Mathematical Analysis and Numerical Methods for Science and Technology || Numerical Methods for Stationary Problems

Chapter XII. Numerical Methods for Stationary Problems

Introduction

In the preceding chapters we have reviewed the results concerning the existence and uniqueness for stationary elliptic problems. These results can provide a valuable insight for the physicist, but they can prove inadequate in so far as the explicit calculation of the exact solution is often out of reach. In particular, the functional transformation methods studied in Chap. III permit only rarely the direct attainment of the exact solution in complicated models. It is for this reason that, in general, the exact problem is approximated by a discrete problem formulated in a finite dimensional space, thus leading to the solution of a linear system.

These methods which we term numerical are the subject of this chapter. We lay great stress in particular on the finite element method which has undergone a considerable expansion in the course of the last thirty years and which is now used in most numerical methods in engineering science. Orientation. The finite element method arose in the years 1950-1955 for the solution of problems posed in the mechanics of deformable continuous media. Since then, this method has been considerably developed and utilised, not only for the solution of problems in mechanics, but also for the solution of general boundary value problems. We wish to recall here the origins, the successive developments and the fields of applications of the finite element method, emphasising, in particular, its advantages compared with the finite difference method. Next we present the different aspects of the methods analysed in this chapter, then we recall the Lax-Milgram theorem and the results on Sobolev spaces which we use constantly in the later sections.

1. The Basic Ideas of Finite Difference Methods and Finite Element Methods

The Continuous Problem. Numerous physical problems are formulated as follows (see Chap. I): (0.1) { Au = f on

Bu = g on D, r,

R. Dautray et al., Mathematical Analysis and Numerical Methods for Science and Technology Springer-Verlag Berlin Heidelberg 2000

Introduction 161

where Q is a subset of ~n, r the boundary of Q, A and B differential operators,! and g are given and u is the solution of the problem. The disadvantages associated with this type of formulation are essentially due to the excessive regularity demanded of the different parameters, as for example:

i) the possibility of the data f and g and the unknown u having to be defined at every point of Q or r, ii) the possibility of u having to be differentiable in the classical sense in order to calculate Au and Bu. Consequently the formulation (0.1) is poorly adapted to take into account discon-tinuous or irregular data or solutions. It was for this reason that in Chap. VII, we introduced for problems of type (0.1) the following variational formulation or weakformulation: find u belonging to a space of admissible functions V such that: (0.2) a(u, v) = f(v) for all v E V , where the functions a(. ,.) and f(.) are bilinear and linear forms using integrals on Q and r. This second type of formulation is often more conformable with physical reality: for example in elasticity, the quadratic form a(v, v)/2 represents the deformation energy of the medium, associated with a "virtual" displacement v, whilst f(v) represents the potential energy of the exterior forces applied to the medium, associated with the same displacement v (see Chap. VII, 2.7.4). Furthermore, during the years 1945-1950, the theory of distributions was being developed, especially the notion of differentiation in the sense of distributions (see Schwartz [1]), and, during the years 1937-1938, the theory of Sobolev spaces (see Sobolev [1]). These theories allow one to prove that the formulation (0.2) of the problem admits a unique solution in a space of admissible functions sufficiently general to take into account most of the problems encountered in practice (see Chap. VII, Lions-Magenes [1] and Duvaut-Lions [1]). Naturally, the exact solution of equations (0.1) or (0.2) is generally impossible and one is led to seek approximations of the solution.

The Approximate Problem. Different approximation methods for the solutions of equations (0.1) and (0.2) have been successively used. We distinguish two periods: i) During the first half of the twentieth century, almost all of the approximations methods consisted of calculations "by hand" based essentially on series expansions (Fourier series, for example). ii) Beginning in the 1950s, the use of electronic calculators was developed, which completely modified the approximation methods. Two types of methods are essentially distinguishable: in chronological order, the finite difference methods and the finite element methods. Among other methods we cite the collocation methods and the least squares methods.

The Finite Difference Method. The use of this method employs the differential formulation (0.1) of the problem and consists of replacing each of the differential operators by a difference quotient.

162 Chapter XII. Numerical Methods for Stationary Problems

Suppose we wish to solve, for example, the Dirichlet problem (see Chap. 11):

{ - Au =_1 in the square Q = {(x,y); 0 < X,Y < 1} (0.3) ulr = oU - 0 .

Let N be an integer ~ 1 and h = 1j(N + 1). We then define a mesh on the square Q as the set of points (Xi = ih, Yj = jh), 0 ~ i,j ~ N + 1, called the nodes of the mesh (see Fig. 1).

I

0 1

Fig. 1. Mesh on the square ]0, 1[ x ]0,1[( N = 4, h = D The finite difference method consists of obtaining an approximation to the solution U at the points (Xi' Yj), 1 ~ i,j ~ N, and it rests on the following Taylor formula: for all U E ~4(Q), we have:

(0.4) J - Ju(x" YJ} ~ :'[;[ij o~U Ui+l,j - Ui-l,j -o4U~,j+l - Ui,j-l]] l + 12 OX4 (Xi + ih, Yj) + oy4 (Xi' Yj + Ojh) with uij = u(xi , Yj) and IOil < 1, 1 ~ i,j ~ N. The two basic principles of the method are then:

i) neglecting the rest of the expansion in which the coefficient ~; is "small", ii) requiring that equation (0.3) is satisfied at all the points (Xi' Y), 1 ~ i,j :E; N, of the mesh, the quantity - Au(xi , Yj) being approximated, in conformity with i), by the difference quotient:

1 h2 [4uij - Ui+l,j - Ui-l,j - Ui,j+l - Ui,j-l] .

Introduction 163

Putting fij = f(x j, Yj), we get in this way N 2 equations in N 2 unknowns Ujj, 1 ~ i,j ~ N, i.e.: (0.5) :2 [4Ujj - Uj+ l,j - Uj-l,j - Uj,j+ 1 - Uj,j-l] = jjj' 1 ~ i,j ~ N . The boundary condition ulr = 0 is taken into account in the equations (0.5) by requiring that:

(0.6) UO,j = UN+l,j = Uj,O = Uj,N+l = 0, 1 ~ i,j ~ N . The system (0.5HO.6) is then written in the form AU = F. In the particular case N = 4, we verify that for an enumeration of the nodes first with i increasing, and then with j increasing, the matrix A has the values(l):

(0.7)

--

4 -1 -1 -1 4 -1 -1

-1 4 -1 -1 -1 4 -1

-1 4 -1 -1 -1 -1 4 -1 -1

-1 -1 4 -1 -1 -1 -1 4 -1

x

-1 4 -1 -1 - 1 -1 4 -1 -1

-1 -1 4 -1 -1 -1 -1 4 -1

-1 4 -1 -1 -1 4 -1

-1 -1 4 -1 -1 -1 4

'- -

This matrix A is tridiagonal by blocks and is regular (i.e. invertible). Naturally, we have noted the basic principles of the finite difference method which may be used in the more general cases: the dimension of the space (Q c IR"), the form of the boundary r, and the definitions of the operators A and Bin (0.1).

(1) In (0.7), every unoccupied place is understood to be O.


The convergence of the approximate solution to the exact solution is studied, in particular, by Collatz [1], Forsythe-Wasow [1] and Mikhlin-Smolitsky [1]. We have already indicated that the formulation (0.1) of the continuous problem often turns out to be insufficient to deal with the general problems of mechanics or physics. Whence the idea of using the more general weak formulation (0.2), and of associating with it a family of discrete problems for which the difference quotients are "compatible" with a variational formulation. This type of method is described by Felippa [1] as the variational approximation method for finite differences. The study of the convergence of the approximate solution to the exact solution is then quite a delicate problem. For a general study of the approximation of elliptic boundary value problems by this type of method, we refer to Cea [1], Aubin [2, 3], and Temam [1]. In the same spirit, Raviart [2] has made a general study of the approximation of boundary value problems of evolution (see Chap. XX). We now continue with a general survey of the successive developments of the finite element method-for more details, one should consult Martin-Carey [l]-as well as the basic ideas, which will enable us to indicate succinctly in Sect. 2 the advantages and disadvantages respectively of each of these methods.

The Finite Element Method. In its beginnings, the finite element method had been used to solve the problems of deformable continuous media. These problems are generally formulated on domains with complicated geometry, and because of this are difficult to approximate by the finite difference method. Among the precursors, we cite Argyris, Clough and Martin (see Argyris-Kelsey [1], Clough [1], Turner et al. [1]). Little known until 1955, the finite element method subsequently received ever increasing attention on the part of engineers confronted with the problems of deformable continuous media. Between 1955 and 1964 most of the finite elements (today called "classical") were defined, together with the principal extensions of the method designed to deal with the essentials of the problems of deformable continu-ous media. Counting from 1965 the number of users of the method substantially increased, especially after the publication in 1967 of the first edition of the book by Zienkiewicz [1], soon to be followed by the publication of many more works. From that. date on, the method has become used more and more for the numerical approximation of boundary value problems of great diversity. It was at this time that the development began of the mathematical analysis of the method (see Zlamal [4]). This analysis confirmed and clarified many of the results formulated in an "empirical" fashion by the engineers and thus allowed the preparation on solid foundations of new applications of the method to problems more and more complex. Let us now illustrate the basic ideas of the finite element method using the simple example (0.3). To this end, we make use of the variational formulation - or weak formulation-of problem (0.3) (see Chap. VII, 1.2.1): find u E V such that:

(0.8) a(u, v) = f(v) , \Iv E V ,

Introduction 165

with

a(u, v) = L (~:~~ + ~;~; )dXdY , (0.9) f(v) = LfVdXdY,

V = Hb(Q) = {v; v E Hl(Q), vir = O} , where Hl(Q) denotes the usual Sobolev space (see Chap. IV, and Sect. 4 below). The finite element method is a particular case of the variational approximation method, also called the Rayleigh-Ritz or Galerkin method, which allows the approx-imation of the solution u as follows. We construct afinite dimensional subspace Vh of the space V, then we define the approximate solution Uh for the solution u to be the solution of the following problem: find Uh E Vh such that: (0.10) We then show: i) that the problem (0.10) admits a unique solution uh; to do this, we use the inclusion Vh C V, which makes it possible to arrive at the existence and uniqueness results for the solution of the continuous problem (0.8); ii) that the convergence of the approximate solutions Uh to the solution u is directly connected to the properties of the approximation of functions u E V by the functions Vh E Vh Thus, the construction of the space Vh constitutes the keystone of the variational approximation method. The object of the finite element method is precisely the construction of the family of spaces Vh which have suitable approximation properties and which lead to a satisfactory numerical implementation. In its simpler form, a finite element method for constructing these subs paces Vh of the space V = Hb(Q), considered in (0.9), is the following. We establish a triangula-tion of the square Q, (see Fig. 2). It is indifferent for what follows whether this triangulation is regular or not. Then, the subspace Vh is made up of the linear functions on each triangle, continuous on Q, and null on oQ. A basis for this space Vh is formed by the set of functions Vhi in Vh which take the value 1 at the interior vertex i of the triangulation and the value 0 at every other vertex. Denote by N the total number of interior vertices. Writing:

(0.11) N

Uh = L UhiVhi , i = 1

the vector OIih = (Uhl' ... , UhN) is then the solution of the linear system: (0.12) where

(0.13) { Ah = [ahij] , with ahij = a(vhi' Vhj) , Fh = [fhJ, with h.j = f(vhj) .


Fig. 2. Triangulation of the square ]0, 1 [ x ]0, 1 [

We therefore obtain a matrix Ah which is symmetric and positive definite. The problem of the convergence of the approximate solutions to the exact solution forms the object of one part of this chapter (see 1, Sect. 5, in particular). The Collocation Method. We have emphasised that the weak formulations of differential problems are more general. The basic principle for obtaining a weak formulation of the problem Au = f in Q is to multiply each side by test functions v(x) and to integrate over Q, i.e.:

(0.14) L (Au)vdx = LfVdX . That is how we get, at least formally, the variational formulation (0.2). The characteristic of this formulation (0.2) is the use, as a space V of test functions, of the space U in which we seek the solutions u, i.e. V = U. We then obtain a symmetric formulation of the problem, at least at the level of the choice of functional spaces. Naturally, one is free to make other choices for the space V of test functions. In particular, if the space V contains all the Dirac distributions

Introduction 167

The expression (0.15), which is a system of Mh equations in Mh unknowns, expresses the fact that the solution Uh must exactly satisfy the original equation Au = f at the M h points of collocation. Naturally, the space Uh in which we calculate AUh must satisfy properties of regularity and approximation more constraining than in the case of the classical variational approximation where Vh = Uh On the other hand, the matrix of the system is easier to calculate and it generally has a narrower bandwidth. The convergence of the error estimate depends not only on the choice of the U h' but also on the choice of the Vh , i.e. the position of the collocation points. This type of method has in particular been studied by Douglas-Dupont [1], Prenter-Russell [1] and Wheeler [1].

The Least Squares Method. Another weak formulation of the problem (0.1) consists of minimising the following functional J(v):

(0.16) J(v) = ~t (Av)vdx - tfVdX , on a suitable space V. It is easy to demonstrate the equivalence (see Chap. VII) of this formulation and the formulation (0.2) whenever the bilinear form a(.,.) is continuous, V-elliptic and symmetric. The least squares method rests on an analog-ous idea. It consists of minimising the functional:

(0.17) K(w) = t (Aw - f)2 dx , over the space W of "admissible" functions. Let us expand the expression (0.17). It becomes:

K(w) = t (AW)2 dx - 2 t (Aw)fdx + t (f)2 dx , whence formally, leaving aside the problems of regularity and boundary condi-tions,

(0.18) K(w) = t (A*Aw)wdx - 2 t (A*f)wdx + t (f)2dx (2). Hence, the least squares method is (formally) equivalent to the solving of the problem: (0.19) A*Aw=A*f We get a problem which is automatically symmetric. But if we compare the formulation (0.19) with the formulation Au = f of (0.1), we notice that the order of the operator A * A is double that of the operator A. This has the effect of complicat-ing the approximation.

(2) A* is the adjoint of A.


This type of method has been studied in particular by Bramble-Nitsche [1], Bramble-Schatz [1], Locker-Prenter [1]. These authors show especially how it is possible to take into account the boundary conditions.

2. Comparison of the Two Methods Field of Applications of the Finite Element Method

In Table 1 below, we assemble the principal characteristics of the two methods. The advantage of the finite element method becomes clear as soon as the problem considered is "at least" fairly complicated (the nature of the problem, the complexity of the data, ... ) which is often the case in the problems encountered in practice. This explains the continually growing success of the finite element method and the choice that we make in analysing it as the approximation method for elliptic boundary value problems. The situation is different in the case of evolution problems (see Chap. XX) and for hyperbolic problems. In Table 1 we use the abbreviations FDM and FEM for the Finite Difference Method and the Finite Element Method respectively.

Field of Applications of the Finite Element Method. The FEM was initially developed to solve the problems of the mechanics of continuous media with complicated geometries. Compared with the FDM, the FEM is an extremely

Table 1. Principal Characteristics of FDM and FEM

FDM FEM

Meshing of the domain Q c [Rn. Triangulation of the domain Q c [Rn. o Simple in the case where the boundary r has o Particularly simple in the case of regular

sides parallel to the axes. polyhedral open sets. o Delicate in the case of a curvilinear boundary o In each of the cases (n = 1,2, 3) there exists

r because it is necessary to determine the working triangulation algorithms (whether points of intersection of the mesh with the the open set Q is polyhedral or not). boundary. A general algorithm is difficult to o Irregular triangulations pose no problems. write.

o Regular meshes are strongly advised.

Definition of the differences schemes. Definition of finite elements. o Simple for the regular parts of the mesh, o Simple for finite elements of class reo used

interior to the domain, regardless of the order in the solution of problems of order 2. of the operators. o More delicate for finite elements of class rem

o Delicate in any neighbourhood whatever of used in the solution of problems of order the boundary, the writing of a general algo- 2m +2. rithm is difficult. o General sub-programs can be written for

o More delicate for difference schemes relative all finite elements. Then their utilisation is to problems with high order. immediate.

o It is possible to construct subspaces of finite elements in Hm(Q) (with an approximation of the domain if the boundary r is curved).

Introduction 169

Table 1 (Contd.)

FDM FEM I

Inclusion of boundary conditions. Inclusion of boundary conditions. o In general, rather delicate. o These are partly incorporated in the varia-

tional formulation: - either they figure in the definition of the v,;, - or they appear in the form of integrals on

the boundary r.

Formulation of the system matrix. Formation of the rigidity matrix. o Simple. o This is particularly simple and is made up o The system matrix is sparse. of two steps:

i) for each finite element, the elementary rigidity of matrix is formed,

ii) the system matrix is the "sum" of the elementary rigidity matrices (assem-blage).

o The system matrix is sparse.

Formation of the second member of the system. Formation of the second member of the system. o Often delicate, in particular if the boundary is o Simple and generally carried out with that

curvilinear or if the boundary conditions are of the rigidity matrix. complicated.

System solution. System solution. o Iterative methods are generally used. o Direct methods are generally used which

today allow one to treat very large systems.

Study of convergence. Study of convergence. o This is delicate. It is necessary to establish o Systematic, comprising steps which are

consistency and stability properties. These independent of each other. studies are not systematic; they are narrowly o Error estimates depend directly on the ap-tied to the data for the problem and the finite proximation qualities of the space of finite difference method chosen. elements being used.

powerful tool for solving this type of problem. In Clough [2] will be found examples of applications to the calculation of dams (thickness, arches, effects of earthquakes), to the analysis of constraints and of plane deformations in rocky media, to the analysis of the flexure of plates and thin shells, to the study of the different structures which compose a nuclear reactor. For the use of FEM in the nonlinear mechanics of continuous media, Oden [1] can be consulted. The FEM is largely used in fluid mechanics as the works of Gallagher et al. [1] show. In Glowinski-Marocco [1], an example will be found of the application of the FEM to the solution of magnetostatic problems. These are just some of the examples of the applications of the finite element method. If one refers to the bibliography of finite elements in Norrie--de Vries [1]


which contains no less than 7,115 references spanning 1956 to 1975, one will be able to judge the range and extreme diversity of the field of applications of the FEM. The bibliography of Whiteman [1] can likewise be consulted.

3. The Different Topics Treated in this Chapter XII

The object of this chapter is both to analyse mathematically the finite element method as a method of approximation for elliptic boundary value problems, and to describe precisely the different aspects of its numerical implementation. To do this, the different aspects of the method which we study are illustrated by concrete examples of elasticity problems and problems of plates and shells - which are directly derived from the elasticity problem. The reasons behind this choice are essentially: i) that the solution of the elasticity problem is very largely responsible for the clarification of the finite element method by the engineers, ii) that the study of this example exhibits the different characteristics of this method. The principal aspects of the finite element method which we analyse are the following: The basic mathematical aspects - i.e. the variational formulation of the continuous problem, the construction of a space of finite elements, the definition of approxi-mate problems with reference to numerical integration, the study of the conver-gence of approximate solutions to the exact solution, the obtaining of error estimates - these as well as a precise description of the numerical implementation are given in 1. In the following sections, we analyse other types of approximation which increase the effectiveness and the diversity of the finite element method. From the start it is a question of working with domains with curved boundaries. We define, in 2, curved finite elements of class Cflo, the method used can be generalised to define finite elements of class Cfl m, m;?; 1. In 3, we analyse a non conforming finite element method, i.e. a method for which the discrete space in which the solution is approximated is not a subspace of the space in which the solution of the continuous problem is defined. With 4, we introduce two very important problems in structural mechanics. These are the problems of plates and shells. The new difficulty is essentially due to the fact that the corresponding operators are of order 4. To produce a conformal approx-imation of the solution, it is therefore convenient to use finite elements of class Cfll. Another extremely important problem in structural mechanics is that of vibration analysis. This is why, in 5, we consider the approximation of eigenvalues and eigenvectors. Naturally, this chapter does not pretend to give a complete description and analysis of finite element methods. In particular, we have deliberately omitted the analysis of dual finite element methods, i.e. the mixed and hybrid methods of finite elements. The numerical analysis of these methods is the object of numerous

Introduction 171

researches and the results acquired appear to be narrowly tied to the the nature of the problems considered (see Thomas [1]). In particular, the analysis of the dual approximations of the elasticity problem is for the moment partial and very technical (see for example Amara-Thomas [1]). The methods of approximation using integral equations, Stokes' equations, the stationary part of the transport equation are introduced in other parts of this work. The same holds for the approximation of solutions of the evolution problems. It is our intention to provide complete proofs of the results announced in 1, relative to the basic mathematical aspects of the method. In the last sections,. we put the accent on providing new arguments for these proofs, referring to 1 for the argu-ments previously used. In all cases, we mention the references where the reader can find the complete proofs as well as other aspects of the problems considered.

4. The Lax-Milgram Theorem and Sobolev Spaces

The brief recapitulations in this Sect. 4 are given for the convenience of the reader. In this section, we recall the Lax-Milgram theorem (Lax-Milgram [1]) established in Chaps VI and VII and the principal results on Sobolev spaces which have been introduced in Chap. IV. For more details about these spaces and their use in the formulation of elliptic boundary value problems, we refer beyond these chapters, in particular to Adams [1], Lions [1], Lions-Magenes [1], Oden-Reddy [1] and Showalter [1].

The Lax-Milgram Theorem. Let V be a real Hilbert space(3), let a(.,.): V x V -. IR be a bilinear form, continuous and V-elliptic, i.e. (0.20) 3 a constant IX > 0 , such that IX IIvl12 ~ a(v, v), "Iv E V , and let f: V -+ IR be a continuous linear form. Then there exists a unique element u E V such that:

(0.21) a(u, v) = f(v) , "Iv E V o

Sobolev Spaces. We recall the definition of the Sobolev spaces W"" P(Q): for every integer m ~ 0 and for every real number p satisfying 1 ~ p ~ + 00, the Sobolev space wm, P(Q) is defined by: (0.22)

(3) Throughout this chapter, the spaces are taken to be real. (4) We use in this chapter the notation a'v to denote the (X-th derivative of v, and DI.l v the Frechet derivative of v; thus:

a'v(X) = DI.lv(x).(e" .. , e l , e2, ... , e2, . .. , e., ... , e.) where the basis vectors of [R., (e;)j = I to. are repeated (Xj times, i = 1 to n.


Theorem 1. (see Nacas [1]). Endowed with the norm:

(0.23)

1 ! IIvllm,p,a = (L r 18

l. Finite Element Method Applied to Elasticity 173

Then,for all integers m > 0, and all real p, 1 ::::; p < + 00, we have the following compact injections (denoted by ~):

(0.27)

W"'.P(Q) ~ Lq(Q) n

1 for all q satisfying 1 ::::; q < p* if m < -P , 11m with - = - --

p* p n W"" P(Q) 2::. Lq(Q) for all q E [1, 00 [, if m = ~ ,

p

W"" P(Q) 2::.~(Q) if ~ < m . p n - < m - s (5). P

In particular, the compact injection obtained with p = q = 2, i.e. (0.28)

o

is due to Rellich [1] and has been established in Chap. IV. Naturally, other inclusions can be obtained by "interchange" of the order of the differentiations in the relations (0.26) and (0.27).

1. Principal Aspects of the Finite Element Method Applied to the Problem of Linear Elasticity

Orientation. In this 1, we shall develop the basic concepts of the finite element method in an example of essential importance in applications, i.e., the problem of three-dimensional linear elasticity formulated on a polyhedral domain. We assume that the continuous medium considered is fixed over a part ro of the boundary r and that it is subjected to a distribution of body forces as well as a distribution of surface forces exerted over the complementary part r 1 = r\ro of the boundary. The extensions of the finite element method to the case involving curved boundaries will be examined in 2.

1. Variational Formulation of the Continuous Problem

One of the fundamental problems of the mechanics of continuous media is that of elasticity (see Chap. lA, 2, and the Appendix "Mechanics"). For a detailed study of the mechanical aspect of these problems, we refer for example, to Germain [1],

(5) We assume here that Q satisfies a "strong local Lipschitz condition" (see Adams [1]).

174 Cnapter XII. Numerical Methods for Stationary Problems

Green-Zerna [1], Landau-Lifschitz [1]. The mathematical analysis of these prob-lems is developed, for example, by Duvaut-Lions [1], Hlavacek-Necas [1,2], and in a very complete manner by Fichera [1, 2]. We limit ourselves here to stating the simplest problem (see Chap. VII, 2). Let Q be the configuration of a continuous elastic medium before deformation. We assume that the open set Q is a connected bounded subset in 1R3 with a Lipschitz-continuous boundary r(6) (essentially, there exists a covering of a neigh-bourhood of the boundary r by a finite number of charts such that, for each of them, the part corresponding to r is described by a continuous Lipschitzian function). Under the action of a field of exterior volume forces (of density) p and exterior surface forces (of density) q, the continuous medium is deformed and takes a new configuration Q *. The problem is then to determine the field of displacements it which associate to the position P of an arbitrary particle of the continuous

medium, before deformation, its position P* in the final configuration. Sub-sequently, we suppose that the Euclidean space 1R3 is referred to an orthonormal

--.

frame (0, e;, e-;, e;) i.e. OP = x1e; + x2e; + X3e;. In this frame, the displace-ment field can be written:

(1.1)

Hence, the three functions Ui: x = (Xl> X 2 , X3)~ ui(x) E IR are the unknowns of the problem. From the knowledge of the displacement field t4 we deduce the components (',jk ( it) of the strain tensor (', (we are limited here to the linear theory):

~ 1 (OU j OUk) . (1.2) (',jk( U ) = 2 OXk + oXj , 1 ~ j, k ~ 3 . Finally, the stress-strain relations (Hooke's law) gives the components (] j ( it) of the stress tensor (]: (1.3) In this relation, and subsequently, we use the summation convention over (1, 2, 3) for repeated indices. The coefficients of elasticity Cjklm are independent of the stress tensor; they have the properties of symmetry:

(1.4) Cjklm = Ckjlm = C'mjk , and ellipticity:

(1.5) Cjklm(',jk(',lm ~ IX(',jk(',jk, IX constant > 0, V(',jk = (',kj The constitutive equation (1.3) corresponds to the case of an anisotropic material. In the case of a non homogeneous material, the coefficients of elasticity are functions of xwhich we suppose measurable and bounded in Q, i.e., the Cjklm belong to L 00 (Q).

(6) With Q locally on only one side of r.


In the case of an isotropic material, the coefficients Cjklm are given by: (1.6) where the scalars .A. and Jl. are the Lame coefficients, and the J jk are the Kronecker symbols. Then relation (1.3) can be written: (1.7)

The Deformation Energy. The energy of deformation associated with a displace-ment field v: has the expression:

(1.8) 1 --+ --+ 1 r --+ --+ "2 a( v, v ) = "2 In O'jk( v )ejk( v )dx . In virtue of the constitutive relation (1.3), this can also be written:

(1.9) a( v: it) = In Cjklm(x)ejd it)e,m( it) dx . The symmetry properties ejk = ekj and those of (1.4) permit the association of the quadratic form (1.9) with the symmetric bilinear form:

(1.10) a( u: it) = In Cjklm (x) ejk ( it)e,m( it) dx , which can further be written:

(1.11) --+ --+ i au j av, a( u, v ) = Cjklm(x)-;--;-dx. n UXkUXm

The Work Due to Exterior Forces. Let r = ro u r1 be a partition of the bound-ary r. We suppose that the continuous medium D is subjected to the action of a field of body forces (of density) p and to the action of a field of surface forces (of density) ~ exerted on the part r i of r. The work of these exterior force fields associated with a displacement field it can be written:

f(it) = f p. itdx + f q. itdy. n 1;

(1.12)

Furthermore, we suppose that the continuous medium is fixed over the part ro of r. These data are illustrated by Fig. 1.

The Variational Formulation of the Problem. Let us start by defining the space --+ V of admissible displacements:

(1.13) V = {v+= (Vi' v2' v3) E (Hi (D))3, itl lO = O} . As the coefficients Cjklm are assumed measurable and bounded on D, the choice of V gives a meaning to the integral (1.11) and, moreover, it takes into account the fixing of the medium on the part ro of the boundary. We suppose from now on that ro is measurable, with strictly positive measure. Endowed with the scalar product


Fig. 1

{ ro = fixed part of the boundary r, r = ro u r, is the boundary before deformation, r* = ro u ri' is the boundary after deformation.

induced by that on (H 1 (Q))3, i.e., (1.14) (( u: it)) = ((u i , v;));, n , the space Vis a Hilbert space. The norm associated with this scalar product will be denoted by: (1.15) 1 II It II = [(( v: it))]2 . In order that the integrals definingf( It), in (1.12) shall have a meaning, we suppose that the force fields p and q satisfy the hypotheses: (1.16) Consequently, following Chap. VII, 2 (see also Duvaut-Lions [1], Chap. 3), the elasticity problem admits the following formulations:

(1.17) for pE(L2(Q))3, for qE(L2(rd)3, find -+ -+ U E V

-+~ -+ -+-+ such that a( u, v J = f( v), V V E V , where the bilinear form a( . , . ) and the linear form f(.) are defined in (1.11) and (1.12) respectively.


As the bilinear form a(.,.) is symmetric and the associated quadratic form is positive definite, we obtain the following equivalent formulation:

{for P E (L 2(Q))3, for q~ (L 2(rl ))3, find it E it

(1.18) minimising the functional

-+ -+ -+ 1 -+ -+ -,+ J: v E V I---i> J( v ) = 2a( v, v) - f( v ) . Theorem 1. The problems (1.17) and (1.18) admit a unique solution. Proof It is clear that the bilinear form a(.,.) is continuous on the space -+ V. In virtue of the continuity of the mapping Hl(Q) G L 2(r1 ) (see Chap. IV) the same holds for the linear form f(.). In order to apply the Lax-Milgram theorem and its conclusion, the ~ellipticity of the bilinear form a(.,.) has then to be established, i.e., there exists a constant P > 0 such that: (1.19) a(V: if) ~ PII if 112 , vilE V. The relations (1.5) and (1.10) imply: (1.20) r

a( v: if) ~ (X J n Bjk( if)Bjk( if) dx . We next use Korn's inequality (see Chap. VII, 2): there exists a constant c(Q) such that:

(1.21)

Since mes (To) > 0, we deduce from the inequality (1.21) that the mapping:

Il E V I---i> c.~ 1 IBij( if)15. n y/2 is a norm, equivalent to the product norm (1.15). This equivalence taken together with the inequality (1.20) then implies the result (1.19). 0 Interpretation of the Solved Problem. Denote by ~(Q) the space of functions infinitely differentiable with compact support in Q, endowed with the Schwartz topology, and by ~'(Q) its dual which is the space of distributions on Q (see Appendix "Distributions" in Volume 2, and Schwartz [1]). If T E ~'(Q) and ({J E ~(Q), the value of Tat ({J is denoted by In g({J dx . Given the inclusion:


the variational equations (1.17) are satisfied for all functions v+ E (.9&(0))3. Conse-quently, by definition of the derivative of distributions, the relation (1.11) can be written:

-+ 3 -+ -+ / a ( auj)) v ({J E (.9&(0)) , a( u, ((J) = \ - aXm Cjklm(X) aXk ,({JI

Furthermore, the relation (1.12) can be written: f( = (PI' ({JI>' '1

1. Finite Element Method Applied to Elasticity 179

In summary, we formally solve the following boundary value problem: o -+

- -;-(Ulm( u)) = P, in Q, I = 1, 2, 3 , uXm

(1.27) -+ -+ u = 0 on To,

which is known as the three-dimensional linear elasticity system. In an entirely analogous manner, on moving inside the plane 1R2, we will obtain the plane linear elasticity system. Observe that in the case of a uniform isotropic material the relations (1.12), (1.3), (1.6) allow the first equation (1.27) to be written in the following vectorial form: (1.28) Remark 1. The above results extend to the case of more general boundary conditions. For this, we refer to Duvaut-Lions [1], Fichera [1,2], and Hlavacek-Necas [1, 2]. 0

2. Construction of Approximation Function Spaces

In a general manner, it is not possible to solve analytically the problem (1.17). It is -+

for this reason that we now construct a sequence of discrete spaces Vh in which we approximate the solution it of the continuous problem. We construct the spaces v: so that the inclusion: (1.29) is satisfied. By definition, we hence realise an internal approximation of the solution It This construction can be subdivided into four stages:

Stage 1: "Triangulation" of the Domain. Throughout this 1 we assume that the domain Q is polyhedral. Then, given a parameter h > 0, eventually to tend towards 0, we associate with h a partition f/h of the domain Q with the aid of tetrahedra. It is this partition which we shall call, by abuse of language, a "triangulation" of the domain, by analogy with the partition of a two-dimensional domain with the aid of triangles. Consequently, we suppose that the triangulation f/h is such that:

(1.30)

i) every tetrahedron K E f/h has a non empty interior and a diameter hK ~ h, where h = max hie ,

KE or,

ii) two distinct tetrahedra K; and K j belonging to f/h have: - either an empty intersection , - or an intersection reduced to a common vertex , - or an intersection reduced to a common edge , - or an intersection reduced to a common face .


,,--------7' ""',, ". ,

",,'" / "",., .... / " " ,

,

/'1 .," I

'........... .--''''ro--'--

Fig. 2. Example of the domain Q

I I I I I I I I

Figures 2 to 4 give an example of a triangulation realised with the aid of tetrahedra. But for certain particular domains, it can be more convenient to realise the triangulation with the aid of other types of sets K. Hence, if the domain Q is a rectangular parallelepiped, it is practical to associate with it a partition with the aid of rectangular parallelepipeds; similarly, if the domain Q is a right cylinder, it is interesting to associate with it a partition with the aid of prisms (see Fig. 5). Stage 2: Finite Elements. Examples. To each ofthe sets K of the triangulation f/h (tetrahedra, rectangular parallelepipeds, prisms, ... ), we associate: i) a space Pk of finite dimension nK, generally polynomial, ii) a set l:K of nK degrees offreedom, i.e., ofnK linear forms


Fig. 3. Subdivision of the domain Q with the aid of "bricks"

such that:

(1.31) The functions Pi' 1 ::;;; i ::;;; nK , are called the basis functions of the finite element because the relation (1.31) implies:

"K

(1.32) Vp E PK , P = L 4>i(P)Pi i= 1

We then define the P K-interpolation operator (denoted by 7tK ) associated with the finite element (K, PK , };K) by:

"K

7tKV = L 4>i(V)Pi i= 1

for every function v (on Q or on K) such that 4>i(V) is meaningful for i Let us now give some simple examples of finite elements.


1" Subt/;"';sion

/ ~ Subt/il1ision

Fig. 4. Each "brick" is subdivided into two parts, each part is then subdivided into three tetrahedra

Lagrange Finite Elements. These are finite elements for which the degrees of freedom set 1: K consists of values of the function at certain convenient points chosen in the set K, called nodes of the finite element. Let us first consider the case where the set K is a tetrahedron. Let aj' 1 ~ j ~ 4, be four non coplanar points in ~3, with coordinates (alj, aZj, a3j), i.e., we suppose the

l. Finite Element Method Applied to Elasticity

/ /

I I

I /

/ /

/

)-

I I I I I )--II ...... _ //: .................

/ I _---/_--t--I I I

----1 __ /"..... .......... ----

/ I ........... / I - ..........

I I _-/ --:-----

I

-- I

I I

I

I

I I I I

/)..---I

Fig. 5. The beginning of a "triangulation" of a right cylinder with the aid of prisms

following matrix to be invertible:

(1.33) a12 a l3 a22 a23

a32 a33 1 1

183

These four points define a non degenerate tetrahedron. Hence, given a point x in the space 1R3, with Cartesian coordinates (Xl' X 2 ' x 3 ), we define its barycentric coordinates ).j = Aj(X) for 1 ~ j ~ 4 by:

(1.34) {Xi = aijAix) ,

j=l 4

1 = L Aj(X) . j= 1

1, 2, 3 ,

As we suppose the matrix .xl invertible, this system admits a unique solution:

(1.35) 3

Aj(X) = L ajix i + Pj , j = 1, ... , 4 , i= 1

184

where

(1.36)


_ (_ 1)i+ j detgHji (J( .. -JI detd '

p. = (_ 1)j detgHj4 . J detd

The matrices gHji' 1 :::;; i, j :::;; 4 are obtained by scoring out the i-th row and j-th column from the matrix d. We note that the volume of the tetrahedron K is given by the relation:

(1.37) 1 Volume (K) = 61 det .911 .

In the examples which follow, we use the following notations: - Pm denotes the space of polynomials in Xl' X 2 , X3 of degree less than or equal

to m, - Qm denotes the space of polynomials in Xl' X 2 , X3 of degree less than or equal to

m with respect to each variable Xi' 1 :::;; i :::;; 3, - in the figures, knowledge of the value of the function at a point - a node of the

finite element - is indicated by a black dot.

Example 1. Tetrahedron of Type (1)(8)

{ PK = PI; dimPK = nK = 4; EK = {p(a,), 1 ~ i ~ 4} ! for a1l4 PEP I , P = L .l.jp(aj) .

i=l a j : vertices of tetrahedron .

Fig. 6. Tetrahedron of type (1)

In this Example 1, the set LK is the set of the four linear forms:

i: p E PK -+ p(a;) E IR, i = 1, ... ,4 . By definition of barycentric coordinates, we get:

i(Pj) = Pj(ai) = Aia;) = hij , whence the P K-unisolvence property is verified.

(8) The "type" corresponds to the index m.


In an analogous way, in the three examples which follow, the explicit giving of basis functions allows the verification of the P K-unisolvence property.

Example 2. Tetrahedron of Type (2)

{ PK = P 2 ;dimP" = nK = 10;

EK = {p(a.), 1 .;; i .;; 4; p(b.), 1 .;; i .;; j .;; 4}

{ Pfo~_a1l4 p E P2 ' L Ai(2Ai - l)p(ai) + L i=1

4

i,j= 1 i


Whenever the domain is a right cylinder whose axis is in the direction of e;, we frequently use prismatic finite elements as described in Example 4. In order to define the basis functions ofthis element it is convenient to use the barycentric coordinates A=(A 1 , A2 , A3 ) of the base triangle and the height X3'

Example 4. Right Prism of Type (1)

as

a6

~ e2 e t a2 {

PK = PI(xI,x2)@PI(x3);dimPK = nK = 6

EK = {p(a;l, 1 ~ I ~ 6}

Fig. 9. Right prism of type (1)


Example 5. Hermite Tetrahedron of Type (3)

{ PK = P3 ; dimPK = 20; kK = {p(aJ, p(b;), I '" i '" 4; Dp(a;)(a j - aJ, I '" i, j '" 4, i - j} (9),

4

i= 1

with

1~j


The space Vis defined by the relation (1.13), i.e., -+ -+ 3-+-+ (1.38) V = {v = (Vi' v2 ' v3 ) E (Hi (0)) ; V lro = O} .

Taking account of this definition, we now construct a space of finite dimension X" c Hi(Q), then, in stage 4, we will associate with it the space Xo" = {v; V EX", vlro = O}. Then the space v: will be given by v: = Xo" X Xo" X Xo,,' Definition of the Space X" c Hi(lJ). The space X" is a space offinite elements, of finite dimension. Its definition depends: - on the triangulation ff" of the domain 0, introduced in stage 1, - on the given finite element (K, PK , L K ) associated with every set K E ff". Given a function v defined on D, and sufficiently regular, its interpolant v" EX" is defined, in a unique manner, as follows:

i) the restriction v" IK belongs to the space PK , 'V K E ff" , ii) the restriction v"I K is entirely determined by the given set of values

(1.39) L K(V) of the degrees offreedom of the function v (this is a conse-quence of the P K-unisolvence property introduced in stage 2) ,

iii) v" E ~O(D) .

It is convenient to observe that the condition "v sufficiently regular" should permit, in particular, the definition of the set of values LK(V). Naturally, even if the space PK consist of very regular functions (which is the case in the Examples 1 to 5), this does not necessarily imply the inclusion X" cHi (D). The following theorem gives a simple and sufficient condition for this to be so:

Theorem 2. Suppose that the inclusions PK c Hi(K) for all K E ff" and X" c ~O(Q) are satisfied. Then the inclusion X" c Hi (D) holds.

Proof (Analogous to the proof of Proposition 2, 4, Chap. IV). Let v EX" be given. Naturally, v E L 2 (D). By definition of the space Hi(D), we must find, for i = 1,2,3, a function Vi E L2(D) such that:

We now show that the function Vi of which the restriction to each set K equals

O~IK) satisfies the required properties. As the sets K considered in the above uXi

examples have a Lipschitz-continuous boundary, we can apply' Green's formula: for each K E ff", we have:

f O(VIK) f 0", i -!3- '" dx = - vlK -;- dx + VIK ",vi,Kdy , K uXi K uXi DK where Vi,K is the i-th component of the exterior normal unit vector along oK.


Summing over all the sets K E ff,., we obtain:

f ViCP dx = - f v ~CP dx + L f vlKcpvi,Kdy J n n Xi K E:T. ~K To conclude, it suffices to remark that the sum L f vlKcpvi,Kdy is null: either

KE:T ~K a part of aK is to be found on r = aD and in this case cP = 0, or the contribution of two adjacent sets K is null, 0 Henceforth, we assume that the domain D is triangulated with the aid of sets K (tetrahedra, rectangular parallelepipeds, prisms, ' .. ) which are of the same type as that associated with the finite elements being used. Let us make quite clear that this simplyfying hypothesis is not in the least essential and that it is possible to assemble convenient finite elements associated with sets K of different types (see Fig. 11). Let us then verify that the spaces X h associated with a triangulation :Th of the domain D, realised with the aid of tetrahedra of type (1) (see Example 1), satisfy X h c ~O(Q). Let Vh be an arbitrary function in the space X h' Naturally, the restrictions of this function to each of the tetrahedra K E :Th are continuous. It remains to verify this continuity property at the intersections between elements. Thus let K 1 , K2 E :Th be two distinct tetrahedra. The hypothesis (1.30) shows that Kl n K2 is: - either empty, - or a common vertex a j : amongst the degrees of freedom of the finite elements

-----+ / /

/ /

/ /

/ /

Fig. 11. Assembly of finite elements associated with sets K of different type


considered figure the linear form v: ai 1-+ v(a;), whence by definition: vhlK, (ai) = vhIK2 (a;) = v(a;) ,

- or a common edge aiaj : the function traces vhlK,' vh lK2 on the edge aiaj are then polynomials in one variable, of degree 1, entirely determined by the data v(a i ) and v(a j ), whence the continuity,

- or a common face aiajak: an argument analogous to the preceding then establishes the continuity.

The same type of reasoning would show that the spaces X h associated with the finite elements described in Examples 2 to 5 likewise satisfy X h c CCO(Q). Theorem 2 then implies:

Theorem 3. The spaces of finite elements X h associated with the finite elements described in the Examples 1 to 5 satisfy the inclusion X h c HI (Q). 0

The finite element space X h is a space of finite dimension Mh to which we can associate the set of degrees of freedom Lh, i.e., (1.40) In the case of Lagrange finite elements, the set L h is the union of the sets L K' K E 'h' In the case of Hermite finite elements, the definition of the set of degrees of freedom L K is often local: this is the case for Example 5 for which the first partial derivatives are calculated in the direction of each edge of the tetrahedron. It is then convenient to associate to this set of local degrees of freedom, an equivalent set of global degrees of freedom, i.e., defined independently of the tetrahedra K con-

sidered, for example here ~P , i = 1, 2, 3. UX i

Then the set L h is the union of the sets of global degrees of freedom of each element. By analogy with the relation (1.31), the basis functions Whi' 1 :s;; i :s;; M h, of the finite element space X h are defined by the relations:

(1.41) Then the basis functions Whi, 1 :s;; i :s;; M h of the finite element space X hare obtained from the basis functions Pj.K, 1 :s;; j :s;; nK, of finite elements (K, PK, LK) as follows: let cPhk E Lh be a degree of freedom associated with the node bk, and let K;., A E A(bk ) be the set of all finite elements having bk as a common node. For each A E A(bk ), let P;. be the basis function of the finite element K;. associated with the restriction of cPu to K;.. Then the function Whk E X h defined by:

(1.42) {P;' on K;., A E A(bk ) , Whk = o otherwise, is the basis function of the space X h associated with the degree of freedom cPu Hence, to every sufficiently regular function v: Q -+ IR in order that the degree of freedom cPhj(V), 1 :s;; j :s;; M h, shall be well defined, we associate the function 1thV, called the Xh-interpolant function (or simply the interpolant in X h) of the


function v, i.e.,

(1.43) Mh 1ChV = L cPhj(V)Whj .

j= 1

191

In conclusion, the construction of the finite element space X h makes evident the three characteristic principals of the method: i) the triangulation of the domain D, ii) the functional spaces PK (K E .9'"h) are polynomial; we shall see that this property is essential for the study of convergence properties; iii) the support of the basis functions are also as "small" as possible, whilst conforming to the relation (1.41). For the examples which we have considered, the "largest" support corresponds to the basis functions associated with a node vertex of the set K; in this case, the support is the union of tetrahedra having this node for vertex.

Stage 4: Taking Account of Boundary Conditions. Let the subspace X Oh C X h be constructed such that:

(1.44) X Oh = {vh; vh E X h; vhlro = O} . To do this, we suppose that the part ro of the boundary r coincides with the union of certain faces of the finite elements of the triangulation considered. Then, in order that the functions Vh of the space X h should satisfy the condition Vh I ro = 0, it suffices to assume that: i) in the case of a Lagrange interpolation, (1.45) for all nodes b situated on ro; ii) in the case of a Hermite interpolation (see Example 5),

(1.46) { vh(a) = vh(b) = 0 , (resp. DVh(a). t = 0) , for all nodes a and b situated on ro (resp. for all the vertices a E r 0 of the tetrahedra K E .9'"h having a face K' c ro and for all the directions t contained in the face K'). The space XOh being constructed thus, we then deduce from it the finite element

-+ h' space Vh soug t, t.e., -+ (1.47) Vh = X Oh X X Oh X X Oh

o Remark 3. In stage 2, we have indicated that the functional space PK is generally polynomial. There exist other possible choices: thus Zienkiewicz (Zienkiewicz [1], Chap. X) gives an example of a two-dimensional finite element for which the functional space PK includes a polynomial part completed by rational fractions. In this direction, we could likewise consult Apprato [1], Apprato-Arcangeli [1], Apprato et al. [1], Gout [1] and Wachpress [1]. 0


Kemark 4. In the examples of finite elements which we have considered, the degrees of freedom are associated with a fixed node of the element. There exist other possibilities, for example the degrees of freedom of the type:

4J: v ....... t v(x) dx . Such finite elements have been used by Crouzeix-Raviart [1], Raviart [1], in dimension 2, for other types of problems. 0

3. The First Approximation Problem (Phi)

IE. the precedin-& section, we have constructed a finite dimensional subspace Vh of the space V. It is then natural to associate with the continuous problem (1.17) the following approximation problem (Phd: (1.48) {for P E (L 2 (.0))3, for q ElL 2 (rd)3, find u: E v: such that:

a( U:, V;) = f( V;), 'if V; E Vh , where the bilinear form a(. , .) and the linear form f(.) are defined in (1.11) and (1.12) respectively. TheOrem 4. The problem (1.48) admits a unique solution.

~ Proof Endowed with the norm induced by that on the space V, the finite dimensional space v: is a Hilbert space. From analogous properties holding in the space v: we deduce the continuity and uniform V:-ellipticity of the bilinear l" ( ~ ~) ~ ~ ~ -+ .. f h 1 lorm uh , Vh E Vh X Vh -+ a( uh , vh ), as well as the contmUlty 0 t e mear form V; E v: ~ f( V;). Then the Lax-Milgram theorem implies the existence and uniqueness of a solution for the problem (1.48). 0 Following the relation (1.43), every element vh E X h can be written: (1.49) Mh

vh = L 4Jhr(Vh )Whr But u: = Uhi e:, on using the convention of summation over repeated indices. Let us suppose that the three components Uhi, i = 1,2,3 are approximated with the aid of the same space X h of finite elements; then the relation (1.49) implies:

Mh (1.50) Uhj = L 4Jhr(Uhj) Whr, j = 1, 2, 3 .

r=l

In particular:

(1.51 ) OUhj Mh ow ~ = L 4Jhr(Uhj) :l hr , j, k = 1, 2, 3 . uXk r= 1 uXk

But in the statement (1.48), the functions u: and V; must belong to the space ~ . Vh = X Oh X X Oh X X Oh where, after relatlOn (1.44),


The relations (1.45)-(1.46) show that taking into account the boundary condition Vh I Fa = 0 allows one to reduce the number of unknown degrees of freedom to M;'. The hypothesis meas (ro) > 0 especially implies M;' < M h If we adopt a number-ing of the degrees of freedom which starts with the M;' unknown degrees of freedom, then the relations (1.11) and (1.12) can be written:

(1.52)

(1.53)

--+ --+ for all u: E Vh , for all v: E Vh Hence, the problem (1.48) is equivalent to solving the linear system:

(1.54)

This system of 3M;' equations in 3M;' unknowns cPhr(Uh) admits a unique solution in virtue of Theorem 4. It is interesting to note that the coefficients of the system's matrix, i.e.,

(1.55)

are null whenever the supports of the basis functions Whr and Whs have an intersection of measure null. For the examples which we will consider, this is the case whenever the basis functions Whr and Whs are associated with nodes which do not belong to the same set K. Hence the system matrix is sparse, which is very important when it comes to numerical solutions.

Remark 5. In practice, we adopt a point of view slightly different: in the relations (1.54), we write:

In ( )dx L r ( )dx , LEy.JK

f () dy = L f () dy , r, K' E r, K'

where K' denotes the faces of the sets K situated on r 1 Then we can calculate the contributions of each element K to the global matrix of the system and to the second global member. This method will be given in detail in Sect. 6 below. 0


4. Numerical Quadrature Schemes and the Definition of the Second Approximation Problem (Pu )

The solution of the system (1.54) supposes the exact calculation of the coefficients (1.55) and of the second member of the system (1.54). Generally this calculation is expensive, and sometimes even impossible. This is why we are led into making use of numerical quadrature schemes. In order to define these schemes, it is convenient to introduce the notion of a reference tetrahedron K: this is a tetrahedron which can be chosen arbitrarily in the space 1R3 and which is such that every tetrahedron K E ff", assumed to have non empty interior, is the image of the reference tetra-hedron K under an invertible affine mapping F K' defined by: (1.56) FK: X E K 1--+ Fx(x) = BKx + bK (where BK is an invertible matrix and bK a vector in 1R3 ), such that: (1.57) F K(d i ) = ai' i = 1, ... ,4 , the points ai' ai being the vertices respectively of the tetrahedra K and K. This notion of a reference tetrahedron, as well as the definitions and properties which follow extend without difficulty to the cases of rectangular parallelepipeds or prisms. Henceforth, we suppose for simplicity that the triangulation is realised with the aid of tetrahedra. Let us consider a numerical quadrature scheme on the reference tetrahedron K (for examples, one can consult Stroud [1], 8.8):

(1.58) fA ~(x) dx '" i W, ~(b,) , lie 1=1 where b" w, denote respectively the nodes and the weights of the scheme(1O). To the function cP defined on the tetrahedron K, we associate the function ~ defined on the tetrahedron K by ~ = cP. F K and inversely cP = ~. F ii, where F K : K -+ K denotes the affine mapping defined by the relations (1.56)-(1.57). We then have:

(1.59) Ix cP(x)dx = det(BK) Ii ~(x)dx , where, without loss of generality, we suppose the Jacobian (constant) of the mapping F K is positive, i.e., det (BK ) > O. Hence the numerical quadrature scheme (1.58) on the reference tetrahedron K, induces the following numerical quadrature scheme on the tetrahedron K:

(1.60) f cP(x)dx '" i (J)1.KcP(b',K) , K 1=1 with:

(1.61) (J)',K = det(BK)w, and b"K = FK(I;,) , 1 ~ I ~ L .

(10) h, is also called a quadrature node.


In parallel, we define the error functionals:

(1.62) EK(cP) = L cP(x)dx - J1 W"KcP(b"K) , E((P) = fA (P(x)dx - WI (1J/) ,

K 1=1 (1.63)

so that:

(1.64) Thanks to the numerical quadrature scheme (1.60), we can evaluate the integrals (over the domain D) which figure in the expressions (1.11) and (1.12). It remains to introduce numerical quadrature schemes permitting the evaluation of the expressions:

(1.65) f -qltdy, F,

which is found in the second member of the relation (1.12). In this connection we observe that the part r 1 ofthe boundary r = aD is the union of certain faces of the finite elements in the triangulation !Th considered, in accordance with the analog-ous hypothesis on the complementary part ro = r\r1 , formulated in Sect. 2, stage 4. As we are assuming that the triangulation !Th is realised with the aid of tetrahedra, the boundary r1 is the union of triangles K', i.e.,

(1.66) r1 = U K'. K' E ff.

By analogy with the notion of a reference tetrahedron, let us introduce the notion of a reference triangle f(': this is a non degenerate triangle which can be chosen arbitrarily in the space 1R2. For simplicity, let us take for example the unit

a,

x,

Fig. 12. Parameterisation of the boundary faces K' c r,


rectangular triangle (0, e;, e;) in the plane. Hence, every triangle K' E .r;, is the image of the reference triangle K', by an invertible affine mapping GK ,: if we denote by ai' j = 1, 2, 3, the vertices of the triangle K', with coordinates (ali' a2i' a3i) respectively, we obtain (see Fig. 12): (1.67)

{

Xl = a 13 + (all - a13 )X I + (a 12 - a13 )X2 , GK ,: X E i' 1-+ Gdx) = X2 = a23 + (a21 - a23 )x I + (a22 - a23 )x2 ,

X3 = a33 + (a 31 - a33 )x I + (a 32 - a33 )x2 . Inversely, from the vectorial relation:

-----+ ---+ A---+ a3M = Xl a3al + X2 a3a2 ,

we deduce the definition of the mapping Gi' I: (1.68)

1 -- ____ __ [A(K')]2 {(a3M 1\ a3a2)(a3al /\ a3a2)}

1 -- ____ __ [A(K')]2 {(a3M 1\ a3ad(a3a2 1\ a3 al )}

where we put:

(1.69) A(K') = 1a;ti7 /\ a;a; I (11) we observe: i) that the hypothesis k #- 0 implies A(K') #- 0, which justifies the definition (1.68), ii) that the relations (1.68) allow us to find easily the expressions for Xl and x2 as linear combinations of the coordinates X I' X2' X3 of the point M. Let us consider a numerical quadrature scheme on the reference triangle ie (for examples, we can consult Laursen-Gellert [1], Lyness-Jespersen [1], Stroud [1], 8.8, or refer to Fig. 15 below):

f L' ~ ~(x) dx '" L w; ~(6D , K' 1= I

(1.70)

where 6; and wi denote respectively the nodes and weights of the scheme. To the function

I. Finite Element Method Applied to Elasticity 197

where L1(K') is given by the relation (1.68), namely:

(1.72) t, 4>(x)dy = L1(K') Ii' c,b(x)dx1 dX2 Consequently, the numerical quadrature scheme (1.70) on the reference triangle K' induces the following numerical quadrature scheme on the triangle K':

(1. 73) t, 4> (x) dy - J1 wi, K' 4>(b;' d , with

(1.74) Wi,K' = L1(K')wi and bi,K' = GK,(bl) , 1 ~ I ~ L' . By analogy with the relations (1.62)-(1.63), we define the error functionals:

(1.75) f L' Eid 4 = K' 4> (x) dy - 1~1 wi,K' 4> (bi, d , (1.76) so that:

(1.77) Eid4 = L1(K')1f'(c,b) . Use of the numerical quadrature schemes (1.60) and (1.73) leads to the definition of the following approximate bilinear and linear forms (compare with the relations (1.11) and (1.12) respectively):

(1.78) ah( U:, 1\) = L WI,K(Cjklm ~Uhj :Vhl) (b"

K) , KE:r.

'=1 X k Xm

L v

(1.79) f,,(V:) = L L W"K(Pjvhj)(b"K) + L L wi,dqjvhj)(bi,d . K E :r. 1=1 K' E :r; 1=1

Whence the statement of the second approximation problem (Ph2): (1.80) {find u: E V; such that, for all v: E V;

ah( U;, v:) = f,,( V:) , 5. Error Estimates

The domain Q being assumed polyhedral, it is legitimate to cover it exactly with families of triangulations fi,. composed, for example, of tetrahedra or rectangular parallelepipeds or prisms. We suppose that these triangulations satisfy the hy-potheses (1.30) stated in the case of a triangulation realised with the aid of tetrahedra. Furthermore, we suppose that the family of triangulations fi,. is regular, i.e.,

U hk (1.81) there exists a constant IX such that: V K E fi,., - ~ IX , h PK


where hK = diam(K) and PK = sup {diam (B); B is a ball contained in K},

(1.82) the quantity h = max hK approaches zero . KE :T.

To each of these triangulations ',., we associate a space of finite elements -+ Vh constructed according to the method described in Sect. 2 above. In this sections, we solve the following two problems:

i) show that the problem (Ph2 ) formulated in (1.80) admits a unique solution. To do -+

this, we must establish that the bilinear form ah(.,.) is uniformly Vh-elliptic;

ii) find sufficient conditions such that lim II it - U;II = 0, where the norm 11.11 has h~O

been defined by the relations (1.14)-(1.15). In fact, we give sufficient conditions such that if the interpolation error I v - 1tK vl,. K on each finite element is of order O(h~), then the error between the solution it of the problem (1.17) and the solution U; of the problem (1.80), in norm II. II, is of the same order, i.e. II it - U;II = O(hfJ). This study comprises the following eight stages:

Stage 1: Abstract Estimate of the Error. To prepare the way to obtaining explicit estimates of the error, we give in the following theorem an abstract majorisation of the error:

Theorem 5. Given a family of discrete problems (1.80) for which the bilinear forms -+

ah(. ,.) are Vh-elliptic, uniformly with respect to h, i.e., there exists a constant (X > 0, independent of h, such that: (1.83) Then there exists a constant C, independent of h, such that: ! -+ -+ [. {-+ -+ la(V;, ~) - ah(V;, ~)I} II u - Uh II :::;; C -!nf -+ II u - Vh II + sup -+ II ~ II (1.84) v. E V. W. E V.

+ If(~) - f,,(~)IJ sup -+ II ~ II ' w. E v.

where it (resp. U;) denotes the solution of problem (1.17) (resp. (1.80)). -+

Proof The hypothesis of Vh-ellipticity (1.83) ensures the existence and uniqueness of a solution U; for ,Ute discrete problem (1.80). Hence let V; be an arbitrary element of the space Vh' We can then write:

(XII U; - V;1I 2 :::;; ah(U; - V;, U; - V;) = a(rr- V;, U; - V;) + [a(V;, U; - V;)

- ah(V;, U; - V;)] + [fh(U; - V;) - f(U; - V;)] , so that the continuity of the bilinear form a(.,.) implies, on denoting by M the


constant of continuity,

V;)I

~ Mil it - V;II w,,)1

On combining this inequality with the triangle inequality:

II it - U:II ~ II it - V;II + II V; - U:II , ~

and taking the infimum with respect to V; E Vh , we then obtain the inequality (1.84). 0 Remark 6. If we do not take numerical integration into account, i.e., if we only consider the family of discrete problems (1.48), then the estimate (1.84) is reduced to the error estimate:

II it - U:II ~ C inf II it - V;II . V:e v:

o In order to obtain an explicit error estimate, it is convenient to verify property (1.83) and to evaluate each ofthe two terms on the right hand side of the inequality (1.84). Successively,:

i) the first term jnf-+ { II it - V;II + sup la( V;, w,,) ~ ah ( V;, w,,)I} takes v. E V. WE v II W h II

. . ~ account of the distance between the solution it and the subspace Vh on the one hand, and the consistency(13) of the approximation scheme of the bilinear form a( . , .) on the other. To get an upper bound for it we can write:

(1.85)

(13) In a somewhat intuitive fashion (but which will be made more precise during the development ofthe method), we will speak of "the consistency of an approximation scheme" if the solutions u. of the first approximation problem p., (see (1.48)) and the second approximate problem p., (see (1.80)) converge (or not) simultaneously to the solution u of the problem (1.17), this "consistency" being directly related to the replacement of a and f by a. and f .


~ -+ on denoting by 7th U = (7thU1 , 7thU2, 7thU3 ) the element of the space Vh whose components are the interpolations of the components U1, U 2 , u3 of the vector itin the finite element space X Oh (see (1.43), (1.44)). Since we have:

3 (1.86) II it - n;;u+ 112 = L L IIUi - 7tKuilll.K ,

i=lKE!r,

we are reduced to the problem of evaluating the local interpolation error Ilu - 7tKulll,K' To solve this problem (see stage 3), it is helpful to introduce the notion of a regular affine family of finite elements (see stage 2). The term

I (~ -+) (~-+)I a 7th u , Wh - ah 7th u , Wh suP.... II ~II

>it. E v, estimates the consistence of the approximation scheme of the bilinear form a(.,.). To get an upper bound for it, we state in stage 4 some sufficient conditions on the numerical quadrature schemes which imply suitable estimates of the error func-tional (1.62). The derivation ofthese estimates rests on the Bramble-Hilbert lemma (see Lemma 1, below).

If(w) - f,(w)1 . . ii) The second term sup h -+ h h estImates the consistence of the WE v II whll

approximation scheme of the linear form f( .). To get an upper bound for it, we use the sufficient conditions from stage 4 which allow one to obtain suitable estimates of the error functionals (1.62) and (1.75).

-+ -+ iii) We deduce the property of uniform Vh-ellipticity (1.83) from the V-ellipticity property (1.19), satisfied by the bilinear form a(. ,. ), on making use of the estimates of the error functional (1.62) stated in ii). This investigation forms the object of stage 5. On gathering together these different results, we obtain in stage 6, the explicit error estimate. Finally, in stages 7 and 8, we give respectively examples of error estimates and examples of numerical quadrature schemes.

Stage 2: Regular Affine Families of Finite Elements. In this and the following stage, we recall the theory of interpolation by finite elements in Sobolev spaces, due to Ciarlet-Raviart [1]. In order to do this, we follow the presentation by Ciarlet [1], 3.1. We make use of the definition of a finite element (K, P, 1') given in stage 2 in Sect. 2 above, assuming for example (it is possible to generalise this), that the set of degrees of freedom l' consists of linear forms of the type:

{ cp?: p 1-+ p(a?) , (1.87) cP~: p 1-+ Dp(al )~ik , where the nodes a?, al belong to the set K and where the non null vectors ~ik are either fixed vectors in 1R 3, or vectors depending on the geometry of the set K. Hence, we introduce the following definition:

Definition 1. Affine-Equivalent Finite Elements. Two finite elements (K, ft, 1:) and (K, P, 1') with degrees of freedom of the form (1.87) are called affine-equivalent if

1. Finite Element Method Applied to Elasticity

there exists an invertible affine mapping: (1.88) F: X E JR3 -+ F(x) = Bx + b E JR3 , such that the following relations are satisfied: (1.89) (1.90) (1.91)

K = F(K) , P = {p: K -+ JR;p = fi.F-1,fiEP} ,

{ af = F(a!) , (X = 0, 1 , ~ik = B(~ik) ,

201

where the nodes af (resp. an and the vectors ~ ik (resp. [ik) derive from the definition of the set 1: (resp. t). 0 We make constant use of these correspondences:

(1.92) (1.93)

X E K -+ x = F(x) E K , fi E P -+ P = fi. F -1 E P (14)

between the points x E K and x E K, and the functions fi E P and PEP associated with two affine-equivalent finite elements. In particular, the relations (1.92)-(1.93) imply:

(1.94) fi(x) = p(x) , \:Ix E K, \:Ifi E P . Let us denote by it (resp. n) the operator of P-interpolation (resp. of P-interpola-tion) associated with the finite element (K, P, t) (resp. (K, P, 1:. Then if (j: K -+ JR, v: K -+ JR denote sufficiently regular functions, associat$!d through the correspon-dence v = (j.F- 1, we have: (1.95) ........... nv = {t(j , a property which proves to be essential in the rest of this section. To verify this relation (1.95), it is enough to observe that the basis functions of the finite elements (K, P, i) and (K, P, 1:) are associated through the correspondence (1.93). Definition 2. Regular Affine Family of Finite Elements. Afamily offinite elements is said to be a regular affine family if the following two conditions are satisfied: i) the family of triangulations fI" is regular, i.e., satisfies the relations (1.81) and (1.82); ii) the finite elements associated with all the sets K E fI" are affine-equivalent to the same finite element (K, P, t), called the reference finite element of the family. 0 Stage 3: Estimate of II u-+- n;;u II. To estimate the first term of the inequality (1.84), the inequality (1.85) shows that we can first estimate the term II ~ - n;;u II. The relation (1.86) reduces this estimate to that of the local interpola-tion error II it - n;;u II"K associated with the finite element (K, P K' 1: K) (14) We have also the correspondence (besides 1.92) and (1.93)) between i and 1::

4>i E i ..... cPi E 1:. with cPi(P) = cPi(fi. r 1) = 4>i(P} (which will give (1.95)).


The study of this problem necessitates the recall of some facts concerning the theory of interpolation in the Sobolev spaces wm P(Q): let us recall (see Sect. 4 in the Introduction, and Chap. IV) that for every integer m ~ 0 and every real number p, 1 ~ p ~ 00, the space wm P (.0) is defined by: (1.96) wmP(.o) = {v; v E U(.o), olZv E U(.o), IIlI ~ m} . Endowed with the norm:

(1.97) {II v 11m. P. n = ( L i I olZvl P dx)llP , 11Z1~m n

Ilvllm.


Proof Let N = dim Pk(Q) and let /;, 1 ~ i ~ N, be a basis for the dual space of Pk(Q). The Hahn-Banach extension theorem implies the existence of continuous linear forms on the space Wk+ l,p(Q), again denoted by /;, 1 ~ i ~ N, such that for all p E Pk(Q), we have /;(p) = 0, 1 ~ i ~ N, if an only if p = O. We shall show that there exists a constant C(Q) such that: (1.104) {N } Ilvllk+1,p,n ~ C(Q) Ivlk+1,p,n + i~ll/;(v)1 , Vv E Wk+1.P(Q) . If this inequality is false, then there exists a sequence {v,} of functions v, E Wk+ l,p(Q), such that: (1.105) VI ~ 1, IIv,IIk+1.p,n = 1, and

,~i~oo {lv,Ik+1,p,n + J1 I/;(V,)I} = 0 . Since the sequence {v,} is bounded in W k + 1, P(Q), there exists a subsequence, again denoted by {v,} and a function v E Wk, P(Q) such that:

(1.106) lim II v, - Vllk,p,n = 0 . l~ +00

This result is a consequence of the theorems of Kondrasov (the case where p E [1, + 00 [, P "" 2), and Rellich (p = 2) and Ascoli (p = + 00) see Necas [1], for example). The relations (1.105) particularly imply:

(1.107) lim Iv,lk+1,p,n = 0 . 1- +00

The space W k + l.P(Q) being complete, the relations (1.106)-(1.107) imply the strong convergence of the sequence {v,} in the space W"+l'P(Q). The limit v of this sequence is such that:

lo"'vlo,p,n = lim lo"'v,lo,p, n = 0, Va with lal = k + 1 , 1- +00

whence o"'v = 0 for every multi-index a such that lal = k + 1. The connectivity of the open set Q and the theory of distributions (see Schwartz [1]) implies that the function v is a polynomial of degree less than or equal to k. Using (1.105), we have:

/;(v) = lim /;(v,) = 0 , I- + 00

whence v = 0, in view of the properties of the linear forms /;. But this result contradicts the equality IIv,lIk+"p,n = 1 for alII. The inequality (1.104) is therefore established. The inequality (1.103) is an immediate consequence of the inequality (1.104): for every function v E Wk+1,p(Q) let us denote by q E Pk(Q) the polynomial such that /;(v + q) = 0, 1 ~ i ~ N. Then the inequality (1.104) implies:

inf IIv + pllk+1,p,n ~ IIv + qllk+1.p,n ~ C(Q)l vlk+1,P, n pePk


The equivalence of the two inequalities (1.102) and (1.103) then follows from the definitions (1.100) and (1.101). 0

We now demonstrate the following general result about approximation:

Theorem 7. Let Q be a bounded open set in 1R3 with a Lipschitz-continuous boundary, let k be an integrer ~ 0 and m an integer with 0 ~ m ~ k + 1. Let W k + l'P(Q) and wm,q(Q) be Sobolev spaces satisfying the inclusions: (1.108) and let n E .!e(Wk+ l,p(Q); wm,q(Q)} be a continuous linear operator satisfying: (1.109) 'rI rEP k' nr = r . Then, there exists a constant C(Q) such that: (1.110) Iv - nvlm,q, n ~ C II I - n 1I..r(w'+1.P(Q); w .. q(Q Ivlk+ l,p, n ,

'rIv E W k + l,p(Q) .

Proof For each v E Wk + 1, P(Q) and each r E Pk , we can write: v - nv = (I - n)(v + r) .

Hence, for all r E Pk , we have:

Iv - nvlm,q, n ~ II v - nv Ilm,q, n ~ III - nll..r(wt+I,P(Q); w .. q(Q)) Ilv + rllk+l.p,n

Therefore, using inequality (1.103), we get:

Iv - nvlm,q,n ~ III - nll..r(WH1.P(Q); w ... q(Q inf IIv + rllk+l,p,n, rePk

~ CIII - nll..r(wH1.P(Q); w .. q(Q Ivlk+l,p,n, for all v E Wk + l,p(Q). o It is now convenient to estimate the error constant which appears in the upper bound (1.110), i.e., CII I - n 1I..r(.,.), as a function of the geometr~c characteristics of Q. To do this, we introduce the notion of the reference open set Q which generalises the notion of the reference tetrahedron K used in Sect. 4 above.

Definition 3. Two open subsets Q, Q, in 1R3 are said to be affine equivalent if there exists an invertible affine mapping F, i.e.: (1.111) F: X E 1R3 -+ F(x) = Bx + b E 1R3 , (where B is an invertible linear mapping from 1R3 into 1R3, and b is a vector in 1R3) such that: (1.112) Q = F(Q) .

o


Hence, to every point x E .6, the bijective mapping F associates the point x = F(x) of Q, and inversely x = F - I (x). Similarly, with a function v defined on Q, we associate the function v defined on .6 by: (1.113) v = v.F , and inversely:

(1.114) v = v.F- 1 The following theorem gives an upper bound for Ivlm,p,n as a function of Ivlm,p, n, and inversely.

Theorem 8. Let Q, Q, be two open subsets oflJl3, which are affine-equivalent. Then, there exist constants C, C such that,for all v E Wm,P(Q),

I

(1.115) Ivlm,p,n ~ CIIBllm Idet (B-1)IPlvlm,p,n, and for all v E W m,P(.6):

1 (1.116) Ivlm,p, n ~ CII B- 111 m Idet (B)p Ivlm,p. n , where B is the invertible matrix defined by the relation (1.111), and liB-III = sup II B -1 ~ II where II. II is the Euclidean norm.

Ilell = 1

Proof Since the space ICm(Q) is dense in Wm,P(Q), it suffices to establish the results for all functions v E ICm(Q). It is convenient to use Frechet derivatives (see Cartan [1], 1.2.6). For every multi-index IX with IIXI = m, we have:

a~v(x) = Dm v(x). (ela, ... , em~) , where the vectors ei~, 1 ~ i ~ m, are the basis vectors (e 1 , e2 , e3) of 1Jl3 repeated respectively 1X1' 1X2' 1X3 times. Hence:

Wv(x)1 ~ IIDm v(x)1I = sup IDm v(x). (~1' ~2' ... '~m) , 11M ~ I

l~i~m

whence the existence of a constant C I depending only on m and such that:

The properties of differentiation of composed functions, applied to the relation

v = v. F, give for all the vectors ~i E 1Jl3, i = 1, ... , m: Dmv(x). (~I' ... , ~m) = Dmv(x). (B~I' ... , B~m) ,

so that:

206

Whence

(1.118)


fn IIDmv(x)IIP dx ::;; IIBllmp LllDmV(X)llPdX IIBllmp ldet(B- l )1 til Dmv(x)llPdx ,

on using the properties of change of variables in the calculus of multiple integrals. But there exists a constant C2 (m) such that:

II Dmv(x) II ::;; C2 (m) max 1t3"v(x)1 , l"l=m

whence we deduce: 1

(1.119) (In IIDmv(x)IIPdx y ::;; C2 (m)l vlm,p,n . We then obtain inequality (1.115) on putting together the inequalities (1.117) and (1.119). The proof of the inequality (1.116) is entirely analogous. 0 To make use of Theorem 8, it is convenient to evaluate the norms IIBII, liB-iii and the expressions Idet (B)I, Idet (B- l )1 as functions of the characteristic geometries of the open sets Q and Q. Let us denote by: (1.120)

(1.121)

h = diam(Q), h = diam(Q) ,

{ p -= sup {diam (~); ~ is an open ball contained in~} , p = sup {diam(S); S is an open ball contained in Q} . Theorem 9. Let Q and Q = F(Q) be two affine-equivalent open subsets in [R3. If P > 0 (which is equivalent to p > 0, or again to Q not being contained in a hyper-plane of [R3), we have these upper bounds:

(1.122)

and:

(1.123)

Proof

IIBII h ~ -; ,

p h

::;; -, p

Idet (B)I = meas (~), Idet (B-l)1 = meas ~~~ . meas (Q) meas

We can write: 1

IIBII = -;- sup IIB~II . p WI={i

By definition of p, there exists a ball of diameter p contained in Q, i.e., 'v' ~ E [R3 with II ~ II = p, 3y, Z E Q such that ~ = y - Z .

Then:

B~ = By - Bz = (By + b) - (BZ + b) = y - z ,


with y, Z E Q. Since h denotes the diameter of Q, we have II y - Z II ~ h, whence: 'rI e E ~3 with II e II = p, IIB~II ~ h .

Hence II B II ~ ~. The second inequality (1.122) is derived in a similar way. The P

equalities (1.123) are direct consequences of the change of variables in the multiple integrals. 0

We are now in a position to establish the principal result of this stage 3: Theorem 10. Let (K, P, ) be a finite element and s the maximal order of the partial derivatives intervening in the definition of the set of degrees of freedom . If the following inclusions hold for certain integers k ~ 0, m ~ 0 and certain real numbers p, q E [1, + 00], (1.124) Wk+ 1,P(K) c; rt'"(K) , (1.125) (1.126)

w k+ l,P(K) c; wm'4(K) , Pk c P c wm'4(K) (15)

then there exists a constant C(K, P, ) such that, for every affine-equivalent finite element (K, P K' l' K) and every function v E Wk+ 1,P(K), we have:

_ _ _ 1 1 h1+ 1 (1.127) Iv - 1tKvlm,q,K ~ C(K, 1', P)(meas (K))q-p P~ Ivlk+ 1,p,K ,

where 1t KV denotes the P K-interpolant function of the function v and

{ hK = diamK , (1.128) PK = sup {diam(S), S is a ball contained in K} . Proof Let it be the P-interpolant operator associated with the finite element (K, P, f). By definition, itp = p, 'rip E P, whence in virtue of the inclusion (1.126): (1.129) itp = p, 'rip E Pk Then let fJ be a function in the space Wk+ 1,P(K). The hypothesis (1.124) allows us to define the degrees of freedom (fJ) of the function fJ. On restricting ourselves to the case s = 1, which allows us to take account ofthe Examples 1 to 5, we can write:

(1.130) itfJ = L fJ(a?)p? + L (DfJ(al)!ii)Pii , i,k

each summation being finite and fJ(a?), DfJ(at )!h. denoting the values ofthe degrees of freedom of the function fJ, and p?, Ph. denoting the associated basis functions. Let us show that it is a continuous linear operator from ~+ 1,P(K) into wm,q(K). The hypothesis (1.126) implies that all the basis functions of the relation (1.130) are in the space wm'4(K). Hence:

lIitfJllm,4,K ~ LlfJ(a?)llIp?lIm,4,K + LIDfJ(al)!h.IIlPh.llm,4,K. i i,k

(IS) Because K and K are closed, we use here and subsequently, for simplicity, the notations W",P(K) and W",P(K) in place of W",P(K) and w .. p(k).


The inclusion with continuous injection (1.124) implies: Iv(a?)1 ~ CJivllk+1.p,K, I Dv(af) !tk I ~ Cllvllk+l,p,K ,

whence

(1.131) Thanks to the relations (1.129) and (1.131), the operator it satisfies the hypotheses of Theorem 7. Consequently, there exists a constant C = C(K, P, i) such that: (1.132) The relation (1.95) shows that nv = ';;:v, where 7tK denotes the PK-interpolation operator associated with the finite element (K, P K' L K)' Hence, v - nv = ~ If we denote by F K the affine mapping which associates K with K, i.e., from (1.88), F K(X) = BKx + bK, Theorem 8 implies:

I (1.133) Iv - 7tKvlm,Q,K ~ C1IBi l ll m Idet(BK)lq Iv - nVlm,q,K .

The same theorem again implies: I

(1.134) Ivlk+ l,p,K ~ C IIBd k + I Idet (Bi I )Iq IVIk+ l,p,K .

On combining the inequalities (1.133), (1.132), (1.134) and the conclusions of Theorem 9, we obtain the upper bound (1.127). 0 In particular, for a regular affine family of finite elements (see Definition 2), Theorem 10 can be written:

Theorem ll. Suppose we are given a regular family of finite elements for which the reference finite element satisfies the hypotheses (1.124) to (1.126). Then there exists a constant C such that, for all finite elements in the family, and for all functions v E W k+ I,P(Q), we have:

(1.135)

Proof It suffices to combine the results of Theorem 10 and the inequality (1.81). 0 We are now in a position to be able to estimate the term II it - n,;u II in the inequality (1.85): Theorem 12. Suppose that the finite element spaces X h defined in Sect. 2 are associated with a regular affine family of finite elements in the sense of Definition 2. Suppose also that the corresponding reference finite element satisfies the hypotheses: (1.136) Pk c P C HI (K) , (1.137) Hk+I(K) c+ CCS(K) , where s is the maximal order of the partial derivatives intervening in the definition of


- ~ the set of degrees of freedom L and that k is an integer ~ 1. Let Vh be the finite dimensional space defined by the relations (1.44) and (1.47), so that we have

~ ~ ~ Vh C V, where the space V is defined by the relation (1.13). Then, if the solution it of the problem (1.17) belongs to the space (Hk+1(.Q))3, there exists a constant C independent of h such that: (1.138) II it - ~ II ~ Chk I itl(HH1(D3 Proof. The relation (1.86) can be written:

II it - ~ II = (. L Ilui - 1tKUdlI.K)1/2 l=lKEff.

In order to evaluate the local interpolation errors Ilui - 1tKUi 111. K it is legitimate to use Theorem 11 with p = q = 2: there exists a constant C, independent of hK' such that:

Ilui - 1tKUJ1,K ~ Chl:lui lk+1,K, i = 1,2,3 . Consequently, using the hypothesis (1.82), we have:

( 3 )1/2

.L L lIui - 1tKui llI.K 1=1 KE ff.

whence the upper bound (1.138). o In the light of this Theorem 12, which furnishes the desired estimate, it is appropri-ate to make the following three observations:

i) the estimate (1.138) is valid for every it E (Hk'; 1 (.0))3; ii) Theorem 2 in the Introduction (Sobolev's continuous injection theorem) shows that the inclusion (1.137) holds if k > s + ~. In particular, for Lagrange finite elements (s = 0), this inclusion is always satisfied (we have assumed k ~ 1); iii) in the particular case where the integrals are calculated exactly, the error estimate (1.84) reduced to:

II it - UZII ~ C inf II it - v: II , V;:EV:

where it (resp. UZ) denotes the solution of the problem (1.17) (resp. of the problem (Phd, i.e. (1.48)). Then Theorem 12 gives the following error estimate: (1.139) This case is not without interest: in practice it is possible to calculate exactly the integrals whenever the following conditions are realised: (1) the space P is a space of polynomials of fairly low degree; (2) the coefficients of the bilinear form a(. , .) defined in (1.11) are constants, which corresponds to the case of an isotropic, homogeneous material, i.e., the coefficients are given by (1.6) with A and J.l con-stants; (3) the fields of force of density p and q which appear in the expressions (1.12) admit very simple primitives.


Subsequently, we take numerical integration into account and we establish suffi-cient conditions on the choice of quadrature schemes in such a way as to obtain an error estimate of the same order as that obtained in the case of exact integrations, i.e., (1.139). These sufficient conditions further assure that the uniform tT;-ellipticity property (1.83) is satisfied. Stage 4: Local Error Estimates. ~The following two theorems allow us to establish the property (1.83) of uniform Vh-ellipticity and to evaluate the two consistency terms in the inequality (1.84). To establish these two theorems, we use the lemma of Bramble-Hilbert [1] in the form given by Ciarlet [1]: Lemma 1. The Bramble-Hilbert Lemma. Let Q be an open subset in ~n, n = 2, 3 with Lipschitz-continuous boundary. For an integer k ~ 0 and a real number q E [1, 00], let f be a continuous linear form on the space Wk+ 1.q(Q) such that: (1.140) 'Vp E Pk , f(p) = 0 . Then, there exists a constant C(Q) such that: (1.141) 'VVE Wk+l.Q(Q) , 11MI ~ C(Q)IIfllt+l.Q.nlvlk+l.q,n, where II. IIt+ l,Q, n is the norm in the dual space of wk+ l,Q(Q). Proof Let v be an arbitrary function in the space wk+ I, Q(Q). Being given by hypothesis that f(v) = f(v + p) for all p E Pk, we can write:

'Vp E Pk , If(v) I = If(v + p)1 ~ IIfllt+l,Q,nll v + pllk+l,Q,n, or again,

If(v)1 ~ IIfllt+l,Q, n inf II v + pllk+l,Q, n .

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Mathematical Analysis and Numerical Methods for Science and Technology || Numerical Methods for Stationary Problems