An Analysis of the Finite Element Method [1 ed.] 0130329460, 9780130329462

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AN ANALYSIS OF THE FINITE ELEMENT METHOD

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MINSKY,

AN ANALYSIS OF THE FINITE ELEMENT METHOD

GILBERT STRANG Massachusetts Institute of Technology

GEORGE J. FIX University of Maryland

PREIMTICE-HALL, INC. ENGLEWOOD CLIFFS, N.J.

77933 5 Library of Congress Cataloging in Publication Data Strang, Gilbert, An analysis of the finite element method. (Prentice-Hall series in automatic computation) Bibliography: p. 1. Finite element method. I. Fix, George J., joint author. TA335.S77

II. Title. 515'.624

72-12642

ISBN 0-13-032946-0

© 1973 by Prentice-Hall, Inc., Englewood Cliffs, N.J.

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215922

PREFACE

The finite element method has been an astonishing success. It was created to solve the complicated equations of elasticity and structural mechanics, and for those problems it has essentially superseded the method of finite differ¬ ences. Now other applications are rapidly developing. Whenever flexibility in the geometry is important—and the power of the computer is needed not only to solve a system of equations, but also to formulate and assemble the discrete approximation in the first place—the finite element method has some¬ thing to contribute. From a mathematical point of view, the method is an extension of the Rayleigh-Ritz-Galerkin technique. It therefore applies to a wide class of partial differential equations. The Ritz technique does not, however, operate directly with the differential equation; instead, the continuous problem is put into an equivalent variational form, and the approximate solution is assumed to be a combination 2 °f given trial functions 0. Then the difference equation at xn is

Since each term is nonnegative, all three must vanish. If the zero-order coeffi¬ cient qn is positive, it follows immediately that Uhn — 0. In any case, the other terms lead to Uhn+\ — Uhn Uhn_Thus these components are also maximal, and the whole argument can be repeated with n - 1 or n + I in place of n. Ultimately, after enough repetitions, it follows that Uhn t/J — 0. Thus LhUh = 0 holds only if Uh is the zero vector, and Lh must be invertible. Allowing inhomogeneous boundary conditions, the same argument would show that no component U* can be larger than both Uh0 and UhN. This is a discrete maximum principle, from which it follows that the discrete Green s function (LhYx is a nonnegative matrix. By the way, a similar proof leads to Gerschgorin's theorem in matrix

22

AN INTRODUCTION TO THE THEORY

CHAP. 1

theory: Every eigenvalue X of a matrix A lies in at least one of the circles

U - A, I < £ IA, Ij*i Choosing A to be either of the matrices hlLh displayed in (15), all eigenvalues satisfy \ X — 2| < 2. Thus the theorem does not rule out the possibility that X = 0 is an eigenvalue, that is, that Lh is singular; it was at this point that the repetitions of the argument at / n — 1, n — 2, . . . , 1 were needed. Gerschgorin’s theorem becomes perfectly useless in a fourth-order problem. The leading coefficients in the simplest case are A.. = 6, Ai l±l = —4, and Ai i±2 -- 1, and the Gerschgorin circles are \X — 6| < 10. Since X = 0 lies inside these circles, the Gerschgorin argument fails to prove even that A is semidefinite. This difficulty simply reflects the absence of a maximum principle for fourth-order problems. If we compare u" = 0 with uUv) = 0, for example, it is obvious that straight lines attain their extrema at the ends of the interval, whereas cubics need not. The maximum principle, when it holds, can be made to yield a simple proof of convergence. We prefer instead to preserve a close analogy between differential and difference equations, by discussing the discrete inequality which corresponds to ||w||2 < C||/||0. The statement of such an inequality requires, first of all, a redefinition of the norm to make it apply to grid func¬ tions. An obvious choice for the discrete energy is ll/*!lo = (I>l/5l2),/2For the square of the 2-norm, we introduce the energy in the function and its first and second forward-difference quotients: II

Uh\\\ = II Uh\\l + II A+t/A||£ -f \\A2+Uh\\l.

These sums extend only over the grid points at which they are defined; the forward-difference quotient A+/,. = (fi+l — ft)/h makes no sense at the last grid point. ' There is one other new feature to be introduced, inhomogeneous bound¬ ary conditions. For the two-point boundary-value problem, the continuous dependence of u on/and on the boundary data is expressed by (17)

II “Hi < Cfl|/||o + | «(0) | + | u\n) |).

For the finite difference equation, in which denotes whichever boundary operator is applied at the right-hand end, the corresponding inequality is

SEC. 1.4.

FINITE DIFFERENCE APPROXIMATIONS

23

For each choice of difference equation, this is the basic estimate to be proved. It is not an automatic consequence of (17), although it implies (17): The continuous inequality is necessary hut not sufficient for the discrete inequality. In this sense the theory of difference equations is the more difficult; there is an enormous variety of possible difference schemes, and each requires a moreor-less new proof of (18). As in the continuous problem, we shall accept the inequality as true and concentrate on its implications; the technique of proof in one-dimensional problems is summarized by Kreiss [K7], The inequality (18) asserts the stability of the difference equation: The discrete solution Uh depends continuously on the discrete data fh, uniformly in h. To establish that \Jh converges to u, we need the most celebrated theorem of numerical analysis: Consistency and stability imply convergence. This theorem is proved in two steps: 1. The error Eh satisfies the same difference equation (13) as Uh, and therefore by stability it depends continuously on its data, which is nothing but the local truncation error: ||^||2 '(x)v/(x) + q(x)v(x)w{x)) dx.

This is the energy inner product. It is defined for all v and w in the admissible space 3Cjj and represents the inner product that is intrinsic to the given prob¬ lem. Our goal in this section is first to prove the theorem described above— that the energy in the error is minimized by the Ritz method—and then to apply this theorem in establishing bounds for the error with linear elements. Theorem 1.1 Suppose that u minimizes I(v) over the full admissible space 5ClE, and Sh is any closed subspace of 3C'E. Then: (a) The minimum of I(vh) and the minimum of a(u — vh, u — vh), as vh ranges over the subspace Sh, are achieved by the same function uh. Therefore

(20)

a(u — uh, u — uh) = min a(u — vh,u — vh). vh in Sh

40

CHAP. 1

AN INTRODUCTION TO THE THEORY

(b) With respect to the energy inner product, uh is the projection of u onto Sh. Equivalently, the error u — uh is orthogonal to Sh:

(21)

a(u — uh, vh) = 0

for all vh in Sh.

(c) The minimizing function satisfies (22)

a(uh, vh) = (/, v'j

for all vh in Sh.

In particular, if Sh is the whole space 3C E, then

(23)

a(u, v) = (/, v)

for all v in 3C'E.

Corollary It follows from (21) that a(u — uh, uh) = 0, or a(u, uh) = a(uh, uh), and the Pythagorean theorem holds: The energy in the error equals the error in the energy, a(u — uh,u — uh) = a(u, u) — a(uh, uh). Furthermore, since the left side is necessarily positive, the strain energy in uh always underestimates the strain energy in u:

(24)

a(uh, uh) < a(u, u).

This whole theorem is fundamental to the Ritz theory, and its three parts are very much interdependent. We can deduce (b) immediately from (c): If (23) holds for all v, it holds for every vh in Sh, and subtracting (22), the result is (21). Also (b) follows from (a): In an inner product space, the function in a subspace Sh closest to a given u is always the projection of u onto Sh. In the opposite direction, to show that (b) implies (a), we compute a(u — uh — vh,u — uh — vh)

a(u — uh,u — uh)

2 a(u

uh, vh)

+ a(vh, vh). If (21) holds, then a{u — uh,u — uh) < a(u — uh — vh, u — uh ~ vh).

Equality occurs only if a(vh, vh) = 0, in other words, only if vh = 0. Thus uh is the unique minimizing function in (20), and (a) is proved.

The problem is now to establish (c), since that will imply (b) and there¬ fore (a). If uh minimizes / over Sh, then for any e and vh. I(uh) < 1(uh + evh).

SEC. 1.6.

THE ERROR WITH LINEAR ELEMENTS

41

The right side is a(uh + evh, uh + evh)

2(/, uh + evh) = I(uh) + 2e[a(uh, v") - (/, «*)] + e2a(vh, vh).

Therefore, 0 < 2e[tf(u\ wA) — (/, vh)\ + e2a(vh, vh). Since this is true for small e of either sign, it follows that a(uh, vh) = (/, vh). This equation expresses the vanishing of the first variation of I at uh, in the direction of vh. In particular, a(u, v) = (/, v), so that at u the first variation vanishes in every direction v. This is the equation (11) derived earlier. Thus (c) is established; it is the equation of virtual work. The choice v = u in this equation leads to an interesting result, that at the minimum, the strain energy is the negative of the potential energy: (25)

I(u) = a(u, u) — 2(/, u) = —a(u, u).

Similarly, I{uh) = —a{uh, uh). In every case I{u) < I(uh), since u is minimizing over a larger class of functions, and therefore a change of sign reproduces the result stated in the corollary, that the strain energy is always under¬ estimated: a(uh, uh) < a(u, u). The proof of the theorem is now complete, except for one point: Neither the existence nor the uniqueness of uh (or, in case Sh is the whole space 5CJr, of u itself) has been properly established. To a functional analyst, such an omission means that the proof has hardly begun. We shall try to mollify him by pointing to the key word in the hypothesis of the theorem: The sub¬ space Sh is required to be closed. This means that the subspace must contain its own limiting functions. If there is a sequence vN in Sh such that a(vN - vM, vN

—

vM) — > 0

as TV, M —»

oo,

then there must exist in Sh a lim it v, such that a(vN — v, vN — v) —► 0

as tV —>

oo.

This will always be true if Sh is finite-dimensional, which is the case contem¬ plated in the Ritz method. In general, one cannot guarantee that there exists a function uh in Sh which is absolutely the closest to u, unless the subspace is closed. We cite the example Sh = 3C£, which is not a closed subspace. It contains functions arbitrarily close to u(x) = x, but there is no closest one; the “projection” breaks down.

42

CHAP. 1

AN INTRODUCTION TO THE THEORY

To prove the existence of u, defined to be the minimizing function over the whole space 3Cj[, we need to see that X.'E is itself closed. This is exactly what was achieved in the completion process, enlarging to the full admis¬ sible space 3Cjj. In particular, it was in completing (or closing) the admissible space that the natural boundary condition u'(n) = 0 was dropped. There was one technical point in that process which we hurried by: the space was completed in the natural energy norm, a(v — vN, v — vN) —>■ 0 as in (10), and yet we have described the completed space in terms of the 5C1 norm. This step is justified by the equivalence of the two norms: there exist constants a and K such that (26a)

a(v, v) < K || v\\\,

(26b)

a(v,v)>o\\v\\\.

The last inequality also yields the uniqueness of u and uh, since it means that the energy is positive definite: a(v, v) = 0 if and only if v = 0. The sur¬ face I(v) is strictly convex, and can have only one stationary point, at the minimum. The first inequality is easy, since

j

[pW)2 +