K that provides PK -multi-solvency of  K .

Let us card   K



M , dim  P K

  N . Further in the text, dimensions N

of the space PK are

varied. For the purpose of differentiation between the spaces PK of different dimensions, the parameter h is introduced into the designation: instead of PK one writes PhK , assuming that

h 1 N.

It is apparent that: 

When N  M , multitude  K cannot be multi-solvent on PhK



When N  M , PK -unique-solvency of  K might take place



Multi-solvency of  K can only occur when N  M

However, there can be invented the examples when the multitude  K does not become multisolvent even if h  0 . In other words, if there are no requirements to the content of the space PhK , then however big is the dimension N of the matrix R , it does not guarantee the equality of

the rank of the matrix to the number M .

The following theorem defines a sufficient condition for the existence of such h0 beginning with, for which  K is PhK -multi-solvent for all h  h0 .

Theorem 3.2 For the multitude  K of boundary degrees of freedom of FE to be PhK -multi-solvent, beginning with some h  h0 , it is sufficient that the family PhK is dense in the space H m  K  .

 Proof - Let us i  p  , 1  i  M be a multitude of the functionals (3.5) that are the boundary degrees of freedom of FE. Let us show that these functionals are linearly independent on the space H m  K  due to the linear independence of the basic functions g rK, j , t of the space G K of boundary functions of FE.

Trace operators

 rK, j 



K r , l

map the space H m  K  onto the space

K r ,l , j

,

0  j  m 1

H

m  j 1 2

 K  r ,l

K r , l

which is densely enclosed into the space L2  Kr  . Let us assume that the functionals (3.5) are linearly dependent on H m  K  , i.e. there exists such a vector

   r , j ,t   R M ,

 0

that for any u K  H m  K  m 1

   r, j, t j  0 K r

t



grK, j , t  rK, j u K d  



m 1

K r

      j  0 K r K r

r, j, t

t

 grK, j , t  rK, j u K d   0 . 

From this follows the existence of the vector   0 for which



r , j ,t

g rK, j , t  0

t

That contradicts the assumption about linear independence of the basic functions g rK, j , t .

Now let us assume that the family of the spaces PhK is dense in H m  K  but there is no such h0 beginning with, for which the multitude of the functionals i  p  , 1  i  M becomes linearly independent on the spaces

PhK . It means that for any

h

there exists a vector

  1, ...M   RM such that p  PhK

M

   p   0 . i 1

i

i

(3.17)

From (3.17) and Theorem 1.2 from [33] it follows that the family PhK is not dense in the space

H m  K  . The achieved contradiction proves the required result. ‫ٱ‬

Virtual finite element In Section 3.1.1, a conformed by subspace FE was defined as a union  K , G K , P K , P  of four multitudes, where: 

K is the element’s domain



G K is a space of boundary functions



PK is a space of approximating functions of FE



P is a subspace of the space PK on which the multitude  K of boundary degrees of freedom is P -unique-solvent

When constructing basic functions of the element, according to the procedure described in Section 3.1.2, it is necessary first to introduce some system of basic functions that generates the initial space PK . Then one has to choose a subspace P  P K on which the multitude  K of boundary degrees of freedom of FE is P -unique-solvent. But in the space PK there can exist a finite multitude of subspaces P on which the multitude of boundary degrees of freedom is P -unique-solvent. Any of such subspaces P of this multitude defines its corresponding FE

 K, G

K

, P K , P  .

If dim  P K

  N,

dim  G K

M

and N  M , then the maximum number of subspaces P

that can be extracted from the space PK is equal to the number of combinations

CNM 

N! . M ! N  M  !

To extract a subspace P  P K one needs to find a non-singular block R with dimensions M  M from the matrix R in the system (3.12). It can be performed by the permutation of the

columns in the matrix R (and respectively the elements of vector  of the initial system of basic functions).

As it will be shown in Section 4, the convergence rate of the approximations is determined by the approximation qualities of the space PK and does not depend on the choice of the subspace P . The choice of P can only influence the magnitude of the constants in the estimates of the approximation error. Thus, all conformed by subspace FE, characterized by the same multitudes K , G K and PK , possess equivalent properties in terms of the convergence order.

In relation to the above said, it makes sense to introduce the virtual FE as a triplet  K , G K , P K  , which contains a finite multitude of conformed by subspace FE conformed by subspace FE have the same multitudes

 K, G

K

 K, G

K

, P K , P  . All these

, P K  but differ each other in

subspaces P .

Homogeneous and composite finite elements

Let us consider a trivial case when the equations (3.4) are fulfilled p i  P i ,

g ri , j , t  G i ,



1  i  N,

g ri , j , t  ri , j p i d   0

0  j  m 1

(3.18)

K 1

From (3.18) follows that for the element K  K 1 , the space PK is a finite-dimensional subspace of the kernel Z  H m  K  of the system of M functionals

i  p  

g

K r , j ,t

 rK, j pd  , 1  i  M .

(3.19)

K



A homogeneous FE is defined as a triplet K , G K , PZ

K

 , where:



K is an element’s domain



G K is a finite-dimensional space of boundary functions



PZK  Z is a finite-dimensional subspace of the kernel of the system of the functionals

(3.19)

Let us consider the procedure for the construction of the basic functions pkZ , 1  k  N  of a homogeneous element. Let us choose a system

 pk k 1 N

of basic functions determined on the FE

K . The system generates the initial space P which is not necessarily a subspace of the kernel Z

, P  Z . Any p  P is expressed as p 

N

b k 1

where bk are some coefficients.

k

pk ,

(3.20)

By substitution of (3.20) into (3.18), the system of homogeneous algebraic equations is obtained N

 b   p   0, k 1

k

i

k

1 i  M,

that can be expressed in a matrix form

RbT  0 ,

(3.21)

where R is a matrix with dimensions M  N ; b   bk k 1 is a vector of coefficients. N

It is assumed that M  N . If the rank r of the matrix R equals to N , then the system (3.21) has the trivial zero solution only, and, therefore, the subspace PZK is empty. If r  N then the dimension of the space of solutions of the system (3.21) is N  r . In this case a square nonsingular block R1 can be extracted from the matrix R by the permutation of the equations and unknowns. The system (3.21) can be rewritten as follows R1b1T  R2b2T  0 ,

(3.22)

where R1 is a non-singular block with dimensions r  r ; R2 is a block with dimensions

r  N  r.

Multiplying the equation (3.22) by the inverse matrix from the left the following complete system of the solutions is obtained b1T   R11R2b2T .

(3.23)

p  1b1T   2b2T ,

(3.24)

Let us rewrite (3.20) as follows

where 1   pk k 1 ;  2   pk k  r  1 . r

N

Substituting (3.23) into (3.24) one obtains

p    2  1 R11 R2  b2T

p  PZK  P K



N r

b k 1

k r

r r  1 2   pk  r   pl  rlj rjk  l 1 j 1  

From the expression (3.25) it follows that the basic functions

p  Z k

N

k 1

,

(3.25)

N  N  r

of the space PZK are defined as follows r

r

l 1

j 1

pkZ  pk  r   pl  rlj1rjk2 , where: 

rlj1 , 1  l , j  r are the elements of the matrix R11



rjk2 , 1  j  r, 1  k  N  are the elements of the matrix R2

As one can see a homogeneous element has only internal degrees of freedom and does not have any boundary degrees of freedom. It will be shown below that the homogeneous FE generate non-converging external approximations Vh , ph , rh  of the space H m    on both fixed and variable partitions of the domain  . However, these elements are useful when constructing the composite FE described below.

Let us consider two FE:







a homogeneous FE Kˆ , Gˆ K , PˆZK ,



a conformed by subspace FE  K , G K , P K , P  .

Let us the following relations hold:



K

Kˆ  K ,

(3.26)

Gˆ K  G K ,

(3.27)

PˆZK  PZ   .

(3.28)



Then the element K , G K , PC , P , where:

PCK  P  PZ  PˆZK

(3.29)

is referred to as a composite FE. A composite FE is a superposition of a homogeneous and a conformed by subspace FE. The equality (3.26) means the same geometry of the elements. Equality (3.27) implies the property PˆZK  P  0 .

(3.30)

The properties (3.28) and (3.30) provide the expansion of the space PK due to the addition of new functions from the space PˆZK . It can be useful for improvement of the approximation qualities of FE.

Spaces of finite elements

Let us characterize the spaces of the FE defined above. As it was assumed in Section 3.1, the support of every basic function g r , j , t , from the space Gh of the boundary functions, is either a sub-domain of an inter-element boundary r , i.e. supp  g r , j , t    r , or a sub-domain of the face K r coinciding with the boundary  of the domain  , i.e. supp  gr , j , t   K r (here and below are used the symbols accepted in Section 3.1).

Every basic function g r , j , t with a support in r is a pair g r , j , t   g 1r , j , t , g r2, j , t  whose components satisfy the relation (3.2). Herewith, in accordance with Section 3.1, for the adjacent FE K 1 and K 2 the following holds

p1  P1 ,

 g

1 r, j, t

p 2  P 2 ,

0  j  m 1

 r1, j p1  g r2, j , t  r2, j p 2  d   0

r

that can be rewritten as follows

 r1, j , t  p1    r2, j , t  p 2  , where:

 r1, j , t  p1    r2, j , t  p 2  

g

r



r

1 r , j ,t

 r1, j p1d  

  g r2, j , t r2, j p 2 d   

(3.31)

are the boundary degrees of freedom of the elements K 1 and K 2 respectively. Then the spaces X h of conformed by subspace FE that approximate the space H m    are defined by the

characterization

 X h  v   pK 



r



K 

  PK |

g r , j , t  Gh  Hˆ 

K

g 1r , j , t  r1, j p1 d     g r2, j , t  r2, j p 2 d  , r

 0  j  m  1 

The characterization of the space X 0h that approximates the space H 0m    is

 X 0h   v   p K 

g

1 r, j,t

r







K 

1 r, j

  PK |

g r , j , t  Gh  Hˆ 0

K

p d     g r2, j , t  r2, j p 2 d  , 1

r

g r , j , t  rK, j p K d   0,

K r

 0  j  m  1 

Basis of the space Gh generates the multitude h of boundary degrees of freedom of the space of FE. Every degree of freedom from the multitude h is a pair   r1, j , t ,  r2, j , t  whose elements are determined by the relations (3.31). Any FE also possesses a possibly empty multitude of internal degrees of freedom: B K  i , 1  i  N K  M 

where (see Section 3.1.1):

N K  dim  P K  M  card   K   dim  P 

The multitude of internal degrees of freedom of FE space is denoted through Bh : Bh   B K K

K   , card  Bh   N h  M h

where N h 

N

K

, M h  dim  Gh  .

K

Any element from the space X h of conformed by subspace FE can be uniquely defined by multitudes h and Bh of boundary and internal degrees of freedom.

The basis of the space X h consists of the basic functions of two types:

wrˆ, ˆj , tˆ  X h



 g

 ri , j , t wrˆ, ˆj , tˆ 

r

i r, j, t





 ri , j wrˆ, ˆj , tˆ |K d   i

i 1 when   1   0 otherwise

wkZ  X h ,

r  rˆ,

(3.32)

j  ˆj , t  tˆ, i  1, 2

 ri , j , t  wkZ   0,

i  1, 2 .

(3.33)

Here the symbols: 

r , rˆ identify the inter-element boundaries r , rˆ of the adjacent FE



j , ˆj identify the orders of the derivatives that participate in the definition of boundary

degrees of freedom 

t , tˆ identify the basic function from the spaces GrK, j

The equalities r  rˆ, j  ˆj , t  tˆ mean the coincidence of: 

faces r   rˆ ,



orders of differentiation



IDs of basic boundary functions

Basic functions of the space X h are constructed from the basic functions of FE in the following way. Let us: 

    1 ,  2    h is a boundary degree of freedom



g is a basic boundary function that generates 



K 1 and K 2 are adjacent FE on whose common face is defined the function g



pi  Pi , i  1, 2 are the basic functions from the spaces Pi of the elements K i that

correspond to the boundary degree of freedom 

Then the function w  X h , defined by the condition    pi  x  , w    0, 

x  K i , i  1, 2 x  Ki ,

is the basic function of the space X h corresponding to the boundary degree of freedom  and satisfying the conditions (3.32).

Let us p Z  PZ designates a basic function from the space PZ of the element K corresponding to the internal degree of freedom  . Then the function wZ  PZ defined as: Z   p  x, w    0, Z

xK xK

is the basic function (3.33) associated with the internal degree of freedom  .

(3.34)

Thus, the support of any basic function wr, j , t  X h consists of two adjacent FE (see Fig. 3.2). A support of a function wkZ is a single FE. Functions wkZ and wr, j , t are linearly independent and they generate the whole space X h .

Let us now examine the spaces of homogeneous FE. For such FE the basic functions of the space Gh of boundary functions do not generate boundary degrees of freedom of FE. The space X h of

homogeneous FE is characterized as follows

 X h   v   p K 



r



K 

  PZK |

g r , j , t  Gh  Hˆ 

K

g 1r , j , t  r1, j p1 d     g r2, j , t  r2, j p 2 d   0, r

 0  j  m  1 

Any element from the space X h can be uniquely expressed through the multitude

Bh   BK  bl , 1  l  Lh  K

of internal degrees of freedom of elements. Herewith dim X h  Lh .

If p Z  PZK designates a basic function of a homogeneous FE corresponding to the internal degree of freedom b , then the basic function w Z  X h is defined by conditions (3.34). Its support is a single FE.

A composite FE is a superposition of a conformed by subspace FE and a homogeneous FE. Therefore, the space X hC of the composite FE is a direct sum of the corresponding spaces X hC  X h  X h .

(3.35)

Herewith dim X hC  dim X h  dim X h .

A function from the space X hC is uniquely defined through the multitudes Bh and Bh . Basic functions from the space X hC are constructed from the basic functions of the conformed by subspace and homogeneous FE.

Figure legends for Section 3:

Fig. 3.1 To the definition of boundary functions with local supports

Fig. 3.2 Supports of basic functions from FE spaces

4.Error estimates and convergence of external approximations of Sobolev spaces Operators of approximations Let us consider a conformed by subspace FE  K , G K , P K , P  , where: dim  P K

  N,

dim  G K

and let us introduce the function v  H m  K  ,

M

v : K  R.

Let us

AK v   K v  SK v

(4.1)

designates a PK -approximant of the function v on the domain K . Here, the mapping

 K : v  p  P is defined as follows K v 

M

 v p

 i

i

i 1

,

(4.2)

and the mapping S k : v  p  PZ SK v 

N M

   v p i 1

i

Z i

is the best approximation in the space L2  K  of the function v by the shift of the subspace PZ  P K , i.e.

p  PZ

 v   K v  SK v, p 0, K

 0.

(4.3)

Because the subspace PZ has the basis

 p , 1  i  N  M , Z i

then the equation (4.3) is

equivalent to the system

v  

K

v  SK v, piZ 

0, K

0

1i  N M

(4.4)

from which the coefficients i  v  , 1  i  N  M of the best approximation are determined.

The approximation (4.1) of the function v is unique by virtue of the P -unique-solvency of the multitude  K of boundary degrees of freedom of FE on the subspace P and because of the linear independence of the system of equations (4.4). Because the boundary degrees of freedom of FE i , 1  i  M are determined on the space H m  K  , then the domain of determination of the operator of PK -approximation is the space H m  K  , i.e. dom AK  H m  K  . On the space P K  dom AK the operator of PK -approximation is identical operator, i.e. p  P K

AK p  p

(4.5)

Let us space X h is the space of the conformed by subspace FE with the multitude of boundary degrees of freedom h and with the internal degrees of freedom Bh . Herewith

dim X h  Nh , card  h   M h , card  Bh   Lh .

With an arbitrary function v  H m    , v :   R is associated its X h -approximant Ah v   h v  Sh v .

Here

(4.6)

 hv 

Mh

   v w j 1

j

 j

,

(4.7)

where wj , 1  j  M h are the basic functions (3.32), and i   h , 1  j  M h are the boundary degrees of freedom of the space X h .

The operator S h is defined by the relation

 v   v  S v, w  h

h

Z i

0, 

 0,

1  i  Lh ,

(4.8)

where wiZ , 1  i  Lh are the basic functions (3.33).

The function  h v  X h is characterized by the following property

 j  h v    j  v  , 1  j  M h .

Because the basic functions wiZ , 1  i  M of the space X h coincide with the basic functions piZ ,1  i  N  M of the spaces PZ  P K , and the support of the function piZ is a single

element, then the system (4.8) is equivalent to the finite set of systems of sort (4.4) formulated for all K   . Therefore, the function S h v is characterized by the following property

i  Bh , Sh v  X h

i  Sh v   i  v  , 1  i  Lh ,

and, for any function Ah v  X h , the following properties take place:

 j  Ah v    j  v  ,

1  j  Mh

 i  Ah v    i  v  ,

1  i  Lh

The domain of determination of the operator Ah of X h -approximation is the space H m    , i.e.

dom Ah  H m    .

Between the operators AK and Ah there exists the dependence determined by the following theorem.

Theorem 4.1 Let us v be an arbitrary function from the space H m    . Then restrictions v K belong to the space H m  K  and K  

 Ah v  K

 AK  v K  .

(4.9)

 Proof - Relations (4.9) follow from the definitions of the multitudes h and Bh . 

Let us define the approximation operators for a homogeneous FE

 K, G

K

, PZK  . The operator

AK* of PZK -approximation can be defined as an operator of orthogonal projection in the space

L2  K  of the function v  H m  K  onto the space PZK  H m  K  , i.e. p  PZK

If

 p , Z k

 v  A v, p  * K

0, K

 0.

(4.10)

1  k  N * is a basis of the space PZK , then the relation (4.10) is equivalent to the

following equation system

 v  A v, p  * K

Z k

0, K

 0,

1  k  N* ,

(4.11)

where AK* v 

N*

 b v p k 1

Z k

k

.

The uniqueness of PZK -approximation of the function v follows from the linear independence of the system (4.11).

The following relations also hold dom AK*  L2 ( K )  H m  K  , p  PZK

AK* p  p

Let us X h* be the space of homogeneous FE. X h* -approximant of function v is associated with an arbitrary function v  H m    , v :   R . The operator of X h* -approximation Ah* : v  Ah*v  X h*

is defined by the relation

 v  A v, w  * h

* i 0, 

 0,

1  i  Nh* ,

where wi* , 1  i  N h* are the basic functions of the space X h* . Because the support of each basic function wi* is a single FE, then any function Ah*v  X h* is characterized by the following property bi  Ah* v   bi  v  ,

1  i  N h* .

Let us consider a composite FE

 K, G

K

, PCK , P  , where PCK  P  PZ  PˆZK . The PCK

-approximant of the function v  H m  K  is AKC v   K v  S KC v ,

(4.12)

where the mapping  K : v  p  P is defined by (4.2), and the mapping S KC : v  p  PZ  PˆZK

defines the best approximation in the space L2  K  of the function v by the shifts of the space PZ  PˆZK , i.e.

p  PZ  PˆZK

v  

K

v  SKC v, p 

0, K

 0.

(4.13)

Let us  piZ  , 1  i  N  M be a basis of the space PZ , and let us  pˆ Zj  , 1  j  Nˆ * be a basis of the space PˆZK . Then the equation (4.13) is equivalent to the system of the equations:

v   v  

K

v  S KC v, piZ 

C ˆZ K v  S K v, p j 

0, K

0, K

 0,  0,

1 i  N  M  . 1  j  Nˆ *  

(4.14)

Herewith the PCK -approximant of the function v is unique because of the P -unique-solvency of the multitude  K of boundary degrees of freedom of FE and because of the linear independence of the system (4.14).

The following relations hold: dom AKC  H m  K  p  PCK

AKC p  p

(4.15)

Let us X hC be the space of composite FE. For an arbitrary function v  H m    , its X hC -approximant is: AhC v   h v  S hC v ,

(4.16)

where the operator S hC is determined by the relations

v   v  S v   v  S h

C h

v, wiZ

h

C h

v, wˆ iZ

 

0,  0, 

1  i  Lh    1  i  Nˆ h*  

 0,  0,

and the operator  h is defined according to the representation (4.7). Herewith

dim X hC  Nh  Nˆ h* .

It is easy to verify that the operators AhC and Ah* possess the property determined by the Theorem 4.1 for the operator Ah .

Let us  Vh , ph , rh  ,  Vh* , ph , rh*  , VhC , ph , rhC  be the external FE approximations of Sobolev spaces, constructed respectively on conformed by subspace FE, on homogeneous FE, and on composite FE. Restriction operators of the approximations are defined as follows:





rh : v  H m     vh  1 ,...,  M h , 1 ,...,  Lh  R Nh  Vh

    ,...,  ,  ,..., 

rh* : v  H m     vh  1 ,...,  N *  R Nh  Vh* *

h

rhC : v  H m     vh

1

Mh

1

Lh



ˆ*

, b1 ,..., bNˆ *  R Nh  Nh h

     C  Vh  

(4.17)

Then the corresponding operators of approximations can be defined as products of extension operator (1.4) and restriction operator (4.17):

Ah  ph rh , Ah*  ph rh* , AhC  ph rhC .

Convergence of approximations on a fixed partition Convergence properties of external FE approximations of Sobolev spaces on a fixed partition of the domain  are defined by the following theorems.

Theorem 4.2 Let us Vh , ph , rh  be a family of external FE approximations of the space H m    associated with a fixed partition  of the domain  and built on the conformed by subspace FE. Approximations Vh , ph , rh  converge in H m    if, and only if, for each FE K   the family of spaces PhK of approximating functions of FE is dense in the space H m  K  .

 Necessity - Let us assume that the external approximations Vh , ph , rh  converge, i.e.

u  H m   

lim u  ph rh u h0

V

0

where ph rh  Ah is the operator of X h -approximation. Then the density of the family PhK in the spaces H m  K  , K   follows from the definition (1.2) of the norm on the space V and from the dependence (4.9) between operators of X h - and PhK -approximation.

Sufficiency - Let us determine the X h -approximant of the function u  H m    Ahu   hu  Shu  ph rhu

where the operators  h and S h are defined by the relations (4.7) and (4.8) respectively. For any

K   and for any h , the PhK -approximant of a function v K  u K is defined according to (4.1) AK , h v K   Ah u  k 

Mh

Nh  M h

i 1

i 1

 i, h  v K  pi, h 



i , h  v K  piZ, h ,

(4.18)

where: 

i , h  v K  , i , h  v K  are the boundary and internal degrees of freedom of FE that are linear and continuous functionals on H m  K 



pi, h , piZ, h are the basic functions of FE



N h  dim( PhK ) and M h is the number of boundary degrees of freedom of FE

From the family of spaces PhK let us chose such sequences of functions pˆ hK  PhK that

K  

lim v K  pˆ hK h 0

m, K

 0.

(4.19)

These sequences exist because, according to the assumptions, the families of the spaces PhK are dense in the spaces H m  K  . From the triangle inequality one has

v K  AK , h v K

 v K  pˆ hK

m, K

m, K

 pˆ hK  Ak , h v K

which yields

u  Ah u here

V

 u  uˆh

V

 uˆh  Ah u V ,

m, K

uˆh 

 pˆ

K h

,

K  .

K

The Theorem will be proven if it can be established that

lim pˆ hK  AK , h v K h 0

m, K

 0.

(4.20)

According to the Theorem 3.1, the element pˆ hK  PhK can be uniquely expressed as follows

pˆ hK 

From (4.19) follows:

Mh

Nh  M h

i 1

i 1

 i, h  pˆ hK  pi, h 

lim  i , h  v K  pˆ hK h0

lim  i , h  v K  pˆ hK h0



i , h  pˆ hK  piZ, h .

(4.21)

  0, 0

Therefore

i , h  pˆ hK   i , h  v K  i , h  pˆ hK   i , h  v K  and the relation (4.20) is the implication of the representations (4.18) and (4.21).

Let us note that the density of the family PhK in the space H m  K  is the sufficient condition for the PhK -multi-solvency of the multitude of the boundary degrees of freedom of FE (see Theorem 3.2), as well as it is also necessary and sufficient condition for the convergence of the approximations on a fixed partition of the domain (see Theorem 4.2). Therefore, FE approximations of Sobolev spaces, constructed on conformed by subspace FE and a fixed

partition of the domain, can be external but non-converging. The following Theorem shows that external approximations constructed on homogeneous FE are always non-converging.

Theorem 4.3 External FE approximations  Vh* , ph , rh*  of the space H m    associated with a fixed partition

 of the domain  and built on homogeneous FE are non-converging in the space H m    .

 Proof - Let us consider an element K   . The family of subspaces PhK is dense in the space

H m  K  if, and only if, any continuous on H m  K  functional that equals zero on PhK , equals zero on H m  K  also (see Theorem 1.2 from [33]). For homogeneous FE functionals (3.18) equal zero on the elements from PhK . Obviously, functionals (3.18) do not turn to zero for any function v  H m  K  and, therefore, families of the spaces PhK are not dense in the spaces

H m  K  .

Theorem 4.4 Let us  VhC , ph , rhC  be a family of external FE approximations of the space H m    associated with a fixed partition  of the domain  and built on composite FE. For the approximations

V

C h

, ph , rhC  to be converging in the space H m    it is necessary and sufficient that families

PhK of the spaces of approximating functions of conformed by subspace FE contained in the

spaces PCK, h of composite FE are dense in the spaces H m  K  .

 Proof – Proof of this theorem follows from (3.29), (3.35), Theorem 4.2 and from the definition of the operator of X hC -approximation (see formula 4.16).

Convergence of approximations on a family of partitions Some definitions Let us  h be a family of partitions of the domain  , which satisfies the conditions of regularity (2.1). In classic FEM, the partition  h of the domain  is usually conducted in the way that all FE K   h are linearly equivalent to some initial FE Kˆ , also referred to as common FE (see, for example, [10]). In this case any element K   h can be obtained from the common element Kˆ through the linear mapping

F : xˆ  Rn  F  xˆ   Bxˆ  b  Rn ,

(4.22)

where B is an invertible matrix of dimensions n  n ; b is a vector in R n . The concept of linearly equivalent FE is introduced to obtain the approximation error estimates.

Let us PK be a space of functions p : K  R defined on the domain K , and let us G K be a space of functions g : K  R defined on the boundary K of the domain K . Because the

mapping (4.22) defines one-to-one correspondence between the points of the domains K and Kˆ then the following spaces can be associated with the spaces PK and G K

   g : K  R

P*K  p : K  R G*K



p  pˆ  F 1 , pˆ  Pˆ , g  gˆ  F 1  x  , x  K , gˆ  Gˆ



where Pˆ and Gˆ are the spaces of functions defined on the domain Kˆ and on the boundary Kˆ respectively. Then, due to the linearity of the mapping F P*K  P K , G*K  G K .

So the relations between the points xˆ  Kˆ and x  K and between the functions pˆ  Pˆ and p  P K , gˆ  Gˆ and g  G K are the following

xˆ  Kˆ  x  F  xˆ   K

  pˆ  Pˆ  p  pˆ  F 1  P K    1 K gˆ  Gˆ  g  gˆ  F  x   G , x  K  

(4.23)

From the relations (4.23) it follows that



pˆ  xˆ   p  x 

xˆ  Kˆ , pˆ  Pˆ ,

gˆ  xˆ   g  x 

xˆ  Kˆ , gˆ  Gˆ



Let us Kˆ , Gˆ K , Pˆ K , Pˆ be some arbitrary conformed by subspace FE that is characterized by the condition

 

diam Kˆ  1 .

(4.24)

For any element

 K, G

K

, P K , P  , K   h , through

 Kˆ , Gˆ

K

, Pˆ K , Pˆ



let us designate some

initial element for which:

 

K  F Kˆ

   g : K  R

PK  p : K  R GK

p  pˆ  F 1 , pˆ  Pˆ K



g  gˆ  F 1  x  , x  K , gˆ  Gˆ K



      

(4.25)

Moreover, let us the mapping (4.22) be a similarity mapping, when B is a diagonal matrix with the following diagonal elements:

bii  hK , 1  i  n, hk  diam  K  .

The similarity mapping maintains the normal to FE boundary.

The following relations take place for the similarity mapping:

 Kj v  hK j  Kj vˆ  ˆ

 d   hKn  1 d ˆ  0  j  m  1 

(4.26)

where  Kj is the operator of differentiation of j -order along the normal to FE boundary; d is an elementary measure of a  n  1 -dimensional surface in R n .

Let us modify the definition of the boundary degrees of freedom of FE by adding the multipliers hK 

 n  j 1 

:

  v   hK  n  j 1



K r

g K  Kj vd  ,

where g K is a basic function of the space Gh of boundary functions. Let us note that the multipliers do not change the definition of FE given in the Section 3.1.1. The modification allows to prove the important relation between the operators of Pˆ K - and PK -approximations, which later will be used in establishing the convergence of the approximations.

Theorem 4.5



  K, G

Let us Kˆ , Gˆ K , Pˆ K , Pˆ ,

K

, P K , P  be two similar conformed by subspace FE. If

pˆ i , 1  i  M ,

pˆ Zj , 1  j  N  M

are the basic functions of the element Kˆ and if:

pi , 1  i  M ,

p Zj , 1  j  N  M

are the basic functions of the element K , then for the operators of approximation AK and Aˆ K the following relation takes place

 AK v 

^

 Aˆ K vˆ

for all functions vˆ  dom Aˆ K , v  dom AK that are in correspondence vˆ  dom Aˆ K  v  vˆ  F 1  dom AK .

 Proof – The PK -approximant of the function v is (see Section 4.1)

AK v 

M

 i  v  pi  i 1

N M

  v p j 1

j

Z j

,

where coefficients  j  v  ,1  j  N  M are determined from the system of linear algebraic equations as in (4.4):

N M

  v  p j 1

j

Z j

, ptZ



0, K

  v  

M

 v  p

 i

i

i 1

 , ptZ  , 0, K

1t  N M.

(4.27)

Taking into account that F is the similarity mapping which maintains any normal orthogonal to the FE boundary, and using relations (4.26), one obtains:

i  v   hK  n  j 1



g K  Kj vd  hk 

 n  j  1



ˆ

ˆ

ˆ Kn 1dˆ  gˆ K hK j  Kj vh

Kˆ r

Kr



ˆ ˆ  ˆi  vˆ  . (4.28) gˆ K  Kj vd ˆ

ˆ

Kˆ r

For the initial FE the system (4.27) is: N M

 ˆ  vˆ   pˆ j 1

j

Z j

, pˆ tZ



0, Kˆ

   vˆ  

M

 ˆ  vˆ  pˆ

 i

i

i 1

 , pˆ tZ  , 0, Kˆ

1t  N M.

(4.29)

Taking into account the equality (4.28) and the relation

 f , g 0, K



ˆˆ  fgdx   fgh

n K

K

the system (4.29) can be rewritten as follows N M  hK n  ˆ j  vˆ   p Zj , ptZ   hK n  v  0, K j 1 

dxˆ  hKn

Kˆ

M

 v  p

 i

i

i 1

 fˆ , gˆ 

0, Kˆ

 , ptZ  , 0, K

1  t  N  M . (4.30)

Comparing the systems (4.30) and (4.27) it is apparent that

 j  v   ˆ j  vˆ  , 1  j  N  M . Therefore M

AK v   ˆi  vˆ  pi 

N M

i 1

 ˆ  vˆ  p j

j 1

Z j

.

From this, using the relations (4.23) one can finally obtain

 AK v   ^

M

 ˆi  vˆ  pˆ i  i 1

N M

 ˆ  vˆ  pˆ j 1

j

Z j

 Aˆ K vˆ .

Error estimates for piecewise polynomial approximations In the estimates below C designates a constant independent of the parameter h of the family of approximations and considered herewith functions, Pk    designates a space of the polynomials of k -order determined on the corresponding domain.

Theorem 4.6



Let us Kˆ , Gˆ K , Pˆ K , Pˆ 

 be a conformed by subspace FE. Let us m  1 be the highest order of

normal derivatives that participate in the definition of the boundary degrees of freedom of FE. If, for some whole number k  0 , take place the inclusions:

 

 

H k 1 Kˆ  H m Kˆ ,

(4.31)

 

(4.32)

 

Pk Kˆ  Pˆ K  H m Kˆ ,

then there exists such a constant C that for all similar FE  K , G K , P K , P  and for any function

u  H k 1  K  u  AK u l , K

 hKk  1   C  l  u k  1, K ,  K 

0  l  m,

where AK u is a PK -approximant of the function u and where:

hK  diam  K  , here S is a sphere from K .

K  sup diam  S 

(4.33)

 Proof – From the inclusion (4.32), accounting the fact that the operator of Pˆ K -approximation is identical on the space Pˆ K (see Section 4.5), it follows that

 

Aˆ K pˆ  pˆ .

pˆ  Pk Kˆ

 

Let us uˆ be a function from the space H k 1 Kˆ . Because of the inclusion (4.31), this function

 

also belongs to the space dom Aˆ K  H m Kˆ . The linear mapping

 

 

AˆK : H k 1 Kˆ  H l Kˆ

is continuous because of the inclusions (4.31) and (4.32). Herewith, the operators of Pˆ K - and

PK -approximation are in the relation (4.9) u  dom AK

 AK u 

^

 Aˆ K uˆ .

Then the necessary result follows from the Theorem 3.1.4 from [10].

Theorem 4.7 Let us there be given a regular family of conformed by subspace FE  K , G K , P K , P  . Let us for any FE from this family, inclusions (4.31) and (4.32) take place, in which K is substituted by Kˆ . Then there exists such a constant C that for all FE from the family and for any function

u  H k 1  K  the following inequality holds u  AK u

l, K

 ChKk 1  l u k 1, K ,

0  l  m.

(4.34)

 Proof - For any element K from the family it can be put in correspondence through the similarity mapping some initial element Kˆ that meets the relations (4.24) and (4.25). Then the estimates (4.33), in which constants C vary for different elements, are true. Because the constants C are bounded then the estimate (4.34) follows from the estimates (4.33) and the regularity conditions (2.1). 

Theorem 4.8 Let us there be given a family of external FE approximations Vh , ph , rh  of the space H m    associated with a regular family of partitions  h of the domain  and built on the conformed by subspace FE. Let us there exists such whole number k  0 when m  k allowing the following inclusions to take place

Pk  K   P K  H m  K  .

K   h

Then, there exists such a constant C independent from h , that for any function u  H k 1    the following inequalities hold: 1

 2 2 u  A u  Ch k  1  l u k 1,  , h  l, K   K  u  Ah u V  Ch k  1  m u k  1, 

0l m

where Ahu  X h is X h -approximant of the function u defined by the relation (4.6).

 Proof - Applying the Theorem 4.7 the following estimate is obtained

u  AK u

l, K

K   h , 0  l  m .

 ChKk 1 l u k 1, K

Using the relations (4.9) and the inequalities hK  h, K   h one obtains the following    u  AK u  K

2 l, K

  

1

2

 Ch

k 1 l

  2   u k  1, K   K 

1

2

 Ch k  1  l u k  1, 

Let us also note that    u  AK u  K

2 m, K

  

1

2

 u  Ahu

V

. 

Convergence of approximations Let us determine the conditions of convergence of external FE approximations associated with a family  h of partitions of the domain  , i.e. the conditions when the following holds

u  H m   

lim u  ph rh u h0

where ph rh  Ah is an operator of X h -approximation.

Theorem 4.9

V

 0,

Let us Vh , ph , rh  be a family of external FE approximations of the space H m    associated with a regular family  h of partitions of the domain  and built on conformed by subspace FE. If the inclusions

K   h

Pm  K   P K  H m  K 

take place, then the approximations Vh , ph , rh  converge in the space H m    .

 Proof - Using the Theorem 4.8 the following estimate is obtained

v  H m 1   

v  ph rh v

 Ch v m 1,  .

V

Therefore

lim v  ph rh v h 0

V

 0.

(4.35)

From the triangle inequality for any h and any v  H m 1   

u  ph rhv

V

 uv

V

 v  ph rhv V .

(4.36)

By introducing the function u  H m    and the arbitrary number   0 , one can determine the function v  H m 1    that satisfies the inequality u  v

V



 2

This is possible due to the density of the inclusion H m 1     H m    . According to (4.35), there exists such h0    that h  h0   

v  ph rh v

V



 2

.

Therefore, taking into account the inequalities (4.36), it is obtained

u  H m   

lim inf u  ph vh h  0 v !h Vh

V

 0 .

Theorem 4.10 A family of external FE approximations  Vh* , ph , rh*  of the space H m    associated with a regular family of partitions  h of the domain  and built on the homogeneous FE does not converge in the space H m    .

 Proof - In case of homogeneous FE for any member of the sequence ph uh  X h* the following functionals become zero (see Section 3.2)

g

K

K

 Kj  ph uh  K d   0,

0  j  m  1,

(4.37)

where g K is a restriction on K of the basic function from the space Gh of boundary functions. However, for any function u  H m     V the functionals (4.37) do not become zero. Therefore, according to the Theorem 1.2 from [33], it is impossible to segregate the sequence ph uh from the family of spaces X h* converging to the given function u  H m    in the norm

of the space V .

Theorem 4.11

Let us  VhC , ph , rhC  be a family of external FE approximations of the space H m    associated with a regular family  h of partitions of the domain  and built on composite FE. If the following inclusions take place

K   h

Pm  K   PCK  H m  K 

then the approximations  VhC , ph , rhC  converge in the space H m    .

 Proof – Proof of the Theorem is similar to the proof of Theorem 4.9 because the properties (4.15) and (4.9) of the operators AhC and AKC of the approximation, determined by the relations (4.12) and (4.16), are identical to the properties (4.5) and (4.9) of the operators Ah and AK for the approximations built on conformed by subspace FE.

5. Approximations of boundary spaces Definitions To construct the finite elements described in Section 3 it is necessary to know how to build families of spaces Gh of boundary functions that are dense in the boundary spaces Hˆ  , Hˆ 0 (see characterizations (2.5) and (2.6)). The approximations of the spaces Hˆ  , Hˆ 0 by the spaces Gh are referred to as boundary approximations, and they are examined in this Section.

Here and below C designates various constants independent of parameter h of the family of the approximations and herewith considered functions; Pk    designates the spaces of polynomials of k -order, determined on the corresponding domains.

Let us   R n be an open domain with sectionally smooth regular boundary  , and let us  be a connected patch of the boundary  . Let us assume that  can be represented as a combination of the finite number of domains of the same dimensions, i.e. N

   j,

N 

j 1

and let us assume that each of the domains  j can be continuously and one-to-one mapped onto the domain D j of the Euclidean space R n 1 : x   j t 

 x  ,

t   t1 , ..., tn 1   D j  R n 1 

where

 j  t   C0  Dj  ,

1 j  N.

(5.1)

It is assumed that v  H S    if v   j   H S  D j  . Herewith:  The map (5.1) is called the local map of the surface  

 j ,1  j  N is called the domain of the local map



D j ,1  j  N is called the domain of parameters of the local map

If one map is sufficient to define the surface  , then such surface is called elementary surface. Local maps (5.1) can be built by straightening part  j of the surface  through the introduction of new coordinates. Herewith the coordinates ti  ti  x  are defined in the vicinity of point

x0    in such a way that within these coordinates, part of the boundary  j has the equation tn  0 and part of the domain  , adjacent to it, is located in subspace tn  0 . The coordinates ti ,1  i  n can be introduced in the following way. Smooth section  j is associated with the

Cartesian coordinate system y1 , ..., yn so that  j can be defined through the equation

yn  f  y  ,

y   y1 , ... yn 1 

(5.2)

and variables x are substituted by variables y . Then new variables t are introduced by the formulas

ti  yi ,

1  i  n  1,

tn   yn  f  y  .

Further, for simplicity, only elementary surfaces  are considered. Norm and semi-norm on the space H S    are defined as follows

    D

 s

 v s,      D

 

v

s, 

S



s

Dt v    

Dt v   

2

2

   1   2 2 1  grad f dt        1  grad f dt   

1

2

2

(5.3)

where: 

D is a domain of parameters t   t1 , ..., tn 1  of the map of the surface 



  1 , ..., n 1  is a multi-index



Dt is an operator of differentiation by variables t1 , ..., tn 1

 function f  t  is defined by relation (5.2) 

dt  dt1 , ..., dtn 1 ;



grad f   f dt1, ..., f dtn 1 

The expression

d   1  grad f dt 2

defines the  n  1 -dimensional measure of the elementary surface.

When s  0 , the expression (5.3) defines the norm on space the L2    induced by the inner product

 u, v 0, 

  uvd  . 

Let us be given a domain   Rn with sectionally smooth regular boundary  . Let us a partition  of the domain  into FE K   is defined. In addition to the boundary spaces HT and H T (see characterizations (2.2), (2.4) and (2.5)) the following spaces are defined m 1

W   H K

s j 1

r ,l

K r K r , l j  0

   f W

 K  ,

W0  f  W |  r ,l  M   r ,l  W

 0

|

f



s j  0, f r1,l , j    1

j

 0

    2 f r ,l , j , 0  j  m  1     



(5.4)

where K   , K r  M  K r  , K r , l  M  K r , l  .

In the characterization (5.4), figures 1 and 2 indicate that the functions are defined on the interelement boundaries  r , l of adjacent FE K 1 and K 2 . Herewith, for every  r , l  M   r , l  there are such elements K 1 and K 2 that r ,l  Kr1,l  Kr2,l .

The norm on the space W is:

g

W

m 1        g rK, l , j  K K K j  0 r r ,l 

1

2 s j 1, K r , l

 2  . 

The following semi-norms on W are also useful:

 g j      g rK, l , j  K K K r r ,l  0  j  m 1

2 s j 1, K r ,l

Spaces W  and W0 are the closed subspaces of the space W .

  

1

2

(5.5)

Operators of boundary approximation Let us there be given some partition  of the domain  into FE K   . In Section 3.1 it was assumed that the support of any basic function g r , j , t , 0  j  m  1 of the finite-dimensional space Gh of boundary functions is:  either a connected sub-domain K r , j of the common boundary  r  K r1  K r2 of adjacent FE K 1 and K 2 , i.e. K r , j  supp  g r , j , t    r ,

K r, j  0

(5.6)

 or a connected sub-domain of the face K r* of FE that coincides with the boundary  , i.e. K r , j  supp  g r , j , t   K r* (it is also possible that K r , j   r or Kr , j  Kr* ).

Any of the basic functions g r , j , t with support in r is a pair g r , j , t   g 1r , j , t , g r2, j , t  ,

where g1r , j , t  gr , j , t |K1 , gr2, j , t  gr , j , t |K 2 . r

r

Because of the inclusions Gh  Hˆ   Hˆ 0 the components of the pair satisfy the relation g1r , j , t    1 g r2, j , t , j

Let us introduce the following designations:

0  j  m  1.



g rK, j , t is the restriction of the function g r , j , t onto the face of the element K



GrK, j is the space generated by the functions g rK, j , t



G K is a union of the spaces GrK, j

The space Gh of the boundary functions can be represented as follows: Gh 

G

K

K 

(5.7)

K r, j

(5.8)

,

K

where GK 

m 1

G K r

.

j 0

The assumption (5.6) allows the construction of the spaces of FE whose basic functions have a local support in the domain  . With a specific choice of the functions g r , j , t , it also allows to prove the density of the family Gh of the spaces of boundary functions in the space Hˆ 0 or Hˆ  . It will be shown in the Section 6 that, in order to obtain optimal error estimates of the approximations of variational equations, it is necessary to strengthen the assumption (5.6).

Let us designate through g r , l , j , t a basic function of the space Gh whose support is:  either a connected sub-domain K r , l , j of the smooth section r ,l  Kr1,l  Kr2,l of the common face r of adjacent FE K 1 and K 2 : K r , l , j  supp  g r , l , j , t    r , l ,

K r ,l , j  0

 or a connected sub-domain of the smooth section Kr*, l of the face K r* that coincides with the boundary  , i.e. Kr ,l , j  Kr*,l (it is also possible that K r , l , j   r , l or

Kr ,l , j  Kr* ).

Then the space G K in the expression (5.8) can be represented as follows

GK 

m 1

  G

K r ,l , j

,

(5.9)

Kr Kr , l j  0

where GrK, l , j is the space generated by the basic functions grK, l , j , t .

Let us designate through PrK, j (and through PrK,l , j correspondingly) the finite-dimensional spaces of the basic functions g rK, j , t (and grK, l , j , t correspondingly) that have a common support K r , j (and K r , l , j correspondingly). By definition, the spaces PrK, j and PrK,l , j are not empty. The domains K r , j

and K r , l , j , along with their spaces PrK, j and PrK,l , j , are referred to as surface elements. Therefore, a surface element can be characterized by the pair of multitudes  K r , j , PrK, j  or  K r , l , j , PrK, l , j  .

Let us conduct the partitions  r , j , 0  j  m  1 of the face K r into the surface elements K r , j   r , j , 0  j  m  1 so that the following conditions hold:

Kr , j   r , j , 0  j  m  1



Kr   Kr , j ,



for any K r , j   r , j the domain K r , j is closed and the multitude of its internal points is not

Kr , j

empty: K r , j  0



for various arbitrary Kr1, j , Kr2, j   r , j , K r1, j  K r2, j  



any K r , j   r , j has a sectionally smooth regular boundary

Let us define the spaces of surface elements of the face K r :

GrK, j 

P

K r, j

 L2  K r 

Kr , j   r. j , 0  j  m  1 .

Kr , j

Similarly, by partitioning  r , l , j , 0  j  m  1 of a smooth patch K r , l of the face K r into the surface elements K r , l , j   r , l , j , let us define the spaces of surface elements: GrK, l , j 

P

K r,l, j

 L2  K r , l 

K r , l , j   r , l , j , 0  j  m  1.

Kr ,l , j

When constructing the external FE approximations of the Sobolev space H m    , it is necessary to define m layers of surface elements on the boundaries of FE. It is possible that partitions of these layers do not coincide, i.e. if j  k then

 r,l, j   r,l, k , 0  j  m  1,

 r, j   r, k 0  k  m 1

Let us define the following orthogonal projectors: RrK, j : L2  K r , j   PrK, j

(5.10)

BrK, j : L2  Kr   GrK, j

(5.11)

RrK, l , j : L2  K r , l , j   PrK, l , j

(5.12)

BrK, l , j : L2  K r , l   GrK, l , j

(5.13)

0  j  m 1

Let us also define the orthogonal projector

Bh : Hˆ 0  Gh that can be represented as follows (depending on the patches of boundaries being partitioned into the surface elements): Bh 

m 1

 B

K r, j

K

K r

(5.14)

j 0

or:

Bh 

m 1

   B

K r ,l , j

K

.

(5.15)

Kr Kr , l j  0

The orthogonal projector Bh is referred to as the operator of Gh -approximation. Let us call the components of the operator Bh defined by the expressions: BK 

m 1

 B

K r, j

K r

BK 

(5.16)

j 0

m 1

B

K r ,l , j

K r K r , l j  0

as operators of G K -approximation.

From the definition of the operator of Gh -approximation and the operator of G K -approximation follows their important property that is established by the following theorem.

Theorem 5.1 Let us g be an arbitrary function from the space Hˆ 0 . Then

K  

 Bh g  |K  BK g .



(5.17)

The relation (5.17) is similar to the relation (4.9) between the operators of PK - and X h -approximation.

Approximations on a fixed partition Let us consider a patch  of the boundary of FE K   . Let us assume that  is an elementary surface defined by the map

 : D  R n 1    R n where  is a homeomorphism; D is the domain of parameters t   t1 , ..., tn 1  of the map.

Since the partition  of the domain  is fixed, then the spaces Hˆ  and Hˆ 0 are fixed as well. The spaces L2    are the elementary components of the spaces Hˆ  and Hˆ 0 . Dense in the space L2    families of finite-dimensional spaces can be constructed in a number of ways. In particular, for this purpose, complete systems of algebraic polynomials and other systems of

analytical functions can be used. Below are closely examined the approximations of the spaces

L2    by the spaces of the surface elements.

Let us define a family of partitions    h  of the surface  into the surface elements

K     h  so that the partition requirements described in Section 5.2 are satisfied. Let us also require that the family    h  is regular so: 

there exists such positive constant   that

K     h 

hK K   

(5.18)

where:

hK  diam  K   ,

 K  sup diam  S  , S is a sphere from K   



diameters hK approximate zero

The surface elements K     h  are scaled in the same way as FE K   h in Section 4.3.1. Let us designate for any surface element K     h  through Kˆ  the element that satisfies the following condition diam(Kˆ  )=1 .

Herewith K   F ( Kˆ  ) ,

where F is a similarity mapping:

(5.19)

F (tˆ)  B tˆ  b,

tˆ  (tˆ1 ,..., tˆn 1 ) ,

here B is a diagonal matrix of dimensions  n  1   n  1 with diagonal elements equal to hK ; b is a vector of dimension n  1 .

Let us PK be a space of functions p : K  R determined on the domain K  . Due to the bijection of the mapping F one can associate with the space PK the following space



P  p : K  R | p  pˆ  F1 , pˆ  PˆK



where PˆK is a space of functions determined on the domain Kˆ  . Since F is a similarity mapping, then P  PK . Implication of the relations:

tˆ  Kˆ   t  F  tˆ   K  pˆ  PˆK  p  pˆ  F1  PK is the equality pˆ  tˆ   p  t 

tˆ  Kˆ  , pˆ  PˆK .

Theorem 5.2 Let us

 Kˆ , Pˆ  ,  K , P  

K 



K 

be two similar surface elements. Then, for the orthogonal

projectors:

RK : L2  K    PK

 

RˆK : L2 Kˆ   PˆK holds the relation

 R v K 

 RˆK vˆ

^

  vˆ  L  Kˆ   v  vˆ  F

(5.20)

which is valid for all functions vˆ  L2 Kˆ  , v  L2  K  that are in the relation 2

1 



 L2  K 

◊ Proof – Let us pi , pˆ i , 1  i  N are the basic functions of the spaces PK and PˆK correspondingly. Projection of the function v  L2  K  onto space PK is: RK v 

N

 v  p , i 1

i

i

where coefficients i  v  ,1  i  N are determined from the following system of linear algebraic equations

  v   p , p  N

i 1

i

i

j

0, K

  v, p j 0, K ,

1 j  N.

(5.21)

1 j  N.

(5.22)



For the initial surface element Kˆ  , the system (5.21) is:

 ˆ  vˆ   pˆ , pˆ  N

i 1

i

i

j

0, Kˆ 

  vˆ, pˆ j 0, Kˆ , 

Since F is the similarity mapping, then

p,p  i

j

0, K 





pi p j dt



pˆ i pˆ j hKn  1 dtˆ

K



Kˆ 

 hKn  1  pˆ i , pˆ j 0, Kˆ and the system (5.22) can be rewritten as follows



hK1  n  ˆi  vˆ   pi , p j 0, K  hK1  n  v, p j 0, K , N



i 1

1 j  N.

(5.23)



The comparison of the systems (5.21) and (5.23) shows that

ˆi  vˆ   i  v  , 1  i  N and the required result is obtained.

Theorem 5.3



Let us Kˆ  , PˆK

 be a surface element. If for some whole integer s  0 , the following inclusions

hold

 

 

Ps Kˆ   PˆK  L2 Kˆ 

(5.24)

then, there exists such a constant C , that for all surface elements  K , PK  and any function

g  H s 1  K  the following inequality holds g  RK g

l , K

 C  hKs 1 Kl 

g

s  1, K

,

0  l  s,

(5.25)

where RK g is the projection of the function g onto the space PK , and

hK  diam  K  

 K  sup diam  S  , S is a sphere from K   

◊ Proof - On the space PˆK , the orthogonal projector Rˆ K coincides with the identical operator, i.e. pˆ  PˆK

RˆK pˆ  pˆ

Therefore, according to the inclusions (5.24) pˆ  Ps Kˆ 

 

RˆK pˆ  pˆ .

(5.26)

The required inequality (5.25) follows from (5.26) and the relation (5.20) between orthogonal projectors RˆK , RK , and from the Theorem 3.4.1 from [10]. ‫ٱ‬

Theorem 5.4 Let us there be a regular family of surface elements

K



, PK  and for any element from the

family the following inclusions hold

Ps  K   PK  L2  K  ,

s  0.

Then, there exists such a constant C that for all elements from the family and for any function

g  H s 1  K  the following inequality holds g  RK g

l , K

 ChKs 1 l g s 1, K ,

0  l  s.



(5.27)

◊ Proof - For every element K  there can be put in correspondence some initial element Kˆ  which satisfies the relation (5.19). Herewith the space PK can be characterized as follows



PK  p : K  R |



p  pˆ  F1 , pˆ  PˆK .

Then, for any element K  from the family, the inequality (5.25) holds. The inequality (5.27) follows from the estimates (5.25), the conditions of regularity (5.18), and the bounded constants in the estimates. ‫ٱ‬

Theorem 5.5 Let us    h  be a regular family of the partitions of the surface patch  into the surface elements K     h  . Let us assume that for some whole number s  0 and for any

K     h  Ps  K   PK  L2  K  , where Ps  K  is a space of polynomials of s -order. Then there exists such independent of h constant C that

g  H s 1   

g  BK  h  g

g  L2   

0, 

 Chs 1 g s 1, 

lim g  BK  h  g h 0

0, 

0

(5.28)

where

BK  h  : L2     GK  h  is the orthogonal projector from the space L2    onto the space of surface elements:

GK  h  

P

K 

 L2    ;

K     h 

K

and:

h  max hK , K

hK  diam  K  .

◊ Proof - From the Theorem 5.4, for any K     h  the following inequality holds

(5.29)

g  H s 1  K 

g  RK  h  g

0, K

 ChKs 1 g s 1, K . 

According to the representation (5.29) of the space GK  h  , the following relation exists between the orthogonal projectors RK and BK  h  : g  L2    , K     h 

 B h g  | K 

K

 RK g .

Taking into account the inequality

hK  h,

K     h 

one obtains: g  H s  1     K   g  R g  K

2 0, K

  

1

2

 gB

K 

h g

0, 

 Ch

s 1

  g  K

2 s  1, K 

  

1

2

 Ch s  1 g

s  1, 

Let us s  0 then

g  H 1   

g  BK  h  g

0, 

 Ch g 1, 

(5.30)

and

lim g  BK  h  g h 0

0, 

 0.

The relation (5.28) follows from the relation (5.30) and the density of the enclosure

H 1     L2    .‫ٱ‬

To approximate the spaces H m    , it is necessary to define m layers of the surface elements on the boundaries of FE. The parameter of a regular family of partitions of the face

Kr  M  Kr  (or smooth patch K r , l  M  K r , l  of the face K r ) in general depends on the patch under consideration and on the layer number. Let us designate through  r , j  hr , j  the family of partitions of the face K r for the j -layer of surface elements. Herewith





hr , j  max diam  K r , j  . Kr , j

Similarly through  r , l , j  hr , l , j  , let us designate the family of partitions of a smooth patch K r , l of face K r . Here





hr , l , j  max diam  K r , l , j  . Kr ,l , j

The parameter h of the family of boundary approximations is defined by one of the following relations: h  max  hr , j  , K , Kr , j

where K   ,

Kr  M  Kr  ,

(5.31)

0  j  m  1,

or:

h

max

K , K r , K r , l , j

h  , r ,l , j

where K   , K r  M  K r  , K r , l  M  K r , l  , 0  j  m  1 .

Theorem 5.6

(5.32)

Let us  be a fixed partition of the domain  into FE K   . Herewith each face K r of any FE

is

an

elementary

surface.

Let

us

assume

that

there

are

regular

families

 r , j  hr , j  , 0  j  m  1 of partitions of faces K r of every FE into the surface elements K r , j . Also, let us assume that for any surface element K r , j   r , j  hr , j  the following enclosures hold P0  K r , j   PrK, j  L2  K r , j  ,

0  j  m 1

where PrK, j is the space of approximating functions of the surface element. Then, the family of spaces Gh of boundary functions defined by relations (5.7) and (5.8) is dense in the space Hˆ 0 , i.e.

g  Hˆ 0

lim g  Bh g h 0

HT

 0,

(5.33)

where: 

Bh is the operator of Gh -approximation that is defined by (5.14)

 parameter h is defined by (5.31)

◊ Proof - In the Theorem 5.5 let us assume: s  0,   K r ,    h    r , j  hr , j  , PK  PrK, j , BK  h   BrK, j  hr , j  , 0  j  m  1 ,

where operators BrK, j  hr , j  are defined by (5.11). Then for any K   , Kr  M  Kr  g  L2  K r 

lim g  BrK, j  hr , j  g

hr , j  0

0, K r

 0,

and the relation (5.33) follows from (5.31), (5.14), (5.16) and from the relation (5.17) between the operator of G K - and Gh -approximation. ‫ٱ‬

Theorem 5.7 Let us  be a fixed partition of the domain  into FE K   and let us every patch K r , l of any face of FE be an elementary surface. Let us assume that there are regular families

 r , l , j  hr , l , j  , 0  j  m  1 of partitions of patches K r , l into the surface elements K r , l , j . Also, let us assume that for any K r , l , j   r , l , j  hr , l , j  the following enclosures hold Ps j  K r , l , j   PrK, l , j  L2  K r , l , j  ,

where PrK,l , j are the spaces of approximating functions of the surface elements, and s j  0, 0  j  m  1 .

Then, the family of spaces Gh of boundary functions defined by (5.7) is dense in the space Hˆ 0 , i.e.

g  Hˆ 0

lim g  Bh g h 0

HT

 0,

(5.34)

where the operator Bh and the parameter h are defined by the expressions (5.15) and (5.32) correspondingly. Further, there exist such independent of the parameter h constants C j , 0  j  m  1 that

g  W0

g  Bh g

HT



m 1

C h j 0

j

s j 1

g j,

where the space W0 is defined by the characterization (5.4);  j are semi-norms (5.5)

◊ Proof - Let us assume that in Theorem 5.5:

(5.35)

s  s j ,   K r , l ,    h    r , l , j  hr , l , j  , PK  PrK, l , j , BK  h   BrK, l , j  hr , l , j  , GK  h   GrK, l , j 

P

K r,l, j

,

Kr ,l , j

where operators

BrK, l , j  hr , l , j 

are defined by the expression (5.13). Let us also

s j  0, 0  j  m  1 . Then, from the Theorem 5.5 it follows that for any

K   , K r  M  K r  , K r , l  M  K r , l 

g  L2  K r , l 

lim

hr , l , j  0

g  BrK, l , j  hr , l , j  g

0, K r , l

 0.

(5.36)

The relation (5.34) follows from:  the equality

g  Bh g

HT

 m 1       g  BrK, l , j  hr , l , j  g  j  0 K K K r r ,l 

2 0, K r , l

  

1

2

(5.37)

 the relations (5.36)  the definition (5.32) of the parameter h  the enclosure P0  K r , l , j   Pk  K r , l , j  ,

k 0

Let us s j  0, 0  j  m  1 . According to Theorem 5.5, for any smooth patch K r , l of boundaries of FE K   the following inequality holds g  H

s j 1

 K  r,l

g  BrK, l , j  hr , l , j  g

s 1

0, K r , l

 Chr ,jl , j g

s j  1, K r , l

.

(5.38)

The required inequality (5.35) is the implication of (5.38), (5.37), and the definition (5.32) of the parameter h .‫ٱ‬

Approximations on a family of partitions Let us define a regular family of partitions  h of the domain  into FE K   h . The parameter

h of the family has a meaning of the maximum diameter of FE. When refining the partition, then h  0 and the number of finite elements tends to infinity: card  h    . In this case there are the

families

of

the

boundary

spaces

considered

in

the

previous

sections:

HTh , Hˆ h , Hˆ 0h , Wh , Wh , W0h .

For any FE K   h let us designate through Kˆ a similar FE that satisfies the condition

 

diam Kˆ  1 .

(5.39)

Herewith:

 

   diam  K   hK  n n  F : xˆ  R  F  xˆ   Bxˆ  b  R   K  F Kˆ

where: 

F is a similarity mapping



B is a diagonal matrix of dimensions n  n with diagonal elements equal to hK



b is a vector of dimension n

(5.40)

Let us the patch ˆ of the surface of the element Kˆ be defined by the map

ˆ : Dˆ  R n 1  ˆ  R n ,

(5.41)

where ˆ is a homeomorphism; Dˆ is the domain of the parameters tˆ   tˆ1 , ..., tˆn 1  of the map.

Similarity mapping (5.40) transforms the coordinates of the point Aˆ  ˆ in the global and local coordinate systems according to the following formulas:

xi  hK xˆi  bi , t j  hK tˆj ,

1 i  n 1  j  n 1

In this case the domain of parameters Dˆ  Rn 1 of the map (5.41) is transformed into the similar domain D  R n 1 . Therefore, with the similarity mapping (5.40) is associated the similarity mapping F : tˆ  R n 1  F  tˆ   Btˆ  R n 1 ,

(5.42)

where B is a diagonal matrix of dimensions  n  1   n  1 with the diagonal elements equal

 

hK . Herewith D  F Dˆ .

Let us establish the relations between the points tˆ  Dˆ , t  D

and the functions

 

gˆ  L2 Dˆ , g  L2  D  as follows: tˆ  Dˆ  t  F  tˆ   D

(5.43)

 gˆ : Dˆ  R    g  gˆ  F

1 

The equality

: D  R  (5.44)

gˆ  tˆ   g  t 

holds for all points tˆ and tˆ in the correspondence (5.43) and for any functions gˆ and g in the relation (5.44).

Let us PK be an enclosed space of boundary functions g :   R determined on the patch  . The mapping (5.42) associated with the mapping (5.40) establishes a one-to-one correspondence between points of domains D and Dˆ of patches  and ˆ for similar elements K and Kˆ . Then with the space PK there can be associated the space



P  g:R|

g  gˆ  F1 , gˆ  PˆK



that, due to linearity of the mapping F , coincides with space PK , i.e. P  PK .

Theorem 5.8 Let us PˆK and PK be the finite-dimensional spaces of functions defined on the boundary patches

ˆ and  of similar FE Kˆ and K respectively. Then the orthogonal projectors:

RK : L2     PK

 

RˆK : L2 ˆ  PˆK are in the relation

R g K 

 

^

 RˆK gˆ

for all the functions gˆ  L2 ˆ , g  L2    from (5.44).

(5.45)

◊ Proof - The similarity mapping (5.40) of the element Kˆ is associated with the similarity mapping (5.42). Scales of these mappings are identical. Further course of proof of this theorem is identical to the proof of the Theorem 5.2.‫ٱ‬

Theorem 5.9 Let us Kˆ be a finite element, Dˆ be the domain of parameters of the map of a boundary patch ˆ of Kˆ , and PˆK be a finite-dimensional space of functions determined on Dˆ .

If for some whole number s  0 , the following enclosures hold

 

 

Ps Dˆ  PˆK  L2 Dˆ

(5.46)

then, there exists such a constant C that for all elements K similar to Kˆ and for any function

g  H s 1  D  g  RK g

l, D

 C  hKs 1 Dl  g s 1, D ,

0  l  s,

(5.47)

where: 

RK g is the projection of the function g onto the space PK



hK  diam  K 



 D  sup diam  S  , S is a sphere from D

◊ Proof - To the similarity mapping F : Kˆ  K it can be put in correspondence the similarity mapping F : Dˆ  D which has the same as F scale hK . On the space PˆK the orthogonal projector

 

RˆK : L2 Dˆ  PˆK is an identical operator. Herewith, according to the enclosures (5.46) pˆ  Ps Dˆ RˆK pˆ  pˆ .

 

Then the inequality (5.47) follows from the relation (5.45) between orthogonal projectors Rˆ K and RK and from the Theorem 3.1.4 from [10]. ‫ٱ‬

Theorem 5.10 Let us there be given a family of finite elements, and any of the elements has a surface patch  K with a map

 : DK  R n 1   K  R n . Let us assume that the family of the domains DK of the patches is regular, i.e. there exists such a constant  that DK

diam DK /  DK   ,

(5.48)

where

 D  sup diam  S  , S is a sphere from DK  . K

If, for any FE from the family and for some whole number s  0 the following enclosures hold

Ps  DK   PK  L2  DK  ,

where PK is a finite-dimensional space of functions determined on the domain DK , then there exists such a constant C , that for all elements from the family and for any function

g  H s 1  DK  g  RK g

l , DK

 ChKs 1l g s 1, D ,

0ls

(5.49)

K

◊ Proof - To any element from the family, it can be put in correspondence some similar element

Kˆ defined by the relations (5.39) and (5.40). Therefore, for every FE there exists such a constant

C (see Theorem 5.9) that

g  H s 1  DK 

g  RK g

l , DK

 C  hKs 1  Dl K

g

s  1, DK

.

(5.50)

The inequality (5.49) follows from the inequalities (5.50), the regularity conditions (5.48), and the bounded constants in the estimates (5.50).‫ٱ‬

Theorem 5.11 Let us define a regular family  h of partitions of the domain  into FE K   h in such a way that every face K r of any FE from the family is an elementary surface. Let us assume that for every K   h there are regular partitions  r , j , 0  j  m  1 of faces K r into the surface elements K r , j   r , j . Let us assume that for any surface element K r , j   r , j , the spaces PrK, j of approximating functions contain polynomials of zero degree: P0  K r , j   PrK, j  L2  K r , j  .

Then:

gh  Hˆ 0h

lim gh  Bh gh h 0

HTh

 0,

(5.51)

where Bh is the operator of Gh -approximation defined by (5.14); h  max  diam K  .

◊ Proof - By assuming in Theorem 5.10 that DK  Kr , j , PK  PrK, j , RK  RrK, j , s  0 for any:

K   h , Kr, j   r, j , 0  j  m  1 ,

one obtaines: g  H 1  Kr , j 

g  RrK, j g

0, Kr , j

 ChK g 1, K ,

(5.52)

r, j

where RrK, j is the orthogonal projector (5.10).

Let us define the spaces:

Zh 

m 1

 H  K  ,

K   h , K r  M  K r 

1

r

K

K r j  0

Z 0h   f  Z h |  r  M   r 



f r1, j    1 f r2, j , 0  j  m  1 j

Taking into account the estimate (5.52), the definitions (5.14) and (5.16) of the orthogonal projectors Bh and BK , and the inequality hK  h , one obtains that for any function g h  Z 0h m 1  K      g h  Rr , j g h  K Kr j  0 Kr , j

 g h  Bh g h

Therefore

HTh

2 0, K r , j

  

1

2



m 1   Ch      g h  K K j  0 K r r, j 

2 1, K r , j

  

1

(5.53) 2

g h  Z 0h

lim g h  Bh g h h0

 0.

HTh

Then the relation (5.51) follows from the inequality (5.53) and the density of enclosure

Z0h  Hˆ 0h . ‫ٱ‬

Theorem 5.12 Let us define a regular family  h of partitions of the domain  into FE K   h in such a way that each smooth patch K r , l of the face K r of FE K   h is an elementary surface. Let us assume that for each K   h are defined the regular partitions  r , l , j , 0  j  m  1 of the smooth patches K r , l into the surface elements K r , l , j   r , l , j . Let us allow that for any surface element K r , l , j the spaces PrK,l , j of approximating functions contain polynomials of s j -order  s j  0  :

Ps j  K r , l , j   PrK, l , j  L2  K r , l , j 

Then there exist such constants C j , 0  j  m  1 independent of the parameter: h  max hK , K

K  h

that

g h  W0h

g h  Bh g h

gh  Hˆ 0h

lim gh  Bh gh h 0

HTh



m 1

C h j 0

HTh

j

0

s j 1

gh

j

(5.54)

(5.55)

where Bh is the operator of Gh -approximation defined by (5.15);  j is the semi-norm (5.5)

◊ Proof - By assuming in the Theorem 5.10 that:

DK  Kr , l , j , PK  PrK, l , j , RK  RrK, l , j , s  s j one obtains g  H

s j 1

K 

s 1

g  RrK, l , j g

r,l, j

0, Kr , l , j

 ChKj

g

s j  1, Kr ,l , j

,

where RrK, l , j is the orthogonal projector (5.12). From this, considering the definitions (5.7) and (5.9) of the orthogonal projectors Bh and BK and the inequality hK  h , it is obtained that for any function g h  W0h

g h  Bh g h

HTh

     K K K r r ,l 

m 1



j  0 Kr ,l ,l

gh  R

K r,l, j

gh

2 0, K r , l , j

  

1

2



m 1

C h j 0

j

s j 1

gh j .

The relation (5.55) follows from the inequality (5.54) and the density of the enclosure

W0h  Hˆ 0h . ‫ٱ‬

6. External approximations of variational equations Construction and total error estimate Let us  be a domain in the space R n with a sectionally smooth regular boundary  . Let us assume that V  H m    , V0  H0m    and let us consider the continuous on V  V bilinear form a  u, v  

m

  a  x D pq

p , q 0 

p

uD q vdx ,

where a pq  x   L    .

Generated by the bilinear form formal operator  is defined by the relations:

 u, v   a  u, v  u 

m

  1

p , q 0

q

v  V0 , D q  a pq  x  D p u .

The differential operator  has order 2m . The space V     u

u  H m    , u  L2   

is the domain of determination of the operator  .

Elliptic boundary value problems for the operator  are equivalent to the variational equations defined on enclosed subspaces W from H m    such that H0m     W  H m    . The proof of this fact is based on the existence of Green formula that corresponds to the given boundary

value problem. For instance, for a boundary value problem with mixed boundary conditions the space W can be defined by the characterization



W  u  H m  |



 u 0 0

where  is the trace operator (1.7); 0 is a bounded open sub-domain of the boundary  .

The equivalency means that if the function u  V    satisfies some boundary conditions and the differential equation

f  L2   

u  f ,

(6.1)

then the function satisfies the variational equation as well

a  u, v   f  v 

v  W

(6.2)

where f  v  is some linear form on W .

The external approximations of the equation (6.2) are constructed in the following way. Let us conduct the partition  of the domain  into FE K   and let us define the spaces V , V0 , H , T by the relations (1.1) and (1.6). Let us introduce the continuous on V  V bilinear

form a u, v  

m

   a  x D K

where u K  u

K

p , q 0 K

pq

p

u K D q v K dx

(6.3)

, v K  v K . It is apparent that the form a  u, v  is the restriction of the form

a  u , v  onto the space V  V .

The space X h  V of FE approximates the space V . Discrete variational problem consists in the determination of the element uh  Vh  R

vh  Vh

N  h

that satisfies discrete variational equation

a  ph uh , ph vh   f  ph vh  ,

(6.4)

that is the external approximation of the equation (6.2).

Theorem 6.1 Let us the element f  V  in the equations (6.2) and (6.4) is given and the following conditions hold: 

the form a  u, v  is W -elliptic, i.e. C1  0 v  W



a  v, v   C1 v

V

a  v , v   C2 v

V

2

(6.5)

the form a  u , v  is V -elliptic, i.e. C2  0 v  V

2

(6.6)

Then the variational problems (6.2) and (6.4) have unique solutions u -and uh correspondinglyand the following inequality holds

u  ph uh

V

  C  inf u  ph uh  vh Vh 

V

 sup

wh Vh

a  u, ph wh   f  ph wh     ph wh V 

(6.7)

◊ Proof – The existence and the uniqueness of the solutions of problems (6.2) and (6.4) follows from: the inequalities (6.5) and (6.6); the enclosure ph vh  X h  V for any h and for any vh  Vh ; the Theorem 1.1.3 from [10]

From V -ellipticity and the continuity of the bilinear form a  u , v  , it follows: C2 ph uh  ph vh

 a  ph uh  ph vh , ph uh  ph vh  

2 V

 a  u  ph vh , ph uh  ph vh   f  ph uh  ph vh   a  u , ph uh  ph vh 

Then

C2 ph uh  ph vh

V

 C3 u  ph vh

V

 sup

wh Vh

f  ph wh   a  u, ph wh  ph wh

.

V

From this, taking into account the triangle inequality

u  phuh

V

 u  phvh

V

 phuh  phvh

V

it is obtained the estimate (6.7). ‫ٱ‬

The first term in the right-hand side of the estimate (6.7) satisfies the following inequality inf u  ph vh

vh Vh

V

 u  Ah u

V

    u  AK u  K

1

2 m, K

 2  , 

(6.8)

where Ah is the operator of X h -approximation and AK is the operator of PK -approximation determined in Section 4.1. Then one can see that the first term depends on the smoothness of the exact solution u of the variational problem (6.2) and on the properties of the approximating functions p  P K of the finite elements. The magnitude of this term is referred to as an internal

error of approximation. For the approximations built on a family of partitions of the domain and conformed by subspace FE, the estimates of the internal errors were obtained in Section 4.3.

Let us call as functional of the external error of approximation the magnitude

dh  a  u, ph wh   f  ph wh  . The norm of the functional is referred to as an external error of approximation

dh

V

 sup

a  u, ph wh   f  ph wh  ph wh

wh Vh

.

(6.9)

V

External error estimates and convergence of approximations In order to obtain the estimate of the external errors let us derive the corresponding Green’s formula for a partition of the domain.

Generated by the form (6.3) formal operator  is defined by the following relations

 u , v   a  u , v  v  V , u    u , K   , 0

K

K

K

K uK 

m

  1

p , q 0

q

D q  a pq  x  D p u K .

Let us  be the trace operator (1.7) for the partition of the domain. For the spaces V , H , V0 , T and for the operator  hold the assumptions of the Theorem 6.2.1 from [33] and therefore, there exists a unique operator  such that it holds the following Green’s formula: a  u , v    u , v    u ,  v ,

u , v  V

(6.10)

where ,  is a duality relation on T   T and where:

    K

m



 rK, l , j ,

(6.11)

K r K r , l j  2 m 1



 rK, l , j  L H m  K  , H

m j  1

2

 K   r ,l

(6.12)

m  j  2m  1

Theorem 6.2 Let us the variational problem (6.2) is approximated with FE that are built by the necessary and sufficient sign of the external approximations established in the Theorem 2.1 (i.e. conformed by subspace, homogeneous, or composite FE). Then there exists such independent of parameter h constant C that the external error of the approximation satisfies the inequality

dh

V

 C  u  Bh u

HT

,

where: 

HT is the space (2.2)



Bh : g  HT  Bh g  Gh is the operator of Gh -approximation



Gh is a space of the boundary functions

(6.13)

◊ Proof - From the relation (6.10) it follows that a  u, ph wh    u, ph wh    u,  ph wh .

Let us multiply the equation (6.1) by the function ph wh and integrate it over the domain  , so

 u, ph wh    f , ph wh  . The following equality holds

 u, ph wh    u, ph wh  , and then a  u, ph wh   f

 ph wh    u,  ph wh

.

Let us identify the duality relation ,  on T   T with the unique expansion of the inner product

 ,   in

HT . According to the Theorem 2.1

 gh ,  ph wh H

g h  Gh ph wh  X h

 0, T

where X h is a space of FE. Then:

u  V    , g h  Gh a  u, ph wh   f  ph wh     u ,  ph wh  H    u  Bh u,  ph wh  H

T

  u  Bh u

T

HT

   u  g h ,  ph wh  H

 ph wh



(6.14)

T

HT

Here, the operator  is continuous and it maps the space V into the space HT . Therefore, there exists such a constant C that v  V

and then

v

HT

C v

V

,

ph wh  X h  V

 ph wh

HT

 C ph wh

V

.

(6.15)

The inequality (6.13) follows from the inequalities (6.14) and (6.15). ‫ٱ‬

The external error is caused by two reasons:  by the discontinuity of the approximating functions from the space of finite elements on the inter-element boundaries  by approximate fulfillment of the essential boundary conditions of the boundary value problem

From the Theorem 6.2 it follows that, when the approximations are built by the necessary and sufficient sign, then the external error depends on the smoothness of the element  u  H T and on the approximating qualities of the spaces Gh of boundary functions.

Let us consider the approximations for which the spaces Gh are the spaces of surface elements introduced in the Section 5. There were formulated two approaches to define the layers of the surface elements: 

in the first one it is assumed that the faces K r of FE undergo the partition into the surface elements (see Theorems 5.6 and 5.11)



in the second one it is assumed that smooth patches K r , l of faces K r of FE undergo the partition into the surface elements (see Theorems 5.7 and 5.12)

By assumption, each patch K r , l is an infinitely differentiable manifold in R n of dimension (

n  1 ).

Let us the solution u of the variational problem (6.2) belongs to the space

H 2m  k    , m  0, k  0 . Boundary operator  defined by (6.11) and (6.12) maps the space H 2m  k    onto the product of spaces m

   K

Kr Kr ,l j  2 m 1

H 2 m  k  j 1 2  Kr , l  (6.16)

Then, independently on smoothness of the exact solution u of the variational problem, the functions:

u*j 



r ,l , j

u,

m  j  2m  1

K r , l

belong only to the space L2  Kr  even if their components are smooth enough:

 r , l , j u  H 2 m  k  j 1 2  K r , l  .

(6.17)

Functions u*j , m  j  2m  1 are approximated by the surface elements from the space Gh of the boundary functions. If faces K r undergo the partition into the surface elements, then due to the insufficient smoothness of the function u*j , the estimates (6.13) will not depend on the smoothness of the solution u of the variational problem. Different situation occurs when into the surface elements are partitioned smooth patches K r , l of faces. In this case the surface elements approximate each component (6.17) of the product (6.16) that allows obtaining the estimates of

the external errors depending on the smoothness of the exact solution u and on the type of the boundary functions.

Theorem 6.3 Let us the solution

u

of the variational problem (6.2) belongs to the space

H 2m  k    , m  0, k  0 . Let us assume that the external approximations are built with the spaces Gh of the boundary functions that meet the conditions of Theorem 5.7 (or Theorem 5.12). Then there exist such independent of the parameter h constants C j , 0  j  m  1 that

dh

V

!



m 1

C h j 0

j

s j 1

u j ,

(6.18)

where s j  k  j  1 , and where  j are the semi-norms (5.3).

◊ Proof - For the components of the operator  it holds the enclosure

 rK, l , 2m  j 1  L  H 2m  k  K  , H k  j  Kr , l   ,

0  j  m  1.

Let us s j  k  j  1 then

  L  H 2 m  k    , W0  , where the space W0 is defined by the characterization (5.4). Then inequality (6.18) follows from Theorem 5.7 (or Theorem 5.12). ‫ٱ‬

Theorem 6.4 If the approximations of the variational problem (6.2) are built on FE that satisfy the sufficient sign of the external approximations defined in Theorem 2.2, then the external error satisfies the inequality dh

V

 u

HT

sup

 ph wh   ph vh ph wh

wh Vh

HT

,

(6.19)

V

where functions ph vh are defined according to the assumptions of the Theorem 2.2, i.e.

phvh  Yh  H m   

(6.20)

◊ Proof - From the enclosure (6.20) and Lemma 1.1, one has

ph vh  Yh

  u,  p v  h h

HT

 0.

Then the required result follows from the inequality

a  u, ph wh   f  ph wh     u,  ph wh   ph vh H

 u

T

HT

 ph wh   ph vh

HT

.‫ٱ‬

From the inequality (6.19) it follows that, if the approximations are built by the sufficient sign, then the external error depends on the approximating quality of the spaces Yh  H m    of the conformed approximating functions. Similarly one can obtain the estimate of the external error for the approximations built by the sufficient sign established in the Theorem 2.3.

Theorem 6.5 Let us the conditions of the Theorem 6.1 hold. Let us the FE approximations of the space V are the converging external approximations of the space W . Then

lim u  phuh h0

V

0

where u and uh are the solutions of the variational problems (6.2) and (6.4) correspondingly.

◊ Proof - When h  0 , the internal and the external errors in the right-hand side of the inequality (6.7) tend to zero, since by the assumption the approximations are external and converging in the space W . ‫ٱ‬

Convergence of approximations in case when bilinear form is not V -elliptic Let us analyze the solvability and the convergence of the approximate solutions of a discrete variational problem in case when the solution of the continuous variational problem (6.2) exists and is unique, but the expansion a  u , v  of the bilinear form a  u, v  onto the space V  V does not satisfy the condition (6.6) of V -ellipticity.

Theorem 6.6 Let us a partition  of the domain  into FE K   be given and the following conditions hold: 

A finite-dimensional subspace Gˆ  Hˆ 0 (see characterization (2.5)) is generated by such boundary functions gˆ rK, j , t , 0  j  m  1 that for any face Kr  M  Kr  of any FE

K   the continuous on H m  K  functionals

ir  u  



K r

gˆ rK, j , t  rK, j ud  ,

1i  M

(6.21)

are linearly independent on the space Pm 1  K  of polynomials of  m  1 -order 

The enclosed subspace W  V is defined by the characterization W  u  V |  r  M   r 

 gˆ

1 r, j, t

 r1, j u1d   

r

 gˆ

2 r, j, t

 r2, j u 2 d  ;

r

K      , K r  M   K r   M  K r 



gˆ rK, j , t  rK, j u K d   0

(6.22)

K r

where   and M   Kr  are some not empty multitudes.

Then the mapping u W  u

is a norm equivalent to the norm u

V

W

    uK  K

2 m, K

  

1

2

(6.23)

.

◊ Proof - On each FE K   , let us define the projectors

rK : u  H m  K   rK u  Pm 1  K  of the space H m  K  onto the space Pm 1  K  by the formula  rK u 

where

p  r i

M

i 1

M

 u  p r i

i 1

r i

,

(6.24)

is some basis of the space Pm 1  K  . Here the symbol r identifies the face

Kr  M  Kr  of FE. The representations (6.24) exist because of the linear independence of the functionals (6.21) on the space Pm 1  K  . Let us

u W

u

W

 0.

Then on each FE

uK  u

K

 Pm 1  K  .

According to the characterization (6.22), for the projector that corresponds to the face

Kr  M   Kr  the following equality holds 1r  u

K1

  0,

which implies

u

K1

 0.

(6.25)

Let us consider the element K 2 that has some common with the element K 1 face K r2  K r1   r . From the equality (6.25) and the characterization (6.22) it follows that for this

face

 gˆ

r

1 r, j, t

 r1, j  u

K1

 d   gˆ

2 r, j, t

 r2, j  u

r

K2

 d

 0.

It means that, for FE K 2 , it can be pointed out such projector  2r that  2r  u

K2

  0,

and so

u

K2

 0.

Considering in a similar manner the rest of the finite elements of the partition, one comes to the conclusion that

K  

u

K

 0,

and so u  0 because each FE of the partition has at least one face that it shares with the adjacent FE. ‫ٱ‬

Theorem 6.7 Let us the assumptions of the Theorem 6.1 hold. Let us define the enclosed subspace W  V with the norm (6.23) in accordance with the requirements of the Theorem 6.6. Let us the space X h  W of FE approximates the space W . Let us the form a  u , v  be W -elliptic, i.e.

a v, v   C v

C  0 v  W

2 W

.

Then the variational problems (6.2) and (6.4) have the unique solutions u and uh correspondingly and the following inequality holds

u  ph uh

W

  C  inf u  ph vh  vh Vh 

W

 sup

wh Vh

a  u, ph wh   f  ph wh     ph wh W 

(6.26)

◊ Proof - Proof of the Theorem is similar to the proof of the Theorem 6.1. ‫ٱ‬

Theorem 6.8 Let us there be given a fixed partition  of the domain  . Let us the conditions of the Theorem 6.7 be satisfied. Let us the FE approximations of the space W be the converging external approximations of the space W . Then

lim u  phuh h0

V

0

(6.27)

where u and uh are the solutions of the variational problems (6.2) and (6.4) correspondingly.

◊ Proof - Since approximations of the space W are the external approximations of the space W , then the internal error (6.8) and the external error (6.9) tend to zero when h  0 . Then relation (6.27) follows from inequality (6.26) by taking into account the equivalence of the norms  W and  V on the space W as established in the Theorem 6.6. ‫ٱ‬

Theorem 6.9 Let us there be given a regular family  h of partitions of the domain  . For each partition from the family let us be satisfied the conditions of the Theorem 6.7. Let us FE approximations of the spaces Wh , defined by the characterization (6.22), be the converging external approximations of the space W . Then

lim u  phuh h0

Wh

0

(6.28)

where u and uh are the solutions of the variational problems (6.2) and (6.4) correspondingly.

◊ Proof – The relation (6.28) follows from the inequality (6.26) taking into consideration the tendency towards zero of the internal and external errors of approximations. ‫ٱ‬

The theorems show that even in the case when the bilinear form a  u , v  is not V -elliptic, it is possible to define some minimal set of boundary degrees of freedom generated by the space Gˆ

which provides the unique solvability of the discrete variational problem at any partition of the domain. If the approximations are built on a fixed partition of the domain, and if for any h , spaces Gh of the boundary functions contain the minimal subspace Gˆ defined in the Theorem 6.6, then takes place the convergence of the approximate solutions to the exact one, not only in the norm  W , that is a semi-norm on the space V , but in the norm  V as well. In general, for approximations built on a family of partitions of the domain, the convergence can be established only in the norms  W . h

7. Examples of finite elements Justification of some classic non-conformed finite elements Many of classic non-conformed finite elements have been justified in works [9, 15 - 17, 22, 42 46]. Usually such justification is performed for a certain type of boundary value problem and the features of the differential equations of the problem are essentially used in the justification. Below, the theorems from the Section 2 are applied to prove that all of the examined FE generate converging external approximations of the corresponding spaces H m    , m  1, 2 .

Spaces of the approximating functions of each of the elements under consideration contain the spaces of the polynomials of m -degree. This feature provides the convergence of the corresponding external approximations. Proofs of the convergence are not given here due to their simplicity.

Wilson's element

Wilson's element is used for the approximation of boundary value problems of the second order, whose generalized solutions belong to the space H 1    . The element is a parallelepiped in R n (Fig. 7.1.a.b) with vertices:

ai ,

1  i  4  n  1 .

The space PK of the approximating functions of FE can be defined by two equivalent ways:

 n  P K  P2  K   L   xi   i 1 

P  Q1  K   L  x , 1  i  n K

(7.1)

2 i

where: 

L ... is the space generated by the functions listed in the brackets



P2  K  is the space of polynomials of the second degree



Q1  K  is the space of polynomials of degree, less or equal to one, on each of n variables, i.e. p  Q1  K 

p 

 

i 1 1 i  n

b1 ,... n x11 x22 ...xn n .

The multitude of degrees of freedom of FE is defined as follows

K

 p  ai  , 1  i  4  n  1      2 p .   x 2 dx, 1  i  n  K i 

Let us define a regular family  h of partitions of the domain  into FE. Let us introduce the space X h of FE, whose functions ph vh  X h are defined by their values in the vertices (nodes) of the partition (here ph is an extension operator (1.4)) and by their average values, as follows

 2  ph vh |K   xi2 dx, 1  i  n, K   h K

where ph vh |K  P K .

The functions from the space X h are discontinuous on the inter-element boundaries and are continuous in the vertices of the partition. Therefore X h  H 1    .

Theorem 7.1 Wilson's element generates the external approximations of the space H 1    .

◊ Proof – Let us consider an auxiliary finite element for which

P K  Q1  K 

(7.2)

K   p  ai  , 1  i  4  n  1 (7.3) Let us Yh  C 0 be the space of the conformed FE defined by the relations (7.2) and (7.3), so

Yh  H 1    .

(7.4)

According to (7.1), the space Yh is a conformed subspace of the space X h .

For any function phuh  X h , let us designate through ph vh the unique function from the space Yh that takes the same values as ph uh in the vertices of the partition  h . Let us estimate the

magnitude of

 phuh   ph vh

HT

,

where space HT is defined by (2.2);  is the trace operator (1.7).

One can write

 phuh   ph vh 

    K

K r K r , l

K r , l ,0

p K   rK, l ,0 p*K  ,

where p K  ph uh |K , p*K  ph vh |K , and where  rK, l ,0 p is a restriction of the function p  P K onto the face K r  K r , l of FE. Due to the construction of the function ph vh , the function

 rK, l ,0 p*K  P1  K r , l  is a linear interpolant (when n  2 ), and a bilinear interpolant (when n  3 ) of the function  rK, l ,0 p K  P2  K r , l  . Then, according to the Theorem 3.1.4 from [10], there exists such independent of the parameter h constant C that

 rK, l ,0 p K   rK, l ,0 p*K

0, Kr ,l

 Ch2  rK, l ,0 p K

2, Kr ,l

.

Therefore

lim  phuh   ph vh h 0

HT

0

(7.5)

The inclusion (7.4) and the property (7.5) make the conditions of the Theorem 2.2 fulfilled. ‫ٱ‬

Family of Crouzeix-Raviart ‘s elements Crouzeix-Raviart’s elements are intended to approximate generalized solutions of boundary value problems of the second order. The domain K  R 2 is a simplex with the vertices ai ,1  i  3 ; the space of the approximating functions is chosen in such a way that p  P K

p |Kr ,l  Pk  K r , l 

(7.6)

where K r , l  K r is a face of FE.

The multitude of degrees of freedom of the element is

K   p  bi  , 1  i  3k  , where bi  K r , l , 1  i  3k are the points of Gauss’s integration quadrature of k -order (Fig. 7.1.c).

The space X h of Crouzeix-Raviart’s elements is not a subspace of the space H 1    .

Theorem 7.2 Finite elements from Crouzeix-Raviart’s family generate the external approximations of the space H 1    .

◊ Proof - Let us consider the adjacent FE K 1 and K 2 . In nodes bi ,1  i  k of Gauss's quadrature on the common face Kr1, l  Kr2, l  r , l of adjacent FE, the approximating functions from the space X h are continuous, i.e.

p1  P1 , p2  P2

p1  bi   p2  bi 

1  i  k.

Since Gauss's quadrature of k -order is exact for polynomials of degree 2k  1 , and since the inclusions (7.6) hold, then the following relations hold too

p1  P1 , p 2  P 2 , g  Pk 1  K r , l 





g p1

r ,l

 d   g  p  d

(7.7)

2

K r1, l

Kr2,l

r ,l

Let us associate with the family of partitions  h the family of spaces Hˆ 0h defined by (2.3), and let us define the family of spaces of boundary functions: Gh 

  G

K r,l, 0

K

 Hˆ 0h ,

K r K r , l

where GrK, l ,0  Pk 1  K r , l  . According to the Theorem 5.12, the family Gh is dense in the space Hˆ 0 where the space Hˆ 0 is the limit space for the family Hˆ 0h .

For the adjacent FE K 1 and K 2 , functions from the spaces GrK, l ,0 satisfy the relation

g1r ,l ,0   gr2, l ,0 ,

gri , l ,0  Gri , l ,0 , i  1, 2 .

Using the designations introduced, the relation (7.7) can be rewritten as:



g1r , l ,0 r1, l ,0 p1d  

Kr1, l



gr2, l ,0 r2, l ,0 p 2 d   0 ,

Kr2,l

where  rK,l ,0 are the components of the trace operator (1.7).

Therefore, the following relation holds g h  Gh , ph uh  X h

where  is the trace operator (1.7).

 gh ,  ph uh H

 0, T

Therefore, the conditions of the Theorem 2.1 are met and the approximations built on the elements of Crouzeix-Raviart are the external approximations of the space H 1    . ‫ٱ‬

Let us note that according to the Theorem 6.3, for sufficiently smooth solutions of variational problems, the external error of approximation has order O  h k  .

Morley's element Morley's element is used to approximate the solutions of boundary value problems of fourth order. The domain K  R2 is a triangle with the vertices ai ,1  i  3 . The space of the approximating functions of FE is:

PK  P2  K  .

(7.8)

The multitude of degrees of freedom:

K

 p  ai  , 1  i  3    p  aij  , 1  i, j    n

   3 

where p  aij  n

are the derivatives along the normal in the middles of the sides of the triangle (Fig. 7.1.d).

(7.9)

Let us define a regular family  h of triangulations of the domain  . The approximating functions from space X h of Morley's FE are continuous in the vertices of triangles K   h . Derivatives along normal to the FE boundary are continuous in the middles of the sides of triangles only. Therefore, one has the inclusion X h  L2    instead of required X h  H 2    .

Theorem 7.3 Morley's element generates the external approximations of the space H 2    .

◊ Proof - Let us define a triangular FE K  R2 for which:

PK  P1  K  K   p  ai  , 1  i  3 .

(7.10) (7.11)

The corresponding space of FE is designated through Yh0 . The element defined by (7.10) and (7.11) is conformed, i.e. inclusion Yh0  C 0 holds, so

Yh0  H 1   

(7.12)

Due to the characterizations (7.8) and (7.9) the inclusion Yh0  X h holds as well.

For each function phuh  X h , through ph vh let us designate a function from the space Yh0 that has the same values as ph uh in the vertices of the triangulation  h . Let us introduce the designations

p  ph uh |K ,

p*  ph vh |K .

On each face K r , l of FE K   h , the function  rK, l ,0 p* is a linear interpolant of the function

 rK, l , 0 p , so  rK, l ,0 p   rK, l ,0 p*

0, Kr ,l

 Ch2  rK, l ,0 p

2, Kr ,l

and, therefore

lim  0 ph uh   0 ph vh

ph uh  X h

h0

HT

 0,

(7.13)

where  0 is a trace operator of zero order for the partition of the domain. It is defined by the expression (2.16).

Let us consider the adjacent FE K 1 and K 2 . The derivatives along normal to the common side of the adjacent FE are continuous in the middles of the sides, i.e.

  ph uh |K1 

ph uh  X h

n

a   ij

  ph uh |K 2 n



a . ij

By designating through r ,l  Kr1,l  Kr2,l this common side, one can write:

ph uh  X h

 r1, l ,1 p1   r2, l ,1 p2 ,

(7.14)

where  ri , l ,1 is the operator of differentiation of the first order along outward normal to FE boundary; pi  ph uh |K i , i  1, 2 .

The middle of the common side of the adjacent FE is a point of the Gauss's integration quadrature of the first order that is exact for polynomials of the first degree. Since

 rK, l ,1 p  P1  K r , l  ,

p  P K  P2  K 

then from the relation (7.14) it follows that



 r1, l ,1 p1d  

Kr1, l



 r2, l ,1 p 2 d .

(7.15)

Kr2, l

Let us define the family of spaces of the boundary functions (see characterization (2.18))

Gh1 

1

   K

1j

 GrK, l , j  Hˆ 0,1 h ,

(7.16)

Kr Kr , l j  0

where GrK, l ,1  P0  K r , l  .  According to the Theorem 5.12, the family Gh1 is dense in the space Hˆ 0,1 that is the limit space  for Hˆ 0,1  h  when h  0 .

For the adjacent FE, due to the inclusion (7.16), the following equality holds:

g1r ,l ,1  gr2,l ,1,

gri ,l ,1  Gri ,l ,1,

i  1, 2 .

Then (7.15) can be rewritten as follows



g1r , l ,1 r1, l ,1 p1d 

Kr1, l



gr2, l ,1 r2, l ,1 p 2 d  0 .

(7.17)

Kr2, l

and therefore

g1h  Gh1 , ph uh  X h

 g , 1 h

1

ph uh 

HT

 0,

(7.18)

where  1 is the trace operator of the first order for the partition of the domain. The operator is defined by the expression (2.16).

From the inclusion (7.12), from the relations (7.13) and (7.18) and from the Theorem 2.3 follows that the approximations built on Morley's FE are the external approximations of the space

H 2   . ‫ٱ‬

Fraeijs de Veubeke’s element Fraeijs de Veubeke’s element is used to approximate generalized solutions of the boundary value problems of the fourth order. The domain K  R 2 is a triangle and the space of the approximating functions is:

PK   p  P3  K  |

f  p   0 ,

where f  p  is some functional whose expression is not necessary here. The multitude of degrees of freedom of FE is as follows

 K   p  ai  , 1  i  3;

  1 p p  aij  , 1  i  j  3;  d  ,    meas  K r , l   K n   r ,l

where: 

ai , aij are the vertices and the middles of sides of the triangle correspondingly



p n is the derivative along normal to the side K r , l  K r of the triangle (Fig. 7.1.d)

Let us  h be a regular family of partitions of the domain  into FE K   h . The approximating functions from the space X h of Fraeijs de Veubeke’s FE are continuous in the vertices and the middles of sides of the triangles K   h only, i.e. the inclusion X h  L2    holds.

Theorem 7.4 Fraeijs de Veubeke’s element generates the external approximations of the space H 2    .

◊ Proof - Proof of the Theorem is similar to the proof of the Theorem 7.3. The only differences are that the conformed space Yh0 has to be built on the quadratic triangular FE defined as follows:

P K  P2  K  ,  K   p  ai  , 1  i  3;

p  aij  , 1  i  j  3

and that the relation (7.17) immediately follows from the definition of the degrees of freedom of Fraeijs de Veubeke’s FE. ‫ٱ‬

Elements for external approximations In the classic Finite Element Method, strict requirements are imposed on the element shapes, on the location of the element nodes, and on the approximating functions of the element. This is caused by the necessity to provide sufficient smoothness of the approximating functions from the space of FE on the inter-element boundaries. The only requirement to the geometry of FE for external approximations is that it must have sectionally smooth regular boundary. Such FE can be of arbitrary shape. Boundary degrees of freedom of FE are generated by boundary functions and they are defined independently on the types of approximating functions that represent field variables inside FE. This property is reflected in the definitions of P -multi-solvency of the multitude of boundary degrees of freedom of FE (see Section 3.1). Internal degrees of freedom of FE are defined automatically after boundary degrees of freedom and a space of approximating functions of the FE has been specified.

In this Section are given some examples of FE for the external approximations of 2D problems, axis-symmetrical problems, and 3D problems. Also, the ways of parameterization of boundaries of FE and the approaches to the definition of boundary degrees of freedom are discussed. Herewith it is assumed that spaces of approximating functions of FE are complete spaces of polynomials of k -degree, i.e. PK  Pk  K  . With such choice of spaces PK any function p  P K can be expressed as follows p

k

b x ,  

xK,

0

where   1 , ...,  n  is a multi-index; b are some coefficients. The number k is called the order of internal approximation on FE. Polynomial approximations are convenient in computations and, as at will be demonstrated in Section 8, they provide high accuracy of solutions of boundary value problems.

It is assumed also that a face of the element consists of one smooth patch only, so K r  K r , l . Preference for such approach is explained in Section 6.

2D elements for boundary value problems of second and fourth orders

A smooth patch K r , l of the 2D FE boundary is defined by the mapping

 : D  R  Kr ,l  R2 ,

(7.19)

where D is the domain of determination of the parameter t . Let us assume that the mapping (7.19) is a polynomial of l -degree. When l  1 , the patches K r , l are the curves of second, third, and higher orders. In general, the parametric equations of a curve of l -order can be written in the form

xi 

l 1

 f  t  x  , j 1

j

j

i

1  i  2,

(7.20)

where xi  , 1  i  2, 1  j  l  1 are the coordinates of the nodal points of the curves; j

f j  t  , 1  j  l  1 are some polynomial functions.

Below there are the expressions for the functions when l  3, t  D   1,1 (see Fig. 7.2): 

l  1 (mapping 1 ): f1  1 2  1  t  f 2  1 2  1  t 



l  2 (mapping  2 ): f1  1 2  t  t  1 f 2  1 2  t 1  t  f3  1  t  1  t 



l  3 (mapping  3 ): f1  1 16  1  t   10  9  t 2  1  

f 2   1 16  1  t   10  9  t 2  1   f 3   9 16  1  t 2  1  3t 

f 4   9 16  1  t 2  1  3t 

From practical point of view the expressions above are sufficient. More detailed description of the boundary of the domain is best done through increase the number of smooth patches that approximate the boundary instead of through the increase of the order l of the curves.

Parametric equations of smooth patches of FE boundary are completely defined by the coordinates of their nodal points. Let us recall, however, that in the suggested method, the degrees of freedom are not associated with the nodal points. Nodal points, in this case, are only necessary for defining the shape of the element.

Let us consider how to construct the spaces G K of the boundary functions of the element. Solutions of variational equations, which correspond to boundary value problems of 2m -order, belong to the space H m    . Herewith the order of normal derivatives that appear in the definition of boundary degrees of freedom of FE changes in 0  j  m  1 , and on the boundary of FE it is necessary to define m layers of surface elements.

Let us  r , l , j , 0  j  m  1 be partitions of smooth patches K r , l of FE boundary into the surface elements K r , l , j   r , l , j , 0  j  m  1 . For different values of the index j the identical partitions can be used, i.e.  r , l , j   r , l , i , 0  i  m  1, 0  j  m  1 . On each K r , l , j it is necessary to define spaces PrK,l , j of approximating functions. According to the Theorems 5.7 and 5.12, the sufficient condition of convergence of the approximations is the following inclusion P0  K r , l , j   PrK, l , j , 0  j  m  1 .

Here and below Pi    designates the space of polynomials of i -order defined on the corresponding domain. If the following inclusion holds Ps j  K r , l , j   PrK, l , j , 0  j  m  1

then, as it is shown in the Section 6, the external error of approximation of the variational equation with sufficiently smooth solution has an order O  h s  , where s  min  s j  1 , 0  j  m  1 . j

In the benchmark problems of the Section 8, the following space of polynomials of s j -order is used as the space of approximating functions of the surface elements PrK, l , j  Ps j  K r , l , j  , 0  j  m  1 .

The number s j is called the order of the boundary approximation of j -layer of the surface elements. Herewith the space G K of the boundary approximating functions of the element is defined as follows m 1

G

GK 

K r,l, j

,

(7.21)

K r K r , l j  0

where GrK, l , j 

P

K r ,l , j

 L2  K r , l 

Kr ,l , j

is the space of surface elements of j -layer.

Let us also point out one important class of the spaces G K characterized by that the patches K r , l are partitioned into one surface element, i.e.

K r , l , j  D, card   r , l , j   1, 0  j  m  1 .

(7.22)

Herewith:

GrK, l , j  Ps j  D  , 0  j  m  1 .

Basic functions of the space G K generate boundary degrees of freedom of FE. According to the representation (7.21), the space G K is the union of the spaces:

Ps j  K r , l , j   Ps j  t  , t  Kr , l , j , 0  j  m  1 The simplest system of the basic functions of the space Ps j  t  is the system of monomials

t  i

sj

i 0

. When the relations (7.22) hold, the functions f j ,1  i  l  1 from (7.20), that describe

the shape of FE, can be used as the basic functions of the spaces Ps j  t  as well.

Let us:

 g  t  i j

sj

i 0

be some basis of the space Ps j  t  . Then the corresponding boundary degrees of freedom are defined as follows p  P K

 ij  p  



g ij  t   rK, l , j pd  ,

Kr ,l , j

0  j  m  1, 0  i  s j

When j  0 , the function  rK, l ,0 p is the restriction of the function p onto the patch K r , l :

 rK, l ,0 p  p K . r ,l

When j  1 , then

(7.23)

 rK, l ,1 p 

p p n1  n2 , x1 x2

where n1 , n2 are cosines of the outward normal to the boundary of FE. The surface measure is:

  dx1 2  dx2 2  d         dt     dt 

1

2

dt ,

where x1  t  , x2  t  are the parametric equations of the patch K r , l .

3D elements

Let us consider some examples of the polynomial mapping

 : D  R2  Kr ,l  R3 , that can be used to define a smooth patch of the boundary of 3D element.

In Fig. 7.3 are shown the patches of faces of FE whose parametric equations have the form

xi 

N

 f  t  x  , j 1

j

j

i

where t   t1 , t2  ; f j  t  are some polynomials; xi

j

1  i  3,

(7.24)

are the coordinates of the nodal points of

the patches K r , l . Here are the expressions for the functions f j  t  ,1  j  N for the patches shown in Fig. 7.3: 

K r , l is a flat triangle, N  3 (mapping 1 ):

f1  t1 f 2  t2 f3  1  t1  t2 

K r , l is a triangle of second order, described by the complete polynomial of second

degree, N  6 (mapping  2 ):

f1  t1  2t1  1 f 2  t2  2t2  1 f3  1  t1  t2  1  2t1  2t2  f 4  4t1t2

f5  4t2 1  t1  t2  f 6  4t1 1  t1  t2  

K r , l is a flat quadrilateral, N  4 (mapping  3 ):

f1  1 4  1  t1  1  t2  f 2  1 4  1  t1  1  t2  f3  1 4  1  t1  1  t2  f1  1 4  1  t1  1  t2  

K r , l is a curvilinear surface of third order, N  8 (mapping  4 ):

f1  1 4  1  t1  1  t2   t1  t2  1 f 2  1 4  1  t1  1  t2   t1  t2  1 f3  1 4  1  t1  1  t2   t1  t2  1 f 4  1 4  1  t1  1  t2   t1  t2  1 f5  1 2  1  t12  1  t2  f 6  1 2  1  t22  1  t1  f 7  1 2  1  t12  1  t2  f8  1 2  1  t22  1  t1 

Domains D of the parameters of maps shown on Fig. 7.3 have simple shapes and that simplifies their partition into the surface elements K r , l , j   r , l , j , 0  j  m  1 .

Let us assume K r , l , j  D, 0  j  m  1 i.e. let us consider the whole domain D to be one surface element. In this case the system

t  , 

  sj ,

t   t1 , t2  ,    1 ,  2  can be chosen as the system of the basic functions of the surface element K r , l , j . The system of functions f j  t  , 1  j  N from the parameterization (7.24) also can be used for the same purpose.

Boundary degrees of freedom are defined, as in case of 2D FE, by (7.23). Herewith:

d    g12  g 22  g32  2 dt 1

where:

dt  dt1dt2 g1 

x2 x3 x2 x3  t1 t2 t2 t1

g2 

x3 x1 x1 x3  t1 t2 t1 t2

g3 

x1 x2 x2 x1  t1 t2 t1 t2

Elements for approximations of spaces of vector functions

Let us assume that the solution of some boundary value problem is a vector function of variables x   xi i 1    R n , i.e. n

U  x    ui  x  i 1 , n

(7.25)

and let us assume that each component ui ,1  i  n of the function belongs to the space

H m    . Then the variational equation is defined on the space V  V , where V   H m     is n

a space of vector functions. For example, when solving the problems of Theory of Elasticity in





displacements, the function (7.25) is a displacement vector and V  H 1    . n

In computations, each component of the space V is approximated independently of the others by using the FE described above. Therefore, for each FE are constructed the independent representations of the components uiN 

k

b x ,   i

1  i  n.

0

There can be another choice of boundary degrees of freedom of FE different from that defined by (7.23). It can be caused by the following reasons. Often the boundary conditions are imposed not onto the components ui ,1  i  n of vector in global coordinate system, but onto the components ui ,1  i  n  1 of the vector in natural coordinate system, that has coordinate axes normal and tangent to the boundary of the domain. For example, when solving the problems of Structure Mechanics, one may need to fulfill the boundary condition when the normal to the boundary component of the displacement vector equals to zero: un  0 and the tangent

displacements ui can be arbitrary (slippery support). Similar situation occurs when it is necessary to take into account the cyclic symmetry of the structure. Besides, if the approximations of vector functions on adjacent FE are constructed in different coordinate systems, then when forming the system of equations it is necessary to bring the local boundary degrees of freedom of FE to the global ones.

One of the possible ways of effective solution of the above mentioned problem is the construction of vector approximations on FE. Here the displacement vector of FE is represented as follows UN 

N

 b i 1

i

i

,

where  i ,1  i  N are some vector approximating functions.

Let us, for example, n  2 . Then the simplest choice of representation (7.26) is UN 

 1i x  i b      x  i 1   0  2i  2

k

where  0, i  j 1, i  j

 ij  

Therefore, the space PK of approximating functions is a vector space P K   Pi K  ,

1 i  n

(7.26)

and, as the boundary degrees of freedom, it can be taken the integrals from the normal and tangent displacement components.

When n  2 , taking into account the relations (7.23), the boundary degrees of freedom can be written as follows

p   p1 , p2   P K   P1K , P2K

 0i n  p  



g 0i  t     rK, l , 0 p1  n1    rK, l , 0 p2  n2  d 



g 0i  t     rK, l , 0 p2  n1    rK, l , 0 p1  n2  d 

K r , l ,0

 0i   p  



K r , l ,0

(7.27)

where n1 , n2 are the cosines of the outward normal to the boundary of FE. Elements with the boundary degrees of freedom of type (7.27) are referred to as vector FE. In the Section 8 the vector FE are used when solving the 2D and the axisymmetrical problems of the Theory of Elasticity.

Axisymmetrical elements Let us assume that the domain   R 3 is a body of revolution. In this case, taking into account the axial symmetry, the geometry of the domain can be described by its axial cut in the cylindrical coordinate system. If the boundary conditions of the boundary value problem and the right-hand side of the corresponding differential equation are such that the solution of the problem does not depend on the angle coordinate, then the problem is called axisymmetrical. From the mathematical point of view, axisymmetric problems are 2D problems because the

sought functions depend only on the coordinates x1 , x2 . Herewith the analysis domain  is a half of the axial cut of a body and is located in the semi-plane x1  0 .

In axisymmetrical problems of Theory of Elasticity the displacement vector U   u1 , u2  has the radial component u1 and the axial component u2 . On the axis of symmetry the radial component of displacements turns into zero, therefore the vector U belongs to the product of spaces V1  V2 where:

V1  u1  H 1    |

u1  0 when x1  0

V2  H 1   

(7.28)

Let us partition the domain into FE. The representation of the displacement vector of FE can be in the form (7.26) and the boundary degrees of freedom in the form (7.27). Herewith in the relations (7.27) the surface measure is:

  dx1 2  dx2 2  d   2 x1        dt     dt 

1

2

dt

where x1  t  , x2  t  are the parametric equations of the patch K r , l .

Space of the approximating functions of FE can be of two kinds: 

If FE does not touch the axis of symmetry OX 2 , then the space is:

P K   Pk  K  

2



If FE touches the axis of symmetry then, due to the characterization (7.28), the approximant of the radial component of the displacement vector has to satisfy the condition: u1N  0

when

x1  0 .

(7.29)

The condition (7.29) holds if it assumed that u1N  x1 Pk ( K ) . Therefore, for the FE touching the axis of symmetry OX 2 :

PK  Pk  K    x1 Pk (K )  .

Finite elements for stress concentration In classic FEM, analysis of stress concentration around holes, notches, and other small details of structures, requires significant mesh refinement in domains with high stress gradients. Herewith, the dimension of the discrete problem increase, and the conditioning of the equation system worsens. Here, the approach developed in the previous sections is used to construct special conformed by subspace FE with displacement and stress approximation in complex variables. The FE is intended to simulate stress concentration domains. Unlike some other approaches [53 55], that also use local asymptotic representations of solutions of boundary value problems, in external approximations it is not necessary to construct any particular functionals that take into account the discontinuity of approximating functions on inter-element boundaries. Completeness of the system of complex basic functions in the domain of special FE and the fulfillment of the general criteria of the convergence of the external approximations provide the convergence of approximate solutions to the exact. This is supported by the results of the numerical experiments described in Section 8.

Let us consider a polygonal FE meant for the simulation of displacement and stress field in the vicinity of an elliptic hole or a notch. The FE domain can be either double connected, when the hole is located completely within the FE (Fig. 7.4.a), or single-connected when a part of the contour of the element is an elliptical arch (Fig. 7.4.b). Lengths of the semi-axis of the ellipse can be arbitrary, which in combination with the possibility of changing the FE connectivity, allows to handle a rather large class of stress concentration problems.

Let us use the methods of the theory of functions of complex variable z  x1  ix2 to obtain the local asymptotic representation of displacements and stresses that exactly satisfies boundary conditions on the contour of the hole as well as the equilibrium equations. Using two analytical functions,   z  and X  z  (complex potentials), displacements and stresses in the domain of special FE can be represented as follows

2G  u1  iu2   C1  z   z  z   X   z     11   22  4 Re    z      11   22  2i 12  2  z   z   X   z   

(7.30)

where G  E  2  2  ; E is modulus of elasticity; is Poisson's ratio; C1   3    1    for plane stresses; C1  3  4 for plane strains;    denotes differentiation on z ;    denotes complex conjugate.

Any FE with an elliptical hole can be obtained from some initial FE with a round hole by the conformal mapping

z   

  C 

 m 1  ,

where:

    i ; C 

ab ab ; m 2 ab

The reverse mapping:

 

z   z 2  4C 2 m 

1

2

2C

maps the element with an elliptical hole onto the element with a round hole (see Fig. 7.4.a). Conformal mapping allows to construct the approximation on the complex plane  instead of z and, therefore, the problem of boundary conditions fulfillment at the contour of the elliptic hole is replaced by a simpler problem of doing the same thing for the unit round hole.

Let us substitute the variable in the equations (7.30):

  z          1    X  z   X       X 1    Then:

1       

     X1    Xz        1          1             z   3         X 1          X 1          X   z    3          z  

(7.31)

Substituting expressions (7.31) into (7.30) and leaving out the index 1 one obtains:

2G  u1  iu2   C1       

 11   22  11   22





  

  X              

       4 Re           i 2 12 

 2                          

      

 X           X        

3

            

(7.32)

Any function     , analytical in some circular ring 1    R , can be uniquely represented in this ring by Laurent expansion 

     a0     j  j   j

j

j 1

,

where  j  1 j  i 2 j are complex coefficients. Let us approximate analytical functions     and X     by the finite segments of Laurent expansion

 N    0  X N      0 

   N

j 1

N 2

  j 1

j

j

j

j

  j

j

 

     j j   j 1 N

(7.33)

Let us name the number N here as the order of internal approximation on complex FE. Unlike the work [56], in (7.33) are used the segments of complete Laurent expansion and all the members, up to the particular degree, remain.

Let us assume the contour of the hole to be free from the load. Then at   ei   the following condition must hold:

   

      

  X     0 .        

(7.34)

Substituting the equation (7.33) into (7.34) and reducing at the same degrees of  , the equations for the complex coefficients are obtained:

 0  C   0  m 2  2 2 

 1  C  m1   1   1  m1   2  C  m 0   2  2m 2   j  C  m j  2   j   j  2   j  2 3 j N 2

 j  C  m j  2   j   j  2   j  2  N 1

1 j  N   N  2   N 1   N  2  0

       mj j  ,      mj j  ,     

(7.35)

Making the reverse substitution of equations (7.35) into (7.33) it is obtained the following representations of the complex potentials:

  j 1  N  X N       j C  j  j  2  mj  j    j 1   N    j C  m j  2   j    j 0  N     j C   j j  2  mj j    j 1   N     j C  m  j  2    j   j 1

 N    0 

   N

j

j

  j

j



(7.36)

The complex coefficients in (7.36) are:

 j   j1  i j 2  j   j1  i j 2 where  j1 ,  j 2 ,  j1 ,  j 2 are real numbers. Then relations (7.36) can be rewritten in the form:   j 1  j j    j1   j 2i   2 2   X N      01C  m  1   02iC 1  m    N     j1C  j  j  2  mj  j   j  m j  2    j 1   j 2 j j j 2   j 2iC  j  mj    m      j1C  m  j  2  j j  2  mj j    j       j 2iC    j  j j  2  mj j  m  j  2    

 N      01   02i 

  N

  j   j 2i  j 

j1

(7.37)

Let us denote the complex basic functions in (7.37) through  j   

and  j   

correspondingly, for 1  j  4N  2 , and let us denote the real factors of these functions through b j for 1  j  4N  2 . Let us introduce the vector potential:

     F       X      

Then, the approximant FN    of the vector potential F  

 N    FN        X     N  

 is:

4N 2

 b F   , j 1

j

(7.38)

j

where:

  j    Fj        j      are

vector

complex

basic

functions.

Basic

functions

 F   

4N 2

j

j 1

generate

the

finite-dimensional space  K of vector complex functions that approximate vector potential

F    on FE.

Substituting (7.38) into (7.32) and taking into account that    1  z  it is obtained the representations for the displacement vector of the element on the  -plane and on the z -plane:

  0j      U N   bj  0   j    j 1   0 1  u1N  z   4 N  2   j    z     UN  z    N    bj  0 1 u z  j  1   z      2   j 4N  2  ˆ j  z   4 N  2   bj     b jU j  z   ˆ j  z   j 1 j 1    u1N      N   u     2 

4N  2

(7.39)

where  0j    , 0j    , ˆ j   ,ˆ j   are some complex basic functions.

By the terminology of Section 3.1 the system U j  z  j 1

4N 2

is the initial system of functions

necessary to build basic functions of complex FE corresponding to its boundary and internal degrees of freedom. These functions generate the space PK of the approximating functions of FE. To construct the basic functions of FE it is necessary to determine the explicit form of functions U j  z  ,1  j  4N  2 . This is a rather difficult problem. However, there is a way to calculate stiffness matrix of FE that does not require explicit expressions for the basic functions of FE.

Due to one-to-one correspondence of the elements from spaces  K and PK , for each function

U j  z  ,1  j  4N  2 there is a function Fj    ,1  j  4N  2 whose explicit expression is known. Therefore, instead of finding the representation U N  z  for the displacement vector of the element through its boundary and internal degrees of freedom, it is sufficient to find such representation for the vector potential FN    . In accordance with the procedure described in Section 3.1, it is necessary first - to build the system (3.12) of equations for the factors

b j ,1  j  4 N  2 of the initial representation (7.38) of the vector potential, and second - to

construct the new representation of the potential by the formula (3.16).

Let us consider how the elements of the matrix of the system (3.12) are calculated in this case. Referring to the Section 7.2, boundary degrees of freedom of FE on z -plane are as follows:

 0i n 



g 0i  t   n1u1N  z  t    n2u2N  z  t    d 



g 0i  t   n1u2N  z  t    n2u1N  z  t    d 

K r , l ,0

 0i  

(7.40)

K r , l ,0

where: 

z  t   x1  t   ix2  t  is a parametric equation of the boundary patch



K r , l ,0 is the surface element from zero layer



g0i  t  is the basic function of the surface element



n1 , n2 are cosines of the outward normal to the boundary patch

From expressions (7.39) and (7.40) it follows that to calculate the matrix of the system (3.12) one needs to calculate the integrals: t2

i 0  g0  t   j  z  t   d  , t1

t2

 g  t   z  t   d , i 0

0 j

t1

that can be performed numerically (for example, by Gauss’s quadrature). To calculate values of the functions ˆ j  z  t   and ˆ j  z  t   in the nodes tr ,1  r  M of the quadrature, it is not necessary to know the explicit expressions for these functions. It can be done by the formula

ˆ j  z   iˆ j  z  

ˆ j   1   C1 j         2G  ˆ    

  ˆ      ˆ      j

that is evaluated in the points of  -plane:

 r   1  z  tr   .

Representations (7.30) for displacements and stresses satisfy the equilibrium equations in the domain of special FE when the mass forces are absent. In this case the corresponding expression for bilinear form in the variational equation can be transformed by Green's formula



2

  ij  u   ij  v  dx 

K i , j 1

2

    u  n v d

K i , j 1

ij

j i

where u   u1 , u2  , v   v1 , v2  . Therefore, when calculating the stiffness matrix, integration is performed along the contour and not on the domain of special FE. It reduces significantly the amount of calculations.

Figure legends for Section 7:

Fig. 7.1 Examples of non-conformed FE

Fig. 7.2 Examples of 2D faces defined by polynomial mappings

Fig. 7.3 Examples of 3D faces defined by polynomial mappings

Fig. 7.4 Stress concentration element on z - and  -plane: (a) element with elliptical hole (b) element with elliptical arch

8. Numerical experiments This Section presents the results of numerical experiments for various benchmark problems of Structure Mechanics. The accuracy of the suggested method of external FE approximations (MEFEA) is estimated by comparing the approximate solutions with the exact analytical solutions, as well as with numerical and experimental results obtained by other authors. The influence of various parameters of approximation on the accuracy is closely examined. It is demonstrated that results of the numerical experiments are completely in conformity with the theoretical estimates obtained in the previous sections.

All the experiments are performed using the FE described in Section 7.2. Orders k of the internal approximation are assumed to be different on various FE of a partition. Orders s j , 0  j  m  1 of the boundary approximation are assumed to be the same for all surface

elements of j -layer. Identical partitions of smooth patches of faces of FE into the surface elements are used for different layers, i.e.:

 r ,l , j   r ,l ,i ,

0  i  m  1,

0  j  m  1.

In the figures referring to the problems, boundaries of the surface elements are shown in dashes that are perpendicular to the contour of FE. Smooth patches of faces, on which the surface elements are not defined, are marked by dash line placed inside FE.

In Section 3 it was shown that there are two types of the basic functions from the space of conformed by subspace FE: 

Functions associated with boundary degrees of freedom of FE. The support of the function is a pair of adjacent FE



Functions associated with internal degrees of freedom of FE. The support of the function is a single FE

Because of this property, the internal degrees of freedom can be computed immediately after the stiffness matrix and right-hand side of FE has been built. So they are not included into the final equation system. In conventional FEM this procedure is called the “condensation” of degrees of freedom. In the tables below N is the dimension of final equation system assembled from the condensed stiffness matrices of FE, so in fact it is the number of boundary degrees of freedom in the system.

Torsion Formulation Torsion problem consists in finding shear stresses  23 and  13 in cross-section of a bar when it is in twist (Fig. 8.1). This problem can be formulated as a Dirichlet’s problem in two equivalent forms: 1) Find the function w  V    that meets the equations: w  f

 0w  0 where: 

 is Laplace’s operator

2  2 i 1 xi 2



 0 is the trace operator of zero order

 and

(8.1) (8.2)

V      w w  H 01    , w  L2   

2) Find the function w  H 01    that meets the variational equation 2

w v dx  i xi

  x

 i 1

 fvdx

v  H 01   

(8.3)



For torsion problems f  2 in the right-hand side of (8.1) and the function w is called a stress function. Shear stresses in the section of the bar are determined by the formulas

 13  G

w , x2

 23  G

w , x1

where  is a twist and G is a shear modulus of the material of the bar.

In the external approximation of the equation (8.3) space Gh of the boundary functions consists of only one layer of surface elements.

Model problem Let us consider a model problem when the domain  is a square 2  2 and f  1 . The exact solution of the given problem is known [47]. Due to the symmetry it is sufficient to analyze 1 8 of the square.

In the first experiment the convergence on a family of partitions is under consideration. In Fig. 8.2 are shown the partitions of the domain into triangular FE. For all the partitions and any FE,

the order of the internal approximation is k  1 and the order of the boundary approximation is

s  0 . Let us note that for the given problem the external error of approximation is caused not only by the discontinuity of the approximating functions at the inter-element boundaries, but also by approximate fulfillment of the essential boundary condition (8.2).

In Table 8.1 is given the comparison of the approximate and the exact solutions for the partitions shown in Fig. 8.2. Analyzing the data in Table 8.1 one can see that the convergence rate is not high and in general corresponds to the theoretically predicted order O  h  . When partitioning

1 8 of the square into 16 FE, the maximum deviation of the function w from zero on the boundary is approximately 6.6 % of its maximum value in the center of the square. For the alternative partitions 5 and 6 the results almost coincide. The results illustrate the convergence of

h -version of MEFEA when the mesh of FE is refined without changing the type of FE. Herewith, both the external and internal errors of approximation simultaneously decrease.

In the second experiment the convergence on a fixed partition of the domain is analyzed. Herewith, the problem is solved on the partition 2(see Fig. 8.2) for different orders s  0;1; 2 of the boundary approximation, while the order of the internal approximation for each FE of the partition is kept unchanged ( k  4 for all values s ). In this experiment the order of the internal error of approximation is fixed, but the external error is decreased through the refinement of the boundary approximation.

The results given in Table 8.2 show how approximating qualities of the spaces of the boundary functions affect the accuracy of the solution both within and on the boundary of the domain.

When s  2 , the maximum deviation of the function w from zero on the boundary is 1.8% of the function's value in the center of the square. At the transition from s  1 to s  2 , the error at the center of the square is decreased by 15.5 times while at the transition from s  1 to s  2 it is decreased by 1.6 times only. This indicates that when s  0 , due to the low order of the boundary approximation, the major contributor to the total error is the external error. When

s  1 both the external and the internal errors become commensurable so that further refinement of the boundary approximation does not lead to a valuable improvement of the solution inside the domain but it significantly decreases the error in the boundary condition.

Comparison of the Tables 8.1 and 8.2 shows that, for a given number of equations in the final system, the accuracy of MEFEA, for high orders of the internal and the boundary approximations, exceeds the accuracy when using the simplest FE with linear approximating functions.

Stress concentration at a semi-round notch The ratio of the diameter of the bar to the radius of the notch is 2R r  10 (see Fig. 8.3). Due to the symmetry one can consider only a half of the cross-section by assigning to the symmetry axis the following boundary condition w  0. n

(8.4)

The domain is simulated with two FE; the contour of the domain is approximated with inscribed broken line. The boundary between the elements is a line that is inscribed into the circumference

with radius 2r . Unlike the previous problem, the orders k of the internal approximation are varied, but the order of the boundary approximation s  0 is unchanged.

For the problem, only smooth patches of the inter-element boundary have boundary degrees of freedom that are not equal to zero. For the smooth patches that coincide with the approximated boundary of the domain, the values of the boundary degrees of freedom are equal to zero due to the homogeneous boundary condition (8.2). It is not necessary to create the boundary degrees of freedom on faces where natural boundary condition (8.4) is prescribed. These patches are marked in Fig. 8.3 with a dash line in accordance with the accepted rule. The number of the equations in the final system is equal to 5; it is defined by the number of the inter-element patches. The minimum order k of the internal approximation that provides P -multi-solvency of the boundary degrees of freedom of FE of the partition equals to 4.

In Table 8.3 are given the errors in shear stresses calculated by the formula

  0  100 % , 0 where  is the calculated shear stress and  0 is the exact value from [48].

Table 8.3 shows that at an increase of the orders of the internal approximation on both FE from

k  4 to k  6 , the error in stress in the area of its concentration (point A in Fig. 8.3) decreases from 57% to 2.7%. The refinement of the solution is achieved through the increase of the number of the internal degrees of freedom of FE, while the number of the boundary degrees of freedom remains unchanged. However, to get better solution there is no need for a simultaneous increase

of the order of the internal approximation on all FE of the partition. It is sufficient to do that only for those FE located in the area of the stress concentration. Thus, if the order of the internal approximation on the first FE is equal to 6, then the high accuracy is achieved independently of the order on the second FE (see Table 8.3). It happens because the high stress gradient is located in the domain of the first FE, and on the second FE the stresses field is smooth.

It is also necessary to note that the convergence in stresses is the convergence of the first derivatives of the approximated function w . A high accuracy of MEFEA is achieved despite a rather rough partition of the domain into FE, as well as the approximation of the boundary of domain and of the boundary conditions.

Thin plates bending Formulation A great attention is paid to the development of FE for the problem of thin plates bending. It is sufficient to mention a large catalogue [49] of FE used in various software systems (88 elements). The construction of FE for plate bending is more difficult because the differential equation that describes the bending has the fourth order. Bilinear form of the corresponding variational equation contains derivatives of the second order and therefore, the internal approximations of the problem require inter-element continuity not only of the approximating functions but of their first derivatives as well. This results in complex structures of nodal degrees

of freedom of FE, and in high orders of approximating polynomials even for the elements of the simplest form (see, for example the element of Irons in [10]).

The equation of bending of a thin plate of constant thickness is: 2w 

q D

(8.5)

where: 

 2 is a bi-harmonic operator

4 4 4   4 2 2 2  4 x1 x1 x2 x2 2



w is the function of deflection of the plate



q is the intensity of the normal load



D is the cylindrical stiffness of the plate:

D  EH 3 12 1   2   , here E is modulus of elasticity; is Poisson's ratio; H is the thickness of the plate

Boundary conditions can be of different types. In case when on the patch 0 of the plate boundary  are given the clamping conditions: w   0, 0

w n

0 ,

(8.6)

0

and on the patch 1   a lateral load of intensity Q is applied, then solution of the equation (8.5) is equivalent to finding the function w  W that satisfies the variational equation:

v  W

  2 w 2v 2 w 2v 2 w 2v   D  w  v  1   2   2 dx     2 2 2     x  x  x  x  x  x  x  x 1 2 1 2  1 2 2 1     

 qvdx   Qvd



1

where W  H 2    is a sub-space of functions that satisfy the conditions (8.6).

The bending moments and the torque are calculated correspondingly by the formulas:

M 11 M 22

 2 w 2 w   D  2   2  x2   x1  2 w 2 w   D  2   2  x1   x2

M 12   D 1   

2 w x1x2

For bending problems m  2 and it is necessary to define two layers of surface elements on FE boundary.

Bending of a cantilever plate Long cantilever plate is loaded on the edge with uniform lateral load intensity Q (Fig. 8.4.a). The input data are as follows: l  10 ; b  2 ; H  1 ; Q  1 ;elasticity modulus E  1 ;Poisson's ratio   0.3 . The dimensions of the plate allow to consider it like a beam, and so to take the solution of Beam Theory as a target solution to compare with.

The domain is partitioned into two FE (see Fig. 8.4.b), and the following approximation parameters are accepted:  the orders of the internal approximation on each FE: k  4  the orders of the boundary approximation: s j  1, 0  j  1 (i.e. the spaces of polynomials of the first order are the spaces of boundary functions for both layers of surface elements)

The results of MEFEA presented in Fig. 8.4.b practically coincide with the solution of the Beam Theory. It is expected, since the hypothesis of flat section in the Beam Theory is a special case of the hypothesis of straight normal in Kirchhoff’s Thin Plate Theory. Despite discontinuity of approximating functions there are no jumps of deflections and bending moments on the boundary between the elements.

Bending of a square plate It is interesting to know how the complexity of FE shape affects the accuracy. The necessity to use FE of a complicated shape can be caused by the geometry of the initial problem as well as by the need to change the geometry of the domain at iterative computations (for example, when non-linear problems are under consideration). In Section 8.1 it was demonstrated that the use of FE of complex shape provides a high accuracy. However, the problem of torsion is the boundary value problem of the second order. The main purpose of the following experiment is to study the influence of the FE shape on the accuracy of solution of boundary value problems of the fourth order.

Let us consider the following benchmark problem of bending of the square plate: dimensions 2  2 ;thickness H  1 ;loaded by the pressure of intensity q  1 ;modulus of elasticity E  1

;Poisson's ratio   0.3 . On the contour of the plate is assigned either hinge support condition

w   0 , or clamping conditions (8.6).

Due to the symmetry only 1 / 8 of the plate is simulated. The symmetry condition is that the derivative of the deflection function along the normal to the axis of symmetry being equal to zero. There are three partitions of the domain organized by the degree of their irregularity (Fig. 8.5): 

The first partition includes one triangular and two rectangular FE



The second partition is created by the distortion of the first partition through displacement of the common point of FE to the centre of the plate. In result, one FE of swallow-tail type is created



The third partition contains three FE of the swallow-tail type and one FE of irregular hexagon

For all the partitions and for any FE, the order of internal approximation is k  5 . The orders of the boundary approximation s j , 0  j  1 are varied assuming that s0  s1  s . When s  1 , for hexagonal FE of the third partition, the order of internal approximation is increased to k  6 in order to provide P -multi-solvency of boundary degrees of freedom of the FE.

Table 8.4 details the results of comparison of MEFEA solution with the exact solution from [50] for the clamped plate problem. One can see that the degree of irregularity of the partition insignificantly affects the accuracy. For all the partitions, when s  0 , the error in deflection at the center of the plate is within the range 30 – 60%; when s  1 the error decreases to 0.1 – 0.2%. So the order of the boundary approximation mainly affects the accuracy: increasing s from s  0 to s  1 , the error in deflections in the center of the plate decreases by two orders of magnitude.

Similar results take place in the case of hinge supports of plate edges (Table 8.5). The deviation from zero of the bending moment M 22 on the edge with the hinge is approximately 2.5% of the maximum value of the moments in the center of the plate.

The magnitude of the discontinuity in moments at inter-element boundaries significantly depends on the order of the boundary approximation. For s  1 it is 0.5 – 3% of the maximum moment. When s  1 , deflection w is calculated with high accuracy for both types of constraints. Its jump on the inter-element boundary is 0.1 – 0.5% of the maximum deflection. The degree of irregularity of the partition practically does not affect the discontinuity.

2D elasticity problems Formulation Equations that describe equilibrium inside an elastic body   R n have the form:

u  u       grad div u  f

(8.7)

where: 

u   ui i 1 is the displacement vector



f   fi i 1 is the volume force vector



  0,   0 are elasticity constants expressed through modulus of elasticity E and

n

n

Poisson's ratio  as follows:

  

E , 1    1  2 

 

E 2 1   

 is the differential operator of the Theory of Elasticity

Deformations and stresses are expressed through the components of the displacement vector as follows:

 ij  u  

u j 1  ui   2  x j xi

  , 





n

 ij  u    ji  u       kk  u    ij  2 ij  u  ,  k 1



1  i  n, 1  j  n

where: 1, i  j  0, i  j

 ij   is Kronecker’s delta.

Let us consider a boundary value problem with the following boundary conditions: u 0 1

(8.8)

 n     ij  u  n j   gi ,  j 1  2

1 i  n

(8.9)

where: 

1   is an open sub-domain of the boundary 



  1  2



n j , 1  j  n are cosines of the outward normal to the boundary

Let us define the following spaces: V   H1  Y   L2     V     v



W  v

n

n

v  V , v  Y 



v  V, v   0 1

The problem of finding the function u  V    that satisfies the relations (8.7) - (8.9) is equivalent to the following variational problem: find the function u  W such that

v  W

n

n

   ij  u  ij  v  dx   fi vi dx 

 ij  1

 i 1

n

  g v d

2 i  1

i i

,

(8.10)

where n

f  Y,

 1  g   H 2  2   .  

The external approximations of the equation (8.10) are constructed on vector FE described in the Section 7.2.3.

Extension of a square plate The load of intensity q is uniformly distributed along the edges of the plate (Fig. 8.6). The exact values of stresses are 11  12  0,  22  q . Taking into account the symmetry, only 1 4 of the plate is analyzed at different orders of the internal k and boundary s approximations. The exact solution of the problem is obtained for all combinations of k and s changing in the range

1  k  5, 1  s  3 . Herewith displacements and stresses are continuous at the inter-element boundaries. Therefore, in the example, the external and the internal errors of approximation are equal to zero for all the combinations of the approximation parameters.

Bending of a circular bar The input data: cross-section of the bar H  H  0.01  0.01 ;radius R  0.2 ; E  0.1 ;Poisson’s ratio  0 . One end of the bar is loaded with total force P  105 , and the other end is constrained (Fig. 8.7.a).

The curved contour of the bar is approximated by two inscribed broken lines with 16 straight sections each. Three partitions of the domain into FE are considered: 

the first partition contains 8 FE, each is a fragment of arch (Fig. 8.7.b)



the second and third partitions are constructed by dividing each FE of the first partition into 2 and 4 FE correspondingly

The order of the boundary approximation is fixed s  1 , and the order of the internal approximation is variable: k  2, 3 . The bandwidth of the equation system for all the partitions is M  7 .

MEFEA solution is compared to the solution of the Theory of Circular Bars in [50]. The results of the comparison are summarized in Table 8.6. One can see that, for the first partition at k  2 the error is significant (up to 80% in stress). It is expected, because stresses in the arched FE with 22.5o span (see Fig. 8.7.b) cannot be satisfactory simulated by the polynomial of the first order. Increasing the order of the internal approximation to k  3 , the error in both the displacements and stresses decreases to 10 – 15%. Decreasing the size of FE, the error decreases and for the third partition at k  2 it does not exceed 1%.

One can see that the accuracy is improved either through an increase in the number of FE (i.e. decrease of the internal error) or through an increase in the order of internal approximation. The bar is simulated by one layer of elements only. So the increase in the number of FE at fixed s does not reduce the external error since the size of the inter-element boundaries remains constant. The results indicate that for s  1 the external error, which is caused by the displacement discontinuity on the inter-element boundaries, is small. The main contribution into the error is made by the internal error. This is supported by the fact that displacements are practically continuous. The discontinuity in stresses does not exceed 0.1% of their average value at the discontinuity point.

Extension of a plate with a round hole

The exact solution of the problem of stress concentration around the hole in the infinite plate is known [48]. In computations, the infinite plate is replaced with a square plate with the ratio between the side length and the diameter of the hole a d  8 (Fig. 8.8).

If the classic FEM is applied to a stress concentration problem, then the width of the surrounding the hole layer of elements must be 0.05 – 0.1 of the radius of the hole [51, 52]. Let us use the rather rough partition of 1 4 of the plate (see Fig. 8.8). In this partition, the width of the layer of the adjacent to the hole FE is 1.25 of the radius of the hole. As it follows from the analytical solution of the problem, the area of stress concentration spreads approximately onto just a half of the width of the layer of FE adjacent to the hole.

At a fixed partition of the domain into FE, the number of equations of the system is defined only by the order s of the boundary approximation, and for s  0 the system contains 60 equations while for s  1 it contains 120 equations. Bandwidth of the system at s  0 is M  33 while at

s  1 the bandwidth is M  55 .

The results for the stress concentration factor are detailed in Table 8.7 for various combinations of the approximation parameters. Increasing the orders of the internal approximation for the adjacent to the hole FE without increasing the order of the boundary approximation leads to the increase in discontinuities of displacements on the inter-element boundaries (especially in the region, where the stress reaches its maximum). Because of this, the error in stress does not

decrease (see Table 8.7). Instead, at the transition from s  0 to s  1 the error significantly decreases to 1.4%.To increase the accuracy in stress at a fixed partition of the domain it is necessary to increase the orders of both internal and boundary approximations, which agrees with the theoretical results of Section 6.

Special elements in stress concentration This Section contains the results of numerical experiments with the special element developed in the Section 7.2.5 for the simulation of stress concentration regions.

Extension of a plate with an elliptic hole Orientation of long semi-axis of the ellipse, relatively to the direction of the extension, is defined by the angle  , and the ratio of the semi-axes is a b  2 (Fig. 8.9.a).

Square plate has ratio 2a L  0.032 between the long axis and the side of the plate. The plate is partitioned into 9 FE, one of which is a regular octagon with an elliptic hole (Fig. 8.9.b). At one side of the plate is applied a uniformly distributed load while the other side is constrained against vertical displacements.

The order of internal approximation on the complex FE with the hole is k  5 . For the other polynomial FE the order is k  3 . The order of boundary approximation is s  0 . The equation

system contains 33 equations, and the bandwidth of the system is M  23 . The number of Gauss's integration points on faces of the complex FE is N g  8 .

Computations are performed for   30 , 60 , 90 . Orientation of the hole changes without the change in the partition of the domain. For all  the difference between the computed contour stress and the exact stress from the analytical solution for the infinite plate [48] does not exceed 1 – 2%.

Extension of a strip weakened by a square hole with filleted corners

The input data for the problem as shown in Fig. 8.10, are: b d  0.4 ; R b  0.125 ; L d  3 ;

d  0.2 ; the thickness of the strip is 0.006;modulus of elasticity is E  19.6  104 ;Poisson's ratio is   0.3 ;load of intensity q  1 is uniformly distributed along the edge.

Taking into account the symmetry, 1/4 of the strip is analyzed by partitioning it into 13 FE (Fig. 8.10.b). The layer of the adjacent to the hole elements, which width is b 8 , includes: 

two polynomial elements which simulate straight patches of the contour of the hole



complex element that simulates the fillet

Let us denote through n, k the orders of the internal approximation for the complex and polynomial FE respectively. Values k are assumed to be the same for all polynomial FE.

The results for contour stress, for two versions of the approximation, are shown in Fig. 8.11 where, for comparison purpose, is also given the experimental data from [56]. From Fig. 8.11 follows that the result at n  4, k  5, s  1 almost coincides with the experimental data. Despite the fact that only one FE simulates straight patches of the contour of the hole, the calculation proves the experimentally observed fact of the stress decrease in the center of the vertical patch of the contour ( x1  b 2 , x2  0 ). The results nicely agree with the data from work [56], in which 1/4 of the strip is simulated by one complex FE and 81 iso-parametric FE. Decreasing the orders of approximation to n  3, k  4, s  0 , the qualitative picture of the distribution of stresses on contour remains, but the peak stress decreases approximately by 15% . The effect of the stress decrease in the center of the vertical sides of the contour is not observed as well. Therefore, the accuracy at n  3, k  4, s  0 is insufficient for finding detailed distribution of stresses in the concentration area.

At n  3, k  4, s  0 the size of the equation system is N  66 and its bandwidth is M  30 . At n  4, k  5, s  1 one has N  132 and M  61 .

Extension of a perforated plate

The perforation of the plate corresponds to a riveted or bolted joint (Fig. 8.12.a). The input data: the diameter of the holes d  0.02 ; the intensity of the load   1 ; modulus of elasticity E  19.6  104 ;Poisson's ratio   0.3 .

Due to the symmetry, 1/4 of the plate is analyzed. The domain is partitioned into 5 complex FE with the holes and 12 polynomial FE (Fig. 8.12.b). The order of the internal approximation is

n  5 for the complex FE and k  4 for the polynomial FE, the order of the boundary approximation is s  1 .

The stress distribution at contours of the holes is shown in Fig. 8.12.c. From Fig. 8.12.c it follows that the mutual influence of the holes, that are in the central area of the plate, causes some shift of the stress peak towards the neighbor hole. A similar effect is observed for an infinite plate with two rows of holes [57]. However, assuming K   max   4.85 the stress concentration factor for the infinite plate, the actual value for the finite plate is in the range 5.40 – 5.85. For the holes close to the edge of the plate, the shifts of the stress peaks do not occur because of the influence of the edge.

For the problem, the order of the equation system is N  130 , and the bandwidth is M  51 .

Strip with a round hole The input data: the ratio of the radius of the hole to the width of the strip is R b  0.275 (Fig.8.13.a);Poisson's ratio is   0.3 ;one edge of the strip is fixed: u1  u2  0 at x1  0 , and at the opposite edge the load is applied (Fig.8.13.b).

The following load cases are analyzed: 

extension (Fig.8.13.b)



lateral bending (Fig. 8.13.c)



pure bending (Fig.8.13.d)

The partition of the strip includes three FE: one FE is complex (element with the hole), and the other two are polynomial (see Fig. 8.13.b). The number of Gauss’s integration points on the boundary patches of the complex FE is N g  9 .

Stress concentration factor is determined by formula K 

A ,  Bnom

where:  A is the normal stress at point A (see Fig. 8.13.b) calculated by MEFEA;  Bnom is the nominal stress at point B calculated using the elementary formulas of the Theory of Strength of Materials for the section weakened by the hole.

In Table 8.8 are given the stress results for the extension problem depending on the orders of internal and boundary approximations. For comparison purpose, in Table 8.8 are also given the results obtained by classic FEM with 8-node iso-parametric elements (MIFE), by the method of large elements (MLE), as well as experimental data of photo-elasticity method (MPE) [54].

From Table 8.8 it follows that the best conformity of the results of MEFEA with the results of other methods takes place when s  1 . Herewith, the number of elements in MEFEA is 3 times less than in MLE and 90 times less than in MIFE. Similar results take place for the lateral bending (see Table 8.9) and pure bending (see Table 8.10).

Extension of a strip with a cutout

This problem illustrates the effectiveness of the complex FE with an elliptic arch (see Fig. 7.4.b). The input data (see Fig. 8.14.a): the ratio of the depth of the cutout to the width of the strip is

R b  0.25 ; Poisson’s ratio is   0.3 .

Variants of the partitions of the strip are shown in Fig. 8.14.b.c. The data detailed in Table 8.11 indicate that the results of MEFEA are similar to the results of other methods [54]. Herewith the same accuracy is achieved at smaller orders of the equation system.

2D problems of linear Fracture Mechanics Stress intensity factors are the main parameters of Fracture Mechanics to be evaluated. They characterize the stress field at the tip of a crack. They are used to forecast survivability of the structures containing cracks as well as in fatigue analysis. Accurate simulation of the stress field at the tip of the crack is critical to Fracture Mechanics. For a qualitative approximation of the solution in the vicinity of the crack special FE are required that have the singularity in stresses of type r 1 2 [58 - 68]. The problem of construction of such elements is twofold: on one hand, it is necessary to simulate the singularity of a certain type on the element, on the other hand it is necessary to provide the continuity of the approximating functions (including singular) through the boundary of the element.

The simulation of the singularity itself does not pose any major difficulty and it can be done in various ways:



by modifying the conventional technology of MFE (for example, by node shifting in iso-parametric FE [66, 69 - 71])



by employment of auxiliary singular functions [9, 72 - 75]



by use of the local asymptotic representations of solutions for the elements with singularities [76 - 78]

More difficult problem is to provide continuity of the approximating functions through the boundary of singular FE. This problem is solved successfully when using modified iso-parametric FE. However, as it has been demonstrated (see [79 - 81]), the size of the element has a major influence on the accuracy, and it leads to the necessity of a mesh refinement near the tip of the crack. Thus, for example, in work [69] when solving the problem of inclined crack the size of the smallest iso-parametric element was 1 250 of the length of the crack.

Discontinuity of the approximating functions can be handled with special buffer elements that surround the element with singularity [82], or by developing singular asymptotic representations that turns to zero on the boundary [83, 84], or by hybrid schemes of FEM. Hybrid FE are widely used when solving the problems of fracture mechanics. They are based on independent approximation of singular field of stresses and displacements of the boundary of FE [54, 59, 85 90]. Complexity of the construction of conformed singular FE makes some authors use the nonconformed elements in practical calculations without an appropriate theoretical justification [67,

81, 91, 92]. In this case the acceptability of FE is decided through numerical experiments on benchmark problems. However, the benchmark problems cannot cover all the diversity of practical cases and the successful solution of a problem does not guarantee the success in all situations.

Below, the special FE described in the Section 7.2.5 is used to solve fracture mechanics problems. Herewith the length of one of the semi-axes of the elliptic hole is equal to zero, and the space PK of the approximating functions of FE is generated by the terms of the complete asymptotic representation of the exact solution of the problem with a crack in infinite domain. Let us notice that the FE allows the crack orientation change and growth without re-partitioning of the domain. It is convenient when solving the problems of the crack propagation.

A plate with an inclined crack

The exact stress intensity factors for the inclined crack in the infinite plate are calculated by the formulas K I    l sin 2  K II    l sin  cos 

where:  is the stress at the infinity; l is the semi-length of the crack;  is the orientation angle (Fig. 8.15.a).

In the computations, the infinite plate is replaced with a square plate with a ratio of the length of the crack to the side of the square of 2l L  0.032 (Fig.8.15.b). Uniformly distributed load

  1 is applied at the upper edge, while the opposite edge is constrained in normal direction.

The partition includes one complex FE with the crack and 8 polynomial FE (see Fig. 8.15.b). The input data: l  5 ; E  1 ;   0.3 ;   15 , 30 , 60 , 90 .

The crack orientation is changed without re-partitioning the domain. The orders of the internal approximation are: n  5 for the complex FE, and k  3 for all polynomial FE. The order of the boundary approximation is s  0 . The size of the equation system is N  34 , the bandwidth is M  24 .

In Table 8.12 are given the errors in stress intensity factors K I and K II at the tip of the crack for different orientations. Despite the rather coarse partition of the domain and the low order of the boundary approximation, a high accuracy in fracture parameters is achieved. For small  , the error for K I and K II is rather lower than for large  . Thus, at   15 , the error is 2.8% for

K I and 0.8% for K II ; at   30 the error is 0.5% for K I and 0.2% for K II . Values of K I and K II , calculated for both tips of the crack, coincide up to three digits. Error in the load boundary conditions of the problem ranges 0.05 – 0.1%.

The analysis of stress field shows that on the contour of the complex FE takes place the one-axial stress state that corresponds to the extension of the plate without the crack. The domain of high stress gradients is located completely inside the complex FE. This is caused by that the ratio

0.125 between the length of the crack and the diameter of the complex FE is rather small. Interelement boundaries are outside the area of stress disturbance and this is the reason of high accuracy at the low order of the boundary approximation.

Strip with a crack

In the case when the length of the crack is of the same order of magnitude as the width of the strip, the stress intensity factors cannot be calculated by the formulas for the crack in the infinite domain. The finite width of the strip is usually accounted by the correlation factor, i.e. the actual intensity factor is calculated by formula K I  FK I

where K I is the intensity factor for the crack in infinite domain.

The data to the problem: the ratio of the width to the length of the strip is a L  5 18 ;module of elasticity is E  19.6  104 ;Poisson’s ratio is   0.3 ;one edge of the strip is fixed while at the opposite edge the uniformly distributed load is applied (Fig. 8.16).

The strip is simulated by three FE, the element with the crack is complex,the other two elements are polynomial (Fig. 8.16).The order of the internal approximation on the polynomial FE is

k  4 ,and on the complex FE n  5 . The order of the boundary approximation is s  1 .

The length of the crack is changed without re-partitioning the domain. From Table 8.13 it follows that the results of calculation of the correlation factor conform to the data from other theories given in work [93].

Axisymmetrical elasticity problems Formulation In polar coordinate system x1 , x2 , x3 , the equilibrium equations (8.7) have the form  11  12  11   33    f1  0 x1 x2 x1  12  22  12    f2  0 x1 x2 x1

where stresses are defined by the formulas (see Fig. 8.17):  u  u2  u1   11  D  1     x1 1   x2 1   x1 

  u1 u2  u1     x2 1   x1   1   x1

 22  D 

  u1  u2 u1     x1   1   x1 1   x2

 33  D 

 12  D

1  2  u1 u2     2 1     x2 x1 

D

E 1   

1    1  2 

here: E is the modulus of elasticity;  is Poisson's ratio; u1 , u2 are, correspondingly, radial and axial displacements; f1 , f 2 are the radial and axial components of the volume force.

The variational problem (8.10) is transformed as follows: find the function u  W that satisfies the equation

 3   ii  u   ii  v    12  u  12  v   dx  *   i 1   

2

2

  f v dx    g v d 

* i  1

i i

2 i  1

i i

(8.11)

v  W

where:

 ii  v    33 

vi , xi

i  1, 2

v1 x1

12  v  

v1 v2  x2 x1

W  V1  V2



V1  u1  H 1   



u1  0 for x1  0

V2  H 1    and where  is the half of the meridian section of the structure located in the semi-plane x1  0 .

External approximations of the equation (8.11) are built with axisymmetrical elements described in Section 7.2.4.

Hollow sphere under internal pressure

The data to the problem: the radius of the sphere is R  0.1 ; the thickness of the wall is

H  0.04 ; the internal pressure is P  1 ; the material properties are E  2  105 ,   0.3 .

For this problem displacements and stresses are functions of the distance from the center of the sphere. Therefore, for the analysis of the sphere, one can consider just a fragment of its meridian section that is bounded by the central angle of 15 . The domain is the polygon ACDFB (Fig. 8.18.a) with its lines AC and DFB inscribed into the contour of the meridian section of the sphere. The displacements along the normal to sides AB and CD of the fragment ACDBF equal zero. The domain is partitioned into one and two FE (Fig. 8.18.a.b).

The results detailed in Table 8.14 indicate the high accuracy achieved even at the partition of the domain into one FE. Thus, at s  1, k  4 , the error in maximum radial stress  11 is 1.1% and in maximum circumference stress  33 is 6.4%. The radial stress at the outer surface where it must equal zero is 3% of the maximum. The error in circumference stresses on the outer surface is 0.05%. Partitioning the domain into two FE (Fig.8.18.b), the accuracy increases.

Stress concentration around a spherical cavity

The infinite domain around the cavity (Fig. 8.19.a) is simulated by a finite cylinder (Fig. 8.19.b). The data to the problem: the radius of the cylinder is R ; the height of the cylinder is 2R ; the ratio between the radius of the cylinder and the radius of the cavity is R r  8 ;Poisson's ratio is

  0.3 ; the uniformly distributed load of unit intensity is applied to the ends of the cylinder while its side surface is free.

In Fig. 8.19.b is shown the partition of 1/4 of the meridian section of the cylinder. Computations are performed for two partitions of the domain that differ by the width a of the layer of the adjacent to the cavity elements (Fig. 8.19.b). For the first partition a r  1.25 , and for the second one a / r  0.35 .

The order of the boundary approximation is increased from s  0 to s  2 . The order of the internal approximation for all s and for all FE is k  4 .

The results for stresses on the equator and in the pole of the spherical cavity (points A and B in Fig. 8.19.b) are given in Table 8.15 for different partitions and orders of the boundary approximation. One can see that the width of the boundary layer of FE significantly influences the accuracy. Thus, for a r  1.25, s  2 , the error in maximum stress is 17.8% and, for

a / r  0.35, s  2 , it is 0.68%. The error rapidly decreases by increasing the order of the boundary approximation that proves that for k  4 the internal error of approximation is rather small and the accuracy is, in general, determined by the quality of the boundary approximation.

Round plate under pressure

The data for the problem: the radius of the plate is R  0.1 ; the thickness is H  0.01 ; the pressure is q  1 ; the material properties E  106 ,   0.3 .

The domain is partitioned into two and four FE (Fig. 8.20). The order of the boundary approximation for both partitions is s  1 . The order of the internal approximation k is increased from 2 to 4.

From Table 8.16 it follows that increase in the order of the internal approximation from k  2 to

k  3 provides sufficient accuracy of the solution. The further increase to k  4 practically does not cause any changes in the displacements and stresses. So the internal error is significant for s  1, k  2 . For s  1, k  3 the internal and external errors become of the same order of

magnitude and the increase to k  4 does not lead to a valuable change in the accuracy. For

k  4 , the difference between the calculated stress and the Theory of Thin Plates Bending is approximately 0.5% while the difference in the displacements in the center of the plate is 4.1%.

Spherical dome under pressure

The data to the problem: the radius of the middle surface of the dome is R  2.28 ; the thickness is H  0.076 ; the angle is   35 ;Poisson's ratio is   1 6 ;the pressure is q  7 .

The dome is partitioned uniformly with spacing   3.5 (Fig. 8.21). Three variants of the internal approximation are analyzed: 1. k  2 for all FE 2. k  3 for all FE 3. For the FE located at the edge of the dome k  4 while for the others k  3

The order of the boundary approximation is s  1 .

In Table 8.17 by the asterisk is marked the data that correspond to the third variant of the approximation. Here the stresses on the external and internal surfaces of the dome are marked by (+) and (-) respectively. The data in Table 8.17 show that the results in 3D formulation well agree with the theory of thin shells. The increase of the order of the internal approximation for the FE adjacent to the edge of the dome (third variant of the approximation) does not lead to the change in stresses at the center of the dome. At the edge of the dome the stresses also change insignificantly (approximately by 3%).

Sine-shaped shell of infinite length under pressure

Equations of the middle surface of the shell in the polar coordinate system (Fig. 8.22.a) are as follows: x1  R  f 1  cos  x2 



l 2

where  is a parameter.

Considering the periodic symmetry of the structure, only one semi-wave of the shell is analyzed under the displacement boundary conditions: u2  0 at   0,    . The other data:

R  0.225 ; f  0.025 ; l  0.1 ; H  0.0025 ;the material properties E  2  105 ,   0.3 ;the pressure q  1 .

For better approximation of the geometry of the shell, the semi-wave of is partitioned into 18 elements (Fig. 8.22.b). The order of the internal approximation is k  2 for all FE,the order of the boundary approximation is s  1 . The number of the equations in the system is N  58 , the bandwidth is M  7 .

In Fig. 8.23 are shown the results for meridian  1 and circumference  2 stresses on external (+) and internal (-) surfaces. For the comparison purposes, in Fig. 8.23 are also given the results obtained by Timoshenko's model [94] as well as by the Theory of Thin Shells [95]. From Fig. 8.23 it follows that the calculations by different models give similar results.

Shell compensator

In Fig. 8.24 is shown a semi-wave of the compensator and its partition into 14 axisymmetric elements. The semi-wave consists of two torus segments I and III , and a circular plate II . Coordinate  is measured along the meridian of the compensator. The modulus of elasticity is E  2.1  105 MPa, the Poisson’s ratio is   0.3 .

The order of the internal approximation for all FE is k  3 , the order of the boundary approximation is s  1 .

Two load cases are analyzed.

Case 1 - To the upper rim of the compensator compressing axial force P  9810 N is applied and there are no restrictions for the displacements of the rim. The condition of symmetry u2  0 is defined at x2  0 . In Fig. 8.25 are shown the graphs for the meridian stress  1 and for the circumference stress  2 , both on the internal (continuous lines) and on the external (dash line) surfaces of the compensator.

From Fig. 8.25 it follows that the maximum circular stress occur on the upper rim of the compensator, and the stresses at the internal and external surfaces are practically identical. Meridian stress has two obvious peaks: the first peak is located in the second half of the torus segment I while the second peak is located at the center of the torus segment III . Herewith, the amplitude of the second peak exceeds the amplitude of the first peak approximately by 50%.

Case 2 - Often the compensator system consists of few connected shell compensators. The calculations are made for the compensator that is one wave of the compensator system, at the given axial convergence of its rims. Boundary conditions of the problem are defined as follows: 

u2  0 at x2  0



u2  0.001 m at x2  0.027 m (upper rim)

The stress graphs are given in Fig. 8.26. One can see that the stress picture significantly depends on the type of the boundary conditions at the upper rim. For a given convergence of the rims, maximum circumference stress occurs at the tips of the torus segments. At the upper rim the meridian stress is significant. The second peak of the meridian stress is approximately equal to

the amplitude of the first peak and it is located at the center of the torus segment III . The stress graphs shown in Fig. 8.25 well conform to the results for such compensators from [94].

Figure legends for Section 8:

Fig. 8.1 Shear stresses in torsion

Fig. 8.2 Partitions

Fig. 8.3 Partition of the cross-section of the bar

Fig. 8.4 Bending of a cantilever plate: (a) dimensions and loading (b) partition and results

Fig. 8.5 Partitions of 1/8 of the plate

Fig. 8.6 Loading and partition

Fig. 8.7

Bending of a circular bar: (a) geometry and boundary conditions (b) partitions (only 1/8 of the bar is shown)

Fig. 8.8 Dimensions, loading, and partition of 1/4 of the plate

Fig. 8.9 A plate with an elliptical hole: (a) infinite plate (b) finite element model

Fig. 8.10 A strip with a square hole: (a) geometry and loading (b) partition of 1/4 of the plate

Fig. 8.11 Stress at the contour of the hole, where: Broken line is the result for k  3, n  4, s  0 ; Solid line is the result for k  4, n  4, s  1 ; circles is experimental data

Fig. 8.12 A perforated plate: (a) geometry and loading (b) partition (c) Stress distribution at the contour of the holes (circles show the points where the stress reaches its maximum)

Fig. 8.13 A strip with a round hole: (a) geometry (b) partition (c) (d) load cases

Fig. 8.14 A strip with a round cut at its side: (a) geometry (b) (c) partitions

Fig. 8.15 A plate with an inclined crack: (a) infinite plate (b) finite element model

Fig. 8.16 A strip with a crack: geometry, loading, and partition

Fig. 8.17 Stresses in the axisymmetrical problem

Fig. 8.18 A hollow sphere under internal pressure: (a)(b) partitions

Fig. 8.19 Stress concentration around spherical cavity: (a) infinite domain (b) finite element model

Fig. 8.20 Bending of a round plate Fig. 8.21 A spherical dome under external pressure

Fig. 8.22 A sine-shaped shell under internal pressure: (a) geometry

(b) partition of a semi-wave

Fig. 8.23 Stress distribution: Solid line is data from [94], Squares are results from [95], Circles is MEFEA solution

Fig. 8.24 Partition of a semi-wave of a shell compensator

Fig. 8.25 Stresses under axial force

Fig. 8.26 Stresses under axial displacement of the rim

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