..

SOME ASPECTS OF RECENT CONTRIBUTIONS

TO THE MATHEMATICAL THEORY OF

FINITE ELEMENTS

J. T. Oden

Professor and Chairman of EngineeringMechanics, University of

Alabama in Huntsville

Pub lished in

RECENT ADVANCES IN COMPUTATIONAL METHODS IN

STRUCTURAL MECHANICS AND DESIGN

Proceedings of the Second U.S. - JapanSeminar on Matrix Methods of StructuralAnalysis and Design - Berkeley, Calif.

UAH PressUniversity of Alabama in Huntsville

Huntsville, Ala.1972

gs

.~

SYNOPSIS

A discussion of a number of recent contributions to the mathematicaltheory of finite elements is given, with emphasis on results connected withconvergence and error analysis. An attempt is made to illustrate generalmethods-of-attack of these questions for elliptic problems and to put intoperspective certain theoretical results recently published.

INTRODUCTION

One might interpret (perhaps a bit facetiously) the difference betweenapplied mathematicians and engineers or physicists who apply mathematics asthis: the applied mathematician takes a differential or integral equation(often a rather simple one from a physical point of view), studies in minutedetail its various properties and solutions, and then seeks some sort of physi-cal situation in which his results may have some relevance. The applier ofmathematics begins with a physical situation, develops in minute detail adifferential or integral equation to describe it (often very complicated froma mathematical point of view), and then seeks some sort of solution that mayhave some relevance. On the premise that this interpretation is valid, it iseasy to understand why applications of various mathematical methods generallyfar precede the discovery of the complete mathematical structure underlyingthem. Technology demands solutions to physical problems; wh0n the engineer'smathematical techniques fail, they are abandoned and forgotten; when they suc-ceed, they deserve and often get a closer scrutiny by specialists skilled insuch matters. Such is the way with applied mathematics. The history of mathe-matics is briming with examples which substantiate this.

In some respects. the finite element method was almost nn exception tothis rule. Its development began over fifteen years a~o, it progressed throughthe scientific boom of the late 1950's and early 1960's, and its utility as apowerful method of mathematical analysis was firmly established by the mid1960's. Yet, until recent times, all attempts at putting the method on il firmmathematical basis came not from mathematicians hut from appliers of mathe-matics who were primarily interested in specific physical applications (e.g.[l-lOJ). One can find in engineering journals. investigations of convergence,completeness. error estimates, singularities. the effects of corners, etc.

1

which. although sometimes incomplete and often not rigorous by currentstandards, began to appear almost a decade ago.

Perhaps this tardiness among applied mathematicians was due to a tendencyof some researchers to recognize only the similarities of finite elements toother methods and not to appreciate its differences. Whatever the reason, thesituation hegan to change around the time of the First U.S.-Japan Seminar(IQ6Q). During the three year period between the first and second seminar,there has been a mild explosion of mathematical literature on finite elements.Its emphasis has been primarily on questions of error estimation and conver-gence, and the bulk of the work has dealt with linear elliptic prohlems. Inthis short period of time, the mathematical theory as it applies to linearelliptic boundary-value problems has been provided a fairly solid hasis, andprogress in the areas of nonlinear equations and equations of evolution havebeen made.

The present paper contains a brief look at some of these recent contribu-tions, with an aim toward illustrating the methodology and philosophy one findsthere, and toward clarifying the notation, objectives, and results so that theymight prove useful to the appliers of the method. The present account isorganized as follows:

Introduction

Convergence and Error Analysis forLinear Elliptic Problems

Galerkin ApproximationEnergy Error EstimatesRitz Approximations

Finite Element Basis Functions

Construction of Finite Element basesFinite Element InterpolationSpecific Interpolants

Patch Functions and Fourier Analysis

The Patch Function RepresentationFourier Analysis

Perturbation Analysis

Mixed Model.s

Introductory CommentsLocal ApproximationsError Estimates

Initial Value Problems

Nonlinear Problems

Other Topics and Closing Comments

Acknowledgements

References

Under most of the above titles, a survey and crirque of SOOie 01 themathematical literature is given; hut it is necessarily hi<Jsl'dilnd incomplete:biased, hecause, in many cases. insufficient time hlls ('lapsed to assess whichof the recent contributions have lasting impact and v;llue. Thus, I have takena sample of what interested me and of which I fet>l will prove to he important;the sample is also incomplete, because the field is too dynamic at the momentto warrant a detailed review. and because it is felt thnt the space limitationsimposed here would be best used for the objective mentioned above. The inter-ested reader should consult some of the forthcoming books and monographs for a

2

."

more detailed and complete list of references [ll-l4J. For the more ambitiousAkin, Fenton, and Stoddart [lSJ have recently put together a bibliography ofalmost llOO references on the finite element method, and Whiteman [l6J hascompleted an extensive bibliography which references much of the recent mathe-matical literature. See also the expository papers of Zlamal [l7] and Oden[lS].

CONVERGENCE AND ERROR ANALYSIS FOR LINEAR ELLIPTIC PROBLEMS

Galerkin Approximations

In this section we investigate methods for obtaining error bounds forGalerkin approximations of a general class of linear boundary-value problemsof the type

.£u = f in ~ u = 0 on Olt (1)

Here u and f are functions defined on a bounded compact region ~ of n-dimen-sional euclidean space, ~ is the smooth boundary (closure) of ~, and ~ is alinear operator with domain ~(!).

We expandregard u as andepend s on !.(Le., (u,v) '"'

the class of function in which we can seek a solution if weelement of a Hilbert space ~~, the precise structure of whichThen, for example, if (u,v) denotes the L~(~)-inner productI uvdR) , we seek generalized or weak solu~ions in ~ satisfying

(.£u,v) " (f,v) , V v ( U (2)

(3)

and vanishing weakly on oR.

Galerkin's method, of course, involves determining an approximate solutionof (2) by seeking a solution in a finite-dimensional subspace ~ of U. If[~6(~))A-l'~ - (xl,x~, •.• ,xn)( R, denotes a set of G linearly independentfunctions in U, they provide a basis for such a subspace, and each V(x) (ITk isof the form -

G

=I6=l

where the A6

are constants. We then consider the problem of determining u (ITbsuch that

(.ro ,V) 0:: (f, V) V V ( rT\.r. (4)

This leads to the system of linear equations

reo l,2, ... ,G (S)

wherein

3

(7)

(8)

,"

L6r'" (,£cp6'CIT) and I fr'" (f,CIt-) (6)

Upon solving (5) for the coefficients A~, we determine the best approximationof the solution u* of (2) in mG' It is well known that the Galerkin solutionU* of (4) is such that the error in £0*, i.e.

~(u* - U*) = £e, e '" u* - U*

is orthogonal to rna (see, for example, Oden Ll8,l<)J or Zlamal [17J). Indeed,by 6imply setting v - V in (2) and subtracting the result from (4) we get

(~(u* - U*), V) "" (le, V) :: 0 V V € IllG

Energy Error Estimates

We now arrive at the fundamental question of accumcy of the approximate solutionU*. We will Ehow thll:under reasonable asslJIlptionscCXlCeming the operator t, thefundamental error bound for every choice of a subspace mG is of the form

lIell. s; cllu* - villi V V € lTlc (9)

Here 11·11 is a Sobolev norm (to be defined subsequently) of the error e of (7),III

C is a constant, and u* is the "exact" generalized solution of (2). The norm11.11,. is sometimes called the "energy" norm (cf. [19,20J). Nothing stronger canbe said until the space ~ (or, equivalently, its hasis r~~)) is preciselydefined.

To arrive at (9), two assumptions are needed. First, we assume that £is m-elliptic; i.e., there exists a positive constant y such that

(10)

(Generally, ~ is a symmetric partial differential operator of order 2m; see(76)). Secondly, we assume that ~ is bounded in the sense that there exists aconstant M such that

(~, v) s; Mllull~Ilvlllll (11)

V u, v €~. By the notation Ilulllll,we mean the Sobo1ev W;O~) norm; Le.

Ilull~II

(12)

D~

We have employed hereof ordered n-tupleR afollowing conventions

n

the so-called multi-integer notation: if Z; is thl' spacepositive integers ~ = (~l .~~, ... '~n)' ~l = inte~er, thehold

4

CiX--

n..n CitXl

i=l

XCi--n

""ITXlCiIi""l

( 13)

Then the Sobolev norm lIulllllof (12) consists of the sum of the Lo (R) norms ofu and all of its partial derivatives up to and including those of order m.

We next observe that

(~,e) .. (~e,u* - V + V - u*) "" (~,u* - V)

because (~,V - u*) a 0 by virtue of (8) and the fact that V - u* (~. Thus,(10) and (11) give

yllell~S; (,£e,e) ...(:e,u* - V) S; Mllell.Ju* - vllm (14)

Consequently

lie 11m ~ ~llu* - vllm

which coincides with (9) if C ...M/y.

(15)

The beautiful thing about (15) is that_V can be §.!!Y elemcnt of rTlG' Infinite element applications, we choose V = U, the q-interpolant of the exactsolution u*; that is the element U ( ~ whose derivatives of order q coincidewith those of u* at nodal points in some discrete model of~. While wc shalldiscuss this idea in more detail in the next section, we note here that once asuitable basis (finite-element approximation) has been defined, we attempt toestablish an interpolation error estimate of the form

(l6)

where h is (say) the maximum diameter of a finitc elemcnt in the mesh underconsideration, K is a constant independent of hand (hopcfully) s > O. Suh-stitution of (16) into (15) in place of Ilu* - vfl givt's direct Jy the desirederror estimate. If s > 0, then convergence in the 11·11 norm is also provedhecause then lIelim... 0 as h ...o. These results have, of course, hecn greatlygeneralized in recent mathematical literature, particularly, in the work ofStrang and Fix [llJ, Strang [2lJ, Aubin [22], in the series of papers byBabuska (e.g. [23-30]), Ciarlet [31-35], Zlamal [1.7,36l, Yamamoto and Tukuda[37J, Mikkola [38], and many others. We discuss certain of these generaliza-tions elsewhere in this paper.

Ritz Approximations

It is appropriate to comment briefly here on Ritz approximations of (1)as an alternate route to equivalent error estimates. Assuming £ is symmetric(i.e., (~,v) • (£V,u)), we view the solution of (l) as the element in ~minimizing the quadratic functional (see [lS]).

I(u) ...(~,u) - 2(f,u)

5

(17)

.'

The Ritz method involves seeking a minimum to I(u) in ~ rather than U.

Suppose that u* is the exact minimizer of I(u), U-Jr is the element in 11'lGwhich minimizes I(U) among all V (~, and U (rna is the q-interpolant of u*.Then

and

II(u*) - I(U*) I ~ II(u*) - I(U)I (18)

l(u* + w) = l(u*) + (,Cw,w) (19)

since (f;l* + f.w) .. 0, w being an arbitrary variation in u*. Setting wu* - U and making use of (11), we have

II(u*) - I(U) I ~ Mllu* - iill:> (20)III

ThusI I (u*) - I (U*) I ~ Ml\u* -If W'

III

and, once again, we need an estimate of the interpolation error Ilu-Jr- iil\ .111

With a little algebra, we can show that

II (u*) - I (U*) I c: (.1:(u*- U*) I u* - U*) ~ yllu*W'111

(21 )

(22 )

Hence

(23)

which, except for the constant, is of the same form as (15). NOle that weobtain preciselz the same constant if, instead of (20),we derive II(u*) -I (ii) I ~ Mllell" I\u - lill...FINITE-ELEMENT BASIS FUNCTIONS

Construction of Finite-Element Bases

The success of the finite element method is due lnr~ely to the systematicmeans it provides for generating the basis func tions CfltJ. (x) of en for irregu-lar domains. We shall outline here the basis properties·-·o[ lhp ml'thod folLow-ing the viewpoint in [lB,l9,39,40,41J. The development given is standard and,up to the fundamental results given by (37), reproduces the dev~lopment givenin [40J.

We !!egin by considering once again an 0Een h£unded region ~ic I,:n withclosure R = ~ + oR. A finite-element model ~ of ~i is the union of .1 finite- -number E of closed b.2unded subregions R~ of En, bi~ hl'il1~the c)osur.£ of anopen region ~ c En(~ = ~ + o~ ; e = l,2, ... ,E). Th~ subrl'~ions Ii. are

e e - t e ->called finite elements, ~ s referred to as the connected model of ~. and theopen elements R are disjoint in En: -----

eE

~ = U ~ . Ii n Iii = no (':f f (24).' e r 'P'e=l

6

...

-In the connected model ~, we identify a finite number G of points, calledglobal nodes, and we label these consecutively Xl,x~, ••. ,xG. Likewise, weidentify within each finite element R a number-N -of points, called local

e ...., • .nodes, and we label them consecutively xl,X', ... ,XNe; e = 1,2, ... ,E. HSlngthe notation developed in [l9], we referetoea typ~al global node as x (~ =1,2, ... ,G) and a typical local node as xN (N = 1,2, ... ,N ; e = l,2, .•~,E).

-A eWe assume, of course, that the proper correspondence between points in each~ and points in ~ exists, as well as a proper correspondence between nodalp~ints xN in ~ and nodal points x~ in ~, so that the elements fit smoothlytogethe~to fo~m~. Such compatibility conditions are discussed in [l9, p. 361Mathematically, the connectivity and decomposition of a finite element modelare established by the Boolean mappings

(e fixed) (25 )

where ~ - l,2, .•• ,G, N • 1,2, ••• ,N., e ::: 1,2, .•• ,E,

if node xli of ~ is coincident with nude.:xN of W.--AI II

if otherwise,

tll~ telA~ is the transpose of ~ and the arrays A and 0 have the property

N 6r ~ r -G x ,x ( 6tL l&.tIl\ll '"' ~ 2'"T /6- _. It

~~ 6", (e fixed) ; AN( '"~ II

II r -6=1 N=l 0 x ,x I. It (27)- - .Now consider a function U(x) ( Cm(~).

the functionThe restriction u (x) to 6t c ~ is

e •

u (x). -if x (6t

e

if x '~t"

(28)

(29 )

In the finite-element method, a local represe.:ntation of u (x) of order qis defined as the function ~

N

u (x) - ~ ~ aNCeI~) (x)e - ~ ~ N

N= 1 I~S;q -.

wherein a~~ are constant coefficients, a is a multi-integer (ry E Z"), and~)(~) aa the local interpolation funct-ions of order q defined so+as to havethe properties

x ,~"

7

·',

M, N = 1,2, ...,N~ (30)

Here 6: ,O~l, .•• ,o~n are Kf8~ecke~ deatas ang we have again used the multi-integer nfi~l1tion,8au == a -'u/ax11ax/' ...axn n. Therefore, in view of (30),

D ~ (x N) = Cl ~e); 181 ~ q8 .. -.. 8 .-.

The global finite-element representation of lJ(x) is the function

(31)

(32)

when'

(e fixed) (33)

EU

('=1

Ctq1"(x) =A -.

Ctand ~.(~.>are the global interpolation functions

N•2!~w~)(~)N= I

It is easily verified that

Ct( µ. µ'J:Ctl J:.lYnD ~ ~ )oAua ..,us 'e 1 n

(35)

so that

I!I ~q; A,µ. = 1,2 ...• ,G, a.S E Z; (36)

Hence, if we set

and (37)

then the values of u(x) and U(x) and all of their pnrtia1 derivatives of orderS; q coincide at the nCldal points. To he perfectly gcnl'rnJ, we ~;hall n·quirethot (37) hold only for specific values of IL. N, (e), nnd 8, adlllitting, there-by, the possibility that some of the flee arc zero (i.e., not all derivativesof a given order need be specified at a node).

Finite-Element Interpolation

We now follow a procedure descrihed in [l9J (see also L7,IY,4lJ)to cHtab-

lish error estimates for finite-element interpolation. Let u(x) denote a

8

Moreover,function in cr(~) and let U(x) denote aassume that the r + l-st derIvatives oflet h denote the "diameter" of element

Cl

q-interpolant of u(x), q ~ r.u(x) are bounded V-x E \to~e;- 1. c.

We

(8)

The mesh h of the finite element model ~ is the maximum diameter of alL cle-ments in it; h =: max[hl'~".' ,hE J.

oy)D u (x) + R 1 (x,.y, 1I )ry r+L

1~I~ru(~. + r) '"'

The function u(~) can be expanded in a Tnylor' s series of ltl(' form

f){

1

where

1R =: r+l 0 (r+1 (rtl): Y . r+lll ~ + AX) (40)

If III ~h, then

IRr+l I ~ Chr+1[uJr+1 (41)

(42)

t II prescrihl' :ill deriva-U(x) of lI(x) :It theahollt nodl~ "x4 of clcment

I1~I~q

u(x) ..

We shall consider cases in which it iN possihletives of order ~ q of a finite element approximationnodes. In particular, consider the local expansions~e:

The local approximations ~..\(~) helong to a clas~; Jr of finitl' cl(,IlIl'lIl approxi-mations which, for the moment, arc assumed to have tilt, followil1~ propertics:

• The local basis functions ~l(~) are such that

~ .. x/h ; I~(?) I ~ K~

(43)

e, N. and Thus

(44)

. Jr contains a complete po lynomilll p (x) of d('gr('(' q:

9

p(~) "" (45)

Property (44) mere.!y assures that the functions ~'Jarc dimensionally consistentand bounded V x ( ~. Property (35) can be w~akened slightly, .as will be seen .•

By adjusting the constantB c in (45), it is clear that~

lu(x) - p(x)1 S Khq+l (46)

V ~ (~ •. Setting ~"" ~N, we also have

Ipi - D13.u(!<N)I S Khq+1-1~.1 (47)

Now let u (x) denote the q-interpolant of u(x) in J. Then• _ _. r

inequality, IDB(u - li)\ = !OS(u - p +estimates (46).-and (49), <lnd···thefactthe Rame holds globally for D by

p - u" I (p~ - nryuc.~N))I\J.~C.~)I,f\' I S;q

Thus, from (44) lind (47) it Eo llows thnt

ID~.(P - u)1 S; L (Khq+1-I,~I)KO'hl~I-I,S.1 = Khq+l-ls.1N,O'

We now need onlr ap~ly the trianglep - u)1 S; \DS(u - p)1 + IDe(p - u)I, thethat a local--estimate on ii; implies thatvirtue of (33), to obtain [9,39J

1Ds(u - u)\ S; Khq+1-1~1

(48)

(49)

(50)

Theorem l. Let U be a finit~ element model of a [unction u(x) € Cq +1 (6~)which is a q-interpolant of u on hi lind which is generCltl'd from ,I family 3\ offinite element approximations which satisfy conditions (43), (44), and (45).Then (50) holds at every ~ (~J I~I s; q + 1.

We can strengthen this slightly by

Theorem 2. Let U ( Jr, where Jr is generatl·d by functions s<ltisfying (43)and (44). Suppose Jr contains a function p(x) such thut

(51 )

where p is a complete polynomial of degree q. Then (SO) holds.

10

Now if u(x) and U(x) are the functions described in (42) - (52), we can besure that (u --U) (~+~(~). Moreover, in view of the definition of R +1' theconstant K in (50) equals K' IDq+1UI. Thus, (50) suggests the estimate

q

(52)

s • O,l, ... ,q + 1. Turning to (15), we sec that this leads to the errorestimate

(53)

ThUl:l,the finite el~ment method conver~es if q + I ~ 1Tl. Clearly, the require-ment that u(x) and U(x) satisfy the conditions of Theorems 1 and 2 are onlysufficient conditions-for convergence. A more elatorate analysis, l~ading tothe results (52) and (53), was given by Strang and Fix [ll,42] for infinitedomains covered by uniform meshes. We discuss their results more fully inthe next section.

Specific Interpolants

We comment briefly here on Some results obtained for specific types ofelements.

Perhaps the first study of error in finite-clement interpolation can befound in the book of Synge [43J; it concerns piecewise linear approximationsof a function U(Xl,X2)A( C?(~), with bounded third derivatives, over a planetriangulated mesh. If~ denotes the largest angle between the sides of alltriangles in the mesh, h is the maximum diameter, and U is a piecewise linearfinite element interpolant of u which coincides with u at the vertices of thetriangle, the Synge showed that

l~l(U-lbls;~cos .!:. A

2

(54)

A number of additional results were obtained by Key [5J for both two- andthree-dimensional elements. Key considered, for example, the six-node triangleof Fraeijs de Veubeke [4] and obtained estimates of lu - Vi of 0 (h2). Also,for the three-dimensional simplex (tetrahedron), Key showed that

I~( -OK1 U - U)I S; l8Kh (55)

i = l,2, where a is the smallest angle between the face containing e and theother sides of the tetrahedron.

For the Fraeijs de Veubeke-Key six node triangle (one node at each vertexand one at the mid point of each side, IT being a quadratic in XlIX:», Zlamal(36] obtained the estimates

I~(u - U) I s; s~~e h? (56)

while Bramble and Zlamal [44] generalized triangular interpolation to the case

11

in which a general polynomial in Xl ,x:' of degree Itm + 1 is del-ermined byspecifying its derivatives of order 2m at the vertices, its derivatives oforder 2m - 1 at the centroid, and its directional derivatives normal to eachside at r + 1 equally spaced points along each side, with r = 0, l, •.. , m.They show that if such a polynomial P4"~1 (xl ,x~) interpolates u on a triangleT - ~ , then•

( ") II))ry1111~ ) ~I~k L~ (

6ie)

(57)

(58)

For intcrpollltion using rcctangulnr l'll'lIll'nIH, lhl' firsl rl'SIJlls .1ppcar tohe those of Johnson and McLny [61 for: hllinenr approximiltions in I';. Theentire question of finitt'-element intcrpolntion on rectangular clemcnt:; can behandled with Home generality by using tensor products o[ spline spaces. Thisiden wns exploited by Birkhoff, Schultz, And V:lrgn l4'iJ. For example, if L isa linear operator on u(x), x ( [a,bl, we develop a L-spline space g(L,h,z) bypartitioning [a,bJ into elements I.l < Xl < x~ .•. < Xn < b and computing thosefunctions w(x) such that L*Lw = 0 for x ( (a,b) - (XJ}~;;;~' dkw(xt-) = dkw(xt+)for 0 s k s 2m - 1 - z, 0 < i < N. g(L,h,z) is then a llnenr space of dimen-sion 2m + z(n - l) (see, for example, [46,47,48J). InterpoLation spaces ontwo-dimensional domains (a,b) X (a,h) are ohtained by the tensor productg(L,h,z) ~ g(L,h,z). In [45), local approximations of the type

2m-l- - ~ I JU(xl'x~) - L AtJ(x1) (x;"')

i ,J"'O

are obtained, along with the estimates

Ilu - u\1 ~ K h?m-"\'j' (rl) 8~

(59 )

Approximations of the type (59) were lHled by Bogl1('r, Fox, and Schmit [L!9] [orfor m = 3 for flat plate analysis.

The entire theory was generalized by Schult? L')ol who showed that ifu (W;'(/it), where &t." nr"'l(al,bt) C En, nnd if we :In' given:l mesh h '" n~= hibelonging to the class @(~) of rectangular pnrti tions of R, and if gm (hI' La ,btj)

. jis a splinc space corresponding to the ith coordinl.ltc, then there eXlsts aU ( (~~=lgll(ht ,[al ,bIJ) such that

111I - ull\'j~ (Oi) (60)

where 0 S P S min [r,2m - d. An excl'llent ~;lImm[Jry and discussion uf theseand similar results ha~; been compiled hy VargCl [L2J.

12

PATL11 VUNCTIONS AND FOURIER ANALYSI.S

The Patch-Func tion Represcnta tion

In !Ouch of the mathematical literature, the description or thl' hasisfunctions for finite-element approximntion is done in termS of so-c~lled patchfunctions and their translates over an infinite domain of uniformLy spacedelements. Such a viewpoint, promoted by Strang and Fix [ll,21,421 and usedextensively hy Babuska [23-30] leads to the concept of a periodic network ofeLements joined at nodal points ha, where ~ is a multi-integer (ry £ Zn). Atthe origin of this system (a ~ (0-;0,... ,0»)", we construct a colt"c·dion of Nfunctions CPt (x) corresponding to the number of functions we wish to speci fyat each node.~· Each function is normalized by a change of vari.able S = x/h, andeach has compact support: CPt (~) :a 0 for 1.)(.1 > Rh < 00. With the tria·l f~'nctionestablished for the origin, those associated with an arbitrary nodl' withcoordinates ~h are obtained by translation:

xcP (x) = cP (=-h·- ry), i = 1,2, ... , N (6 I )I ,a _ t .

The splice IT\. of finite-element approximations now contains functions of the form

N

U(~) = IA~CP, ,~(x) I

i"l ..

Figure I contllinH an examp Ie a f such n bas is in E2 wi th N = I.

(62 )

(63)

While this particulllr representlltion is llpparpntly useful for Fouril'ranalysis since it involves II periodic mesh, we fec>I that it suffL'rs from ;1

number of serious drawbacks. To mention only a few, the reprpsentation cannotaccommodate irregular geometries, multiply-connected domains, nonuniform meshes"varying boundary conditions, or the use of different types of elements withina network, without gross modifications. More importantly, the most funda-mental notion of piecing elements together to form a particular model of agiven domain is lost, along with the topological features of the method soessential in applications and so intrinsic to the philosophy of the method it-self.

Fourier Analysis

Taking full BdvantEl~e of the ppriodicity of thl' lrnnslatl'd l);Ilchfunclionsand the infinite uniform mesh descrihed previously, StronM ~nd Fix [LI,21 ,42Jdeveloped an elegant analysis of the finite clement llIl'thodvLI Fourier trans-forms. To slImmarize briefLy the essential featun's of their work, let liS

denote the FOllrier transform of a function lI(~) (x (En) hi' dpnotl'lf~(S):~(~)-r (,,)0 -l.&q"En

Here x~ ="x ~,dx = dx1dx..,...dx, and i =/Cl. If u(x) (w~, then one can_,.. L.J I t _ • n •i

13

define a norm equivalent to 11·11 bywp2

Ilull:; - I lu(~)I" (l + 1~1')'d~E

n

(64 )

However, now s can take on nonintegral values. This fact led Babuska [23,2<),301 to the use of the fractional spaces of Aronszajn [5lJ. Strang and Fix [llJprove the following theorem.

Theorem 3. Let ct't (~) f ~ (En) have compact support and p ~ q ~ O. Thenthe followinK are equivalent:

(i) there are linear combinations 0 (x) of the ct' which sntisfy~- t

o (0) "" 1o _ o (2nr;y) =0 V ~.;. 0 E: Z"o ._

(65)

(11) There are linear combinations (J (x) of the ct' which srltisfy0/- t

(iii) there are weights AI such that~

(66)

for s ~ 0,1, ... ,q with

where C(s) and K are independent of 1I.

(67)

(68)

Further results were obtained in the uniform norm 11'lIw~ in [IIJ; inparticular, if ct'J E: ~ have compact support (0 ~ q ~ p) and if u E: W~+l, thenStrang and Fix showed that

Ilu - iill • s; C.hP+1-·lIull p+lw~ Woo

14

(69)

H = 0. l, ... ,q, where u is the so-called qunsi-int~rpolant of u:

U (x) :: L Dryu(~h)h I~I0fY(~ -A)A. (z n

I~IS:p

(70)

nle importance of part (iii) of the theorem in connection with qu~stionsof convergence is obvious. If we select Vex) in (IS) to be the function1: A~q>1 ry in (62) (assuming that the genera~(ized sohlt ion u* of (2) J ics inW8+11En~, p + 1 ~ m) then it follows that

(71 )

An Improved error bound on lIell. when () s: S';; III is given by LltJ (sec L1lso [42J)w~

(7-2)

where r ~ min [p+l-s; 2(p+l-m)).gi v('n in [1L].

PERTURBATION ANALYSIS

Similar reoults in the 11·11 • -norm nrc alsowe<>

To illustrate the importance of having error cstimates of the type on (72~we outline here a generalization of the pertllrbation analysis of Rirkhoff etal [S2J proposed by Fried [53J (see also McLay [S4J). Consider again thelinear elliptic boundary-value problem

.£u ~ f in Ii u = g on ~ (7"3)

where! is the general partial differential operator

r.:: 2 (-l)-~Ory(A0'8(x)08)

I~I ' I~.Is:m

(74)

with AfY8(~) > 0 V .x (b-l. We now consider c<Jses in which the v;Jri<lblc <:oe[fi-cients-''7\"t1'8(~)'the (unction f(~), and the essenti<JI boundary v;IlUL'Sg an' alsoapproximQl.ed in aome way so as to produce in (73) a pl'rturbed prohlem of thetype

(! + 6.1:)(ll + e) ~ [ + 6 fin ~; 11 + 011 = g + 6!!.()II d'li (75 )

when' 0.1:.c, of, llnd og llre perturbations (errors) in L,u,f, :11ll1 g respectively,and

(76)

15

j •

If (5.1) holds, then

o! + !e + oSft :: Of in bt ; OU :: og on ~ (77)

Suppose for the moment that og = 0, that f is approximated appropriatelyby polynomials of order q and the coefficients A~a(x) are approximated bypolynomials so that IIOfII = O(h'l+l) and Ilo!11 = 0(hr-+1'). The finitt' clementmodel of u is formed by polynomials of order p so that, in accordance with (60)or (72), lIelllll co O(hp+l-II). Now take the inner product of (77) with c andrecall that (e,!e) "" Ilell~ ~yllell~, we get

IIe II~ II: ( e ,of) - ( e , o.s:(u + e)) ~ 0 (7 8 )

Noting that (e Of? ~ (e,6.1:(u + e)), ,e,oO s; lIellllorll, (e,6.1:(u + e)) S;

Ilell(llo.Lllliull + rI6!lllell), and Ilell s; y- lIellm, we arrive at the estimat(·

lIellm S; Kl1l5f1l + ~ 1\6.1:11+ K~llo!llllellm (79)

Thus, the approximations of f and! result in no loss in accuracy or in therat c of convergence if q ~ p - m and r ~ p - m, p + 1 .: m.

~If .I: and f are not perturbed, hut 6g ". 0, Fried [')]J showed that Iklll ~116gll. In more recent work [55] he considc>recJ the prohlem ill which the houn-dary oH, itself is approximated by polynomials of order q for th(: (~igenvalueproblem correspondinB to Poi!lsons equation in thl' plane' ~~ = lIull'i/llull~). Ill.'shows that the error in IIull1 is O(h~q) whiLe thnt in lIulI<~ is O(h'l+l) su thatthe perturhed eigenvalue i9 given hy

(SO)

Fried also con!;idered the biharmonic equation "nel round tlwt 1161111~ = o (h:>(q -1\).

We remark that the study of such houndary c>rror:; for fini.ll' l,ll'lIlcntapproximations appears to have he en initiated hy Korneev [56,57J who contribu-ted quite elegant results for both two- and three-climensiona L problc·ms.

MIXED HODELS

Introductory Comments

The use of "equilibrium" finite-element modl'ls elates back to the work ofBest [ssJ in 1963 and was developed further hy PLm [5() I and Frl'ijs de Vcuhekl'[3,1 ..]. Later came the mixed and hybrid models proposed by Jll'rm:llm looJ :mdstudied in a variety of alternate forms hy Pian and Ton~ [61 ,62,o·I,6L, I. In.1completely s('parate effort, the iden of conJuKatl' rinllt, cleml'nt ,lpproxlol;JliollSwas developed by Oden ct Ell [9,65,60,('7,68J and is CJIIUt· c1istilwt rrUIII themixed model idea. However, we rect.'ntly showed thilt ;] quite ~~('n(~r:d lht~ory ofmiXl·d finite element models can he devl'loped from ;J puint uf view similar tothat behind conjugate functions [40J. We slwll <."OllllOl'nthriefly on some or theresults in [40J.

Consider the general linear houndary-value probl<~m

u

o in ~in-J'Tu - S

16

(8l)

Figure l. Generation of finite-element basis functions over ~ uniformmesh in If by translation or a hasic function defin(~d at theorigin. Shown is a case in which the functions have not heennormalized and N=l.

17

Here T is a linear operator from a Hilbert space ~ into a Hilbert space ~,* -dR = ~l + ~,B is a linear operator depending on T; f,g, and S are pres-

cribed, and T is the formal adjoint of T; i.e.

(82)

Here [. ,.J and f' I·} are the inner products in ~ and ~ respectively, nnd[ . ,.)~, [.,. J ~ are corresponding inner produc-ts formed hy restrict ing thedomains of the entries to ca. The operator B* has the property that itsadjoint is the unit operator; i.e.

fB*,~,u}~ = [Bu,vJatlt, B = 1 (83)

A 1lI0re ~eneral ca.e is discussed in [40J. For variational solutions, it isconvenient to put (81) into canonical form (see, for example, [70,7l,n]):

Tu = v T*v + [ = 0

u '" ~ on d&t B*v (84 )

Following the usual finite-element schem~, we idl'nlify finilt·-dimcnsionaiSub:-;pllceR tflG C U and c.f' c ~ spanned hy rune t ions (cp. (x) }~=1 .1nd r w" (x) 16=1respectively. These d"'e1ine-'proJection operators rI: ~:+ rTlr. <lOll I':?r ~c,fH suchthat G G -

IT - ~ a ~ ryu = u .. L [u,qJ }qJQ'(~) :c L (u,qJa}qJ (~)

Q' ry

Pv

H

v .. L [~'.f~6J~ (~)6

11I[y.,.~~t:J~.u6(x)

~

(85)

w..~

GaB ..

Here [qJ<i'(x)! and [w6(x)} are the conjugate basis functions associated with

[<9. } and Tw }: -Q' .'-

Q' ~ ryeqJ .. L G qJs

a

- M· {\. ['GQ'S = (qJQ"ctls) 1 ; II = Iw ,lll j (86)

Returning to (81) and (84), we consider (our types of projections of (81) :Jndeach of (84) into finite-dimensionul subspac('s:

1. f) c 0 I II . P (Til (1I) - P (v) = (J

II. n(T*p(v) + f) = 0 IV. P(TII - P(v» = 0

These lend to the following discrete analogues of (8l) and (8!~):

18

1. '2>ryflAA + [ry ... 0q

KO'S = [TcpO',TcpSJ

II. LT·llb + f = 0Q'!'i 0'

II

·ll IIT = [Tcp •OJ Ja 0'-

f()(

(88)

(89)

('JO)

(9l)

(93)

(n)II 1.LT~ 6.A0' - IH6.rbr = 0

a r~ 6.r !\LII br + g = 0

['

6. (* I:. 1R = n~,Ro'~I

1I('n' ry.B = 1,2, .... C; !\.r c:: l,2 ..... II. Clnd u =LAO'cp.v = LhA1llf.\.0' . L)

111e four mode ls I - 1 V correspond prec i se ly to the lIsua 1 "di. sp Llcemenl"1II0dei (*). mixed models (II llnd III), and the "equilibrium" mod('1 (IV). SinceC -; 1I. in general, (90) and (92) do not possess a soiutton. lIowevl'r, whcnused together they repreStmt a comp lete mixed modl~l of the prob I ('III (81).Indeed, we can solve (92) to obtain

b = ~H T·AAO'r L r6. all,a

Substituting this result into (89) leads to

(95)

(96)

This rPHult !:lhows precisl'ly the differPllcl' in cllarncll'r of lilt' :;l i (I"nl·:;:; 1Il:ltrixK 8 obtnined using the conventionlll displllcement type fOrlnulat lOll:; ;lIId 1111'

s'flffness matrix RO'8 obtcdoed for the mix(·d Illodel:

(97)

We remllrk that the discrete operators derived via tlw theory of l·onju~~tl'functions l65,ll),64J is obtained here by setting 'lJ = '1-<.r~ =~, H = G, etc.

19

Local Approximations

The heauty of the finite-element method is that all of the models (88) -(94) can be generated from a local level. Without elaborating on all of thedetoils (sec [40J), we list the local forms below: Suppose

qJ (x)(X _. (98)

!tI'Here N is the number of nout's of (>lclIIc.:nt(', (~ i:: dd"incd in (Lf» (with I::.rl'rlnc~u hy ry), ~'(x) slltisfy (30), ~I (x) "" 0 rrr x (.;i , and R is somc lnte-.. ~ . ..Ker. We then have locally

M

which lenu to

(9lJ)

(lOO)

(l01)

r'~1 :: f r , ~l }

L ~If ::~Ir~

Q

c

N R

~ t.lll11ali - I F1 Jhli:: 0

N 101 N J

N .I

F1J - [~I ~JJ('1 - 1/11' .• ,

I 'oJ~:: t",1 F-1 tlr-IJ~ Mill J ...

1,.1

k'" II M + p.., "" 0~M (-' ~L

M

~, co [T~ 'f,II8\ J~~ ~' 'i';"

II, III.

R

~ tle,l bCll1+ ~I c: 0L N 1 ~

i:: 1

1.

Convenicntly

(102)

IV. In this case, wc have locally

(103)

where R is regarded as prescribed on all or aH.. We now introduce lh(' localII

quantities

20

1 _where SC&\~-

1 ..then l\., N

~~ are nodal measures such thatlocally

= efe)~N

(104)

(lOS)

while globa lly

IKarawa = Qrya

Here ~~,..L S 1 ~, S J " I I (~I" (e,I K = !~~'M (~~ M .)~ 1 J ~)M ara

i,j e N,M

~ 2181 I~'Q .. rf~l , wN = Cf'lQary a ~ (81 rye N a

Error Estimates

(06)

( L07)

While we flhall not elaborate on the involved question of <.:rror estimatesfor mixed model , we will cite a result obtained in [40J for an intcrpolationof u by polynomials of order p and of v of polynomials of ord<.:r q [or boundedT of order m and for the case of homogeneous boundary conditions on ~. Ifthe mesh size is

1. lIu - vII. :<;; KhP+l-B1

IIIwh i l£'

IV. Ilt - ~*II~IIT(U* - u*) II+ II~- v*11 + IIT(U- u*) IIand I!T(U* - u*)11 ~ KhP+l-.

INITIAL VALUE PROBLEMS

(10!:!)

(l09)

Error estimates of the type in (59) or (72) can often he lIsed dlrect ly instudies of initial-value problems. The exhLlustive analysis of G:llerkin mcthodsfor paraholic problems by Douglas and Dupont [73J (see also [74.7~1) has beenused to study finite-element approximations of certain linear paraholic prob-lems by Strang and Fix luJ and Fix and Nassif [76J. The work of Price ;mdVargn [77J lays much of the groundwork for the lin0ar parabolic prohlem. andthe cssential step6 in the analysis are outlined in the monogrilJlh of V.lrga[12], who points out that related contributions have been made hy Swartz andWendroff [78J.

21

To illustrate an approach toward questions of convergence for finite-element approximations of parabolic problems and to show the role of the esti-mates (59) in such calculations, we shall review some of the steps given in[l2]. Consider the problem of determining the generalized solution of theparabolic problem

au(at'V) + (~,v) m (f,v) (lI("O),V) = (uo'v) (UO)

o 0 wV v € W; (~) and t > 0 (W; (~) being the complet ion of C (En) wl til compilctsupport in~) and D~u = 0 on Ob~ for I~I ~ m - l, t > O. Here II = lI(x,t)

(t:J~(b~), S, is the s"frongly elliptic linear operator desnihcd ill (76), (1I,V) =r lIvdbt,u(x,t) - 0 on ~, and lIo is the initial valul.'of u. WI' llOW IISC' :.('mi-~diActete Ga"lerkin approximations; i.e., the coeffidents A& of (2) (or Ai of(62» are now as:;umed to be functions of time t. Thus, <lmong elements U(x,t)of II finite-dimensional Bubspace IIIc Cl; (~), we wi sh to determiTl(' the specificmember of the form

U (x, t) = I A~ (t )<P-~(~)0/,/1 ~

(ttl)

whose coefficients satisfy

(112)

c~ ~ (~,~), K'tr: = (!~" (4)~ = (f,~) , 6, r:: l, 2 , ••• ,G, 0/,13 € Zn•

(1l3)

(t1'3)

Among clements of the form (lll), we consider two: first, the finite-*element solution U (x,t) whose coefficients satisfy (ll2), and second, a func-

tion U(x,t) which satisfies for all t ~ 0

(.tiL V) = (!u*, V) V V € IT\. (114 )

Such a function has the property, for strongly elliptic !, that !(U - u*) isorthogonal to 11t Proceeding directly to (110), we see that

aU*(at,V) =

- au- U), V) - (at' V)

(f ,V) - (f)J*, V) -

cu*~, V) + (S,(U*

a(at(U* - U*),V) ~

*Hence, setting V ~ U - U, we get

(~t (u - U) ,U* - U) "" (~t (U* - U) ,U* - U) + (S,(U* - U) ,U''"(- U) (lJ 5)

22

* - * - II 7( ·'11Now Inllking lise of the properties (.£(U - U), U - u) :?: (U - U :>, Schwarz f sIneqlll)lity, the inequality labl S; (1/4y)a':' + yb:>, the identity (orp/ot,cp) =~dllcpllf~ (~)/dt, and integrating once hy parts, we get

* uIlu*(',t) - U(.,t)11 S; 11u*(·,O) - U(.,O)II + /T12Y 1I1~ - ~III (116)

Here ° S; t S; T, 11·11 = II·IIL (~), and?

Illulil = sup Ilu(.t)IIL.,(bt) (1l7)OS;tS;T .

oFrom our previous investigations, we know that if Illc W~(\t) then

Ilu* - iill. s; Kl hr, where r = min f p + 1 - s, 2 (p + 1 - o1)}. lience, from thetriangle inequality

The last tc~rm on the right side of (Jl8) is preciseLy thllt given in (lI6);consequently, we need only bound the two terms on the right sidc' of (1l6).The first term is ~ K::, hr Ilu (. ,0) Ilwg+ 1 (W,). I1S Ciln he SC'l'n by invoking again thetriangle inequality and the fact "that 0*(. ,0) is the bc·st ~-:lppr()ximation inIll. The second term is hounded due to the fact that [l2J

(11 Y)

This follows from the observation one can differenti3te (114) to give(NU/at, V) = (.£'0*/ot, V) so that aU/ot is precisely the (;alerkin npproximationof ou*ot in W; (R).

Collecting these results, we have

Ile(. ,til. = Ilu*(. ,t) - U*(. ,t)ll. S; Khr for 0 S; t .,; T (120)

o 0

where r = p + 1 - s, s = g, ... ,m, 1I*(~,t) (W~+l(~t) (p + 1 ~ m), nnd IllCW;(l"t)is spanned hy functions ~.(::) satisfying those conditbns essential for (114)to hold in the static problem.

Rc'sults on hyperholic equations have been scarce. Stl-:lng :l1ld Fix IltJcomment briefLy on Galerkin approximations of systems of hyperhoLic equationsof the type

ouot + r (12 I)

Usc' of some of the reHults of their Fourier analyses of the cJ liptic problemleads them to the speculation that the Galerkin method will he st:lble for areasonably wide class of prohlems.

We mention also the interesting paper of Kikuchi Rnd Ando [7lJJ. T1Hcy con-sider II general class of operators which have the property

23

R (Au - Av, u - v) ~ Yl\u - vll~ V u, v E ~(A)• (122 )

If V> 0, we obtain a so-called monotone operator, to be di~cussed in the nextsection. If y> 0 and A is linear, we have the m-elliptic operators usedthroughout this paper. However, they also consider the possibility that (122)holds when Y ~ O. Kikuchi and Ando then go on to give a penetratin~ analysisof the stability of a variety of finite element approximations of one-dimensional equations of evolution. The full impact of their work is not yetfully appreciated.

We also mention the recent work of Fortin [80J which arrived too late inmy hnndR to Ret n thorou~h examination. Fortin apparently attncks iI numher ofmathemotil.:111questions connected with triLlnKular approximations of transient,nonlincnr [low problems similar to those studied by aden and Wei Iford [81J.Fortin obtains rigorous error estimatcs and presents n rather exhaustive studyof Bingham and Newtonian fluids.

NONLINEAR PROBLEMS

Whi le applications of the finite-element method to nonlinear boundary andinitial-value problems have been fairly extensive (see, for example, aden[18,l9J), the detailed study of convergence, accuracy, and stability of non-linear problems is in its infancy.

Some insight into properties of a class of nonlinenr bOllndllry-value prob-lems is provided in the series of papers hy Ciarlet, Schultz, and Varga [82-86J.Amollg those they consider are one-dimensional problems of the type

m

Li~O

with periodic boundary conditions

f(u(x),x) x € (;j,b) (123)

(124 )m - 1().~i :,.d1u(a) = d1u(b): 0dx dx

with In (x) I < M V x € (a,b), the operator £ in (12"3) being tn-vIliptic, andf(u,x) ~L1tisfying If(u,x) - f(v,x)1 s; vlu - vi V [(u,x) € L.,[3,b], 1I, v €W~[a,bl. With these assumptions, (123) admits a unique sOl~tiOll (eL Varga[12, p. 26J). Ciarlet, Schultz, and Varga [82J HhOWl'd that if U* is theGal(·rkin approximation to (123) obtained using the spline Sp:lCl'~(L, 6., Z),then

(125)

where 0 ~ .1 .,; m, u* being the unique' gencralized solution of thC' prohlem.Whi le these authors did not specifically consider finite (~lemt'nlapproximn-tiona, most of their rcsults apply to such approximations.

24

Oden [19,41,87J showed that most of the criteria for convergence andcompletencss and, indeed, error estimates on the energy that are availablefor the linear theory carryover to a fairly large class of nonlinear problems,provided the nonlinear operators involved are, in a sense, positive definite.For example, consider the nonlinear boundary value problem with appropriatehomogeneous essential boundary conditions

( 126)

where u f~, (} is a nonlinear operator with the following propL'rti('s:

(i)

(11)(iii )

(}(u) hus n linear, continuous G~teaux differential~very point u in some convex set ~ C ~(6(}(u,v),w) = (o(}(u,w,),v)(o(}(u,w),w) ~ y(w,w)

oGl( u , v) at

(l27)

a [lInc tJ ona I J (u) whoseof a solution to (126) (cr.

The first two conditions insure the existence ofgradient is 0 and (iii) guarantees the existence[88J). Thus. we may develop the functional

1

J(u) -J (9(9U),u)d8 - 2(f,u) ; 6J(u,w)

o

((}(II) - f ,w) (128)

which .1SSunWH a minimal value at the generillized solutiontion (1) enRures that Gl slltisfiefl the Lagrange formula

o(r1I of (IZh). Condi-

«(}(u + w) - G>(u) , v) = (6G>(u + Aw, w), v)

which allows us to write, after Some algebra,

1

IJ(U) - J(u*) I • J (869(u* + <w), w)ds < KII_II"o

o < A < I (l2Y)

( 130)

- *Hen' U is the finite-('lement intcrpo lant of \1* lIs(~d previous ly ,.lOci W = U - uThus, we obtain an estimate of the same form as (20) and, consequently, (23)and ultimately (53) of thc linear theory.

The entire development leading to (130) js appilrvlllly ('(juivnl('nt to thatof Melkcs [89J, who discusses finite-element approximat ions of fillite ly ~-tinuous monotone operators (see, e.g., Zarantonello [<JOI or V:Jq-\<l [12]). Anoperator (} 1s (strongly) monotone if

((}(u) - (}(v). u - v) .: {(u - v, u - v) (l3l)

It is finitely continuous on a Hilhert space ~ if ~l.11l (@CPk'v) '" (I.I\I.V) whereCPk E: J;f and lim Cj\ = u. The equation (Gl(u),v) = 0 ,ln~ til<' G.11l'rkin ('(juation«(}(U),V) - 0, then have unique solutions [Yl.! (u,v (U, U,V (~n.c tI). DothMelkes [89J and Ciarlet et al [86J consider fj nitcly continuous ~;t rOllgly

25

monotone operators of the form

@(u) = I \ryl(-1) - D (A (x,u, ... ,Deu))ex ex- I-JI~J~ - '- --where 0 s I~I s:; m - 1. Then one CAn ~eek generaliz(;d solutions to

(132 )

(133)a(u,v)wherea(u,v) := 0 = J ~ A (x, II, .•. ,DQu)D vd~iL ry -- µ ry

6t I~y"1~m-- "--If U* (rile Si satisfic>s a(u*,V) := 0 V V (ITt. then it (;;)11 he ~:llo\.Jn to followtho t

(I·j4)

The rClllDLnder of the ana lysis conl inue6 as hefor(;.

01~ER TOPICS AND CLOSING COMMENTS

We have not yet touched upon [\ lIumher. 0 f important topll's. For exump le,we not~ that since the pioneering paper of Johnson ~nd McLay [bJ on convergenceand error estimation of finite-element opproximalions on domains with corners,a number of pApers have appeared which attempt n more rigorous altack of thesubject. The idea of superimposing clOAlytic and finite-element solutionsproposed by Yamamoto [92] was mod i fied and used by Rao et a I [91 J to developspecial elements for usc in fracture nll'chanic!' :-;ludit:s. Pian, Tong, and Luk[94J developed hybrid dements with built-i.n singularities, and (;nl Lagher [9SJhas prepared .1 rather extensive bihliography on flnJ.t(,-clpmcnl ,'pplicationsin fracture mechanics. Recent mathenwticcll lilt:r:ltIIH' Oil Ilw subject includesthe work of Fix [96]. th(' pllpers hy B:lhusk:l [2')J on dOlllLlins wIth corners (S('('

Also [Y7,9SJ), Birkhorr's work on LJllgular sin~lll[lrilics [991, tIll' n'c('nl p;lperof Wait llnd Mitchell [JOOI, and thL' intt'resting :lnillysis of re-('1I11-:1l11- conwreffects hy Barnhill Ilnd Whiteman [lin!. Adcl1l:ionnl n·fct"cTlLT:: C;lll he found in[15,lfi,9SJ.

Also, since Ergllioudls, Irons, <lnt! Zic'nkiewicz LL021 inlroUIICl'd the ideaof curvL,d isopllrametric finite clements, and since they prov('d to hI' so POWff-

ful in pract1cal nppl1c/Jlions Ll03J, n great dent of interest in these e10-.rents has b('l'n gfmerlltt·d. Apparently the first :Itlempl at st\l(Jylng Lheit-mntlH.:mntic:ll properties WllS that of Korne0V [57J :ll1d Fried L'dl. Fried's worklHls heen summarized in [I04J, and the subject ha:·; been furth('r tli :;clls:;ed byCinrlet ond Raviart rJ),J4J and Zlamnl [10SJ.

Questlun" of convcrRcnce of nonconforming clf'1ll0nts ;)nd Itl tilt· effectsof numerical integration of stiffness matrices Clre apparenlly lliHIct- study,and will probnbly be resolved by the dale of puhlication of lhl'se proceedings.Alon~ these lines we mention the work of Oliveira [106J and Rilbll:;I(;1 :lnd Ziama 1[107L

26

The p<.~netrot ing work of Fried [l08-l LlJ on the condition numher (stahility)of stiffness matrices llnd on accuracy of eigenvalue approximations [lIZ] mustalso he mentioned. If we consider the L::! condition number C::! (K) of thestiffness matrix K: _.

(135)

where 'A:~. and }..K are the maximum and minimum eigenvalues of K,and if M is theconsi8t~nt l11Ds1matrix of the finite element model, Fried snowed that·

max(~8 )Pe n maxs:min (}...t..e )c 1

(136)

andk

mDX 0.._" )~

}..lmax(}..~e)PC n max

max(A:~e)Pmaxe n

~1 m~n (~." )

(137)

where (~ is the element index, A..1-..'.':O/-'-~·) i~ the lTl::lXimUIIl(minimum) cigcnvillueof the locol :-;tiffness and massDmatr!ccx, A1 is the fundamental (,-equency ofthe continuous problem, and P is the maximum numher of clemC'nts 1lIl'C'ting atu node. In addition. Fried o~!~ined t condition numhers. If G(x,~) is ~heGreen's fllllctlon for the continuous prOblem and [" ;: m<Jx G(x,x), Fr·(ed proved

xthat for the N- th order problem

C (K) S:NP rmaxllkC")11CXl .'. mllx e _ <Xl

For simplex representations, (137) leads to estim;!tes of the type C.,(K) =O(h-2) whereas C (K) = 0(h-1) as h .... O. " -

00_

(138 )

Our investigation has taken us to the limits of the space avnilable forthis survey, but many subjects have scarcely been touched upon. It is hoped,however, that the breath of modern developments in the mathematical theory offinite elements has been scanned. along with a few of the basic approaches tosome of the more hasic questions. While the list of IH.'\" reft'rc'nc(~s is long,therC' is still plenty of work to do. The mathemnticlll theory, pilrticularlyin the areas of nonlinear and nonHtationary prohlems, is rich with prohlemswhich challenge researchers for years to come.

ACKNOWLEDGEMENTS

Thanks are due Prof. M. Zlamal, Brno Univer:;ity, Czc'choslovakifl [ormaking available to me an early draft of his article [17J. 'I11is was very help-ful in selecting appropriate topics for this survey. It is ::Ilso a pleasureto acknowledge the benefit I received from recent conversations with Prof. I.Fried of Boston University. Informal discussions with Dr. P. Ciarlet, Paris,were beneficial in preparing Section 2.l. My own work on finite elements hasbeen supported by the U. S. Air Force Office of Scientific Research under

27

Contract F44620-69-C-0124.Wigent receives my sinceremanuscript.

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I am very grateful for this support. Mrs. D.thanks for her painstaking job of typing the

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36