Reprinted fro~

Computer methodsin applied

mechanics and• •engineering

Computer Methods in Applied Mechanics and Engineering 112 (1994) 309-331North-Holland

Optimal h-p finite element methodsJ. Tinsley Oden

TUIU lrutituu for ComputDtionai and Applied Mathemalic.J, the University of Texas at Austin, USA

Received 14 November 1992Revised manuscript received 8 January 1993

" '!I'f .~',.,~ r. .. (I~"

COMPUl'ER ME11fODS IN APPLIED ~NJCS AiID"ENGINEERlNG ~J.1:So ' :EDITORS: J.B. ARGYRIS, STUTl'GART aod LONDON ..~~]; -.J..i~

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Computer Methods in Applied Mechanics and Engineering 112 (1994) 309-331North-Holland

Optimal h-p finite element methodsJ. Tinsley Oden

Texas Institute for Computational and Applied Mathematics, the University of Texas at Austin, USA

Received 14 November 1992Revised manuscript received 8 January 1993

A theory of optimal Petrov-Galerkin, h-p version, finite element approximations is presented. The optimalscheme is defined relative to a fine mesh solution space and relative to an arbitrary symmetric bilinear form. Theoptimal method leads to a symmetric, positive-definite stiffness matrix which is independent of the coefficients of thegiven problem. exhibits 'extra superconvergence' properties, and has a relative error that can be calculatcd exactly,at each point in the problem domain. Various gcncralizations are also discussed. including the connection of thesemethods with certain prcconditioning schemes.

1. Introduction

Early literature on Galerkin formulations of boundary- and initial-value problems made a distinctionbetween the space of trial functions. in which the solution is sought, and the space of test functions,against which the residual acts. The notion of trial function probably had its origin in optimizationtheory. in which one sampled functions from a class of admissible functions in an attempt to find onethat minimized a cost functional. Test functions may be inspired by the theory of distributions, but thenotion of test functions is more primitive. In problems of physics. the fluxes are characterized by theiractions on arbitrary test devices; stress, heat flux. etc .. are only measurable by actions on measuringdevices or transducers (test functions), and the space of test functions represents a mathematicalcharacterization of this notion.

In the mid-1970s, distinctions were made in the finite element literature betwecn Bubnov-Galerkinmethods, in which the spaces of trial functions and test functions coincide. and Petrov-Galerkinmethods in which the spaces of trial and test functions are different. Indeed, beginning with some of themore fundamental literature of the 1970s, the idea emerged of designing special trial or test functions toachieve a particular property of an algorithm.

Among the early Petrov-Galerkin techniques for finite elements was the upwinding scheme ofWahlbin [1] for first order linear hyperbolic problems. The subsequent work of Dendy [21 dealt withstudies of upwinding schemes for linear problems. The paper of Christie et al. [3] dealt with optimalupwinding of linear convection problems and made clear the role of special test functions for stabilizingnumerical schemes for such problems. These schemes were predecessors of the SUPG (steamlineupwinded Petrov-Galerkin) methods of Brooks and Hughes [4], studied in some detail by Johnson eta!. (e.g .. [5]), Barrett and Morton [61 discussed the exact construction of optimal test functions forone-dimensional elliptic boundary-value problems and Demkowicz and Oden [7,81 developed optimaltrial and test functions for both one- and two-dimensional linear parabolic problems. Rccently, thedevelopment of special test functions to deliver specific properties has been exploited by severalauthors. We mention in particular the ELLAM methods (Euler-Lagrange Local Adjoint Methods) otCelia et al. [9] (see also the references in this work). More recently, Brezzi et al. [101 showed the role ofinternal 'bubble functions' in producing algorithms which essentially mimic properties of SUPGschemes. With few exceptions (e.g., [4,8,10]), the papers mentioned deal with one space dimensionand, thus. are not immediately extendable to two- and three-dimensional cases.

Correspondence to: Professor J. Tinsley Oden, Texas Institute for Computational and Applicd Mathematics, 3500 WestBa1cones Centcr Drive. Austin, TX 78759, USA. Tel.: (512) 471-3312; Fax: (512) 471-8694.

0045-7825/94/S07.00 © 1994 Elscvier Science B.v. All rights reserved

310 J. T. Oden, Optimal hop finite demelll methods

In the present exposition, we present a general theory of optimal Petrov-Galerkin methods for hopfinite element approximations of linear boundary-value problems. We introduce the notion of a-optimaltrial functions and a-optimal test functions relative to a fine mesh approximation rh of the space r towhich the exact solution belongs. Here 'a-optimal' refers to optimizations with respect to an arbitrarysymmetric positive-definite bilinear form defined on 'JI, independently of the bilinear form characteriz-ing the boundary-value problem under consideration. The a-optimal Petrov-Galerkin approximationbelongs to a space rH C 'Jl'h and corresponds to a coarse-mesh approximation of the problem.

We prove that such a-optimal hop approximations possess a number of remarkable properties. First,one can calculate the a-optimal solution using a symmetric positive-definite matrix, the entries of whichare independent of the coefficients of the given problem. The exact relative error between the a-optimaland the fine mesh approximation can be precisely calculated anywhere in the domain of the problem, itexhibits 'extra superconvergence' properties in that, independent of the regularity of the exact solution,it vanishes at degrees of freedom shared by the fine- and coarse-mesh approximations. This represents ageneralization of the various one-dimensional constructions designed to produce approximations withzero error at nodal points. In addition, in the present construction, exact expressions are derived for thepointwise relative error at any point in the domain, and, of course, from such expressions, the exactrelative error can be evaluated in any norm. In addition, it is shown that local (elementwise) versions ofa-optimal trial and test functions can be constructed, so that even though a-optimality is a globalproperty valid for the connected finite element mesh, essential properties of such approximations can beproduced by local element shape functions.

Finally, the relationship between matrix condensation and a-optimality is established. It is shown inparticular, that appropriate constructions of Schur complements of submatrices of the global and localstiffness matrices effectively reproduce the a-optimal properties of the optimal Petrov-Galerkinscheme. This explains, in part, why the condensation of bubble functions reported by Brezzi et al. [10\led to schemes similar to upwinded Petrov-Galerkin method.

Following the presentation of the theory of a-optimal approximations in the context of a modelconvection-dominated elliptic problem in two dimensions, several generalizations of the theory arediscussed.

2. Model problem: 2D equilibrium problems with small coefficients

We begin this discussion with the consideration of a model class of steady two-dimensionalboundary-value problems of the form

- 'V . (;'Vu + C • 'VII + (Ill :::::f in n ,Ii = 0 on an, (2.1 )

where the coefficient (; of the diffusion term is assumed to be very small in comparison with the othercoefficients,

0< (;« (]', (;« lei·

Thus, if c "" 0, problem (2.1) is 'convection dominated,' but even if c == 0, the fact that (; can be verysmall in comparison with a leads to notorious computational difficulties. To simplify the discussion, thecoefficients (;, c, (]' arc taken to be constants, although variable coefficients present no conceptualdifficulties in the analysis. We are also interested in the non-convective case (c == 0) since othertechniques for treating convection-dominated problems are also discussed. In (2.1), the domain issimply a rectangular region in ~2 with boundary an and the source term f E L2(n).

In the sequel, we denote by H'(n) the usual Sobolev spaces of functions defined on n withdistributional derivatives of order r ~ 0 in L 2(n) and with norm

J. T. Oden. Optimal hop finite element methods

with dx = dx I dxz and Dav = aal +a2v / ax71 ax;2. Uj = integer;a. 0, i= 1,2. When the domain und'consideration is clear from the context, we simply write IIvll, for Ilvll,./l'

To recast (2.1) in a functional setting, we introduce the space

with norm

'Y = H~(n) = (v = v(x) E HI(n): v = 0 on an} ,

Ivll = {In VV·VV dx} 1/Z ,

(2.2

(2.21

with the boundary values of v interpreted in the sense of traces of H I-functions, and we introduce tIbilinear and linear forms

9A : 'Y x "Y-IR: g)}(u. v) = In (EVU'Vv + c 'Vuv + eTllv) dx.

!t : v -IR: !t(v) = r Iv dx .JllThe weak formulation of (2.1) is thus: Find /I E "Y such that

913(/1.v) = !t(v) 'fIv E 'V .

(2.3

(2.31

(2.·

REMARK 2.1. It is not difficult to verify that !t and ~ are continuous on 'Y. In particular, a consta:fno > 0 exists such that for any It, v E 'Y.

Also, @(. ,. ) satisfies a Garding inequality,

for any v in 'V. where Co is any real number such that 0 < Co < I and Ilv II~ = In VZ dx. Thus. if c="O or

convection is small enough that lelz:s:;; (1 - co)w. then

(2.~

whereas if lei"" 0 and lei> (1 - co)w. then

REMARK 2.2. It is easily shown that a unique solution u exists to problem (2.4) and, therefore. ualso the solution to (2.1) in the sense of distributions.

312 J. T. Odell, Optimal hop fi/lite dement methods

REMARK 2.3. The presence of small s in the definition of the model problem strongly inOuences thestructure and stability of the solution u. If 1c12 ~ (I - Co)HT,

Thus, any change in f is magnified by (Cosr 1 in bounds on the corresponding change in u. If c = 0, theL 2-term, u(u, v), dominates gl) and the addition of small diffusion, s(Vu, Vv), means roughly speakingthat any gradients in the solution contribute to the energy only in terms of order s. In the case of pureconvection (s = 0), the problem becomes hyperbolic. For the time-dependent case, boundary data froma portion of r of an propagate along the characteristics, the lines dx / dt = c. The addition of smalldiffusion then generally produces a continuous solution in which the (possible) jumps across characteris-tics are smoothed and spread out over a region of width 0(Ve). In this layer, u and Vu may undergorapid changes and, depending on the boundary conditions, very thin boundary layers can be developedalong an of width O(e).

3. Discretization of the model problem

We consider two hop finite element approximations of problem (2.4): an approximation uh of u on a'fine mesh' on n, and a coarse mesh approximation ull on a 'coarser' submesh. The terms 'fine mesh'and 'coarse mesh' may have nothing to do with the fineness or coarseness of the FE mesh, but merelyindicate an hop approximation spaces each involving different numbers of degrees of freedom, onebeing a proper subspace of the other. We will ultimately design ul/ so that it is an optimalapproximation of Uh on the coarse mesh, in a way that will be made precise later.

3.1. Shape functions and mesh

Let {Elh} denote a family of partitions of ti into meshes of regular quadrilaterial finite elements. Todefine a mesh (a partition) 2lh, we consider first a master clement f1 which, for present purposes, can betaken to the square, j) = r -I, 1]2. The fine mesh is produced by a family of NIH•EM affine invertiblemaps FK' where

U tiK, nKnnJ =0 if K¥J.l"'K"'NELEM

Here TK is an invertible matrix and OK is a translation vector. Standard properties of these partitionsare assumed to exist, e.g. (J) tiK n tiJ is either a common vertex or side or is empty; (2) iftiK n tiJ = rKJ is a common side, then F~l and F;I coincide on this side; (3) vertices of n coincide withthose of some nK' ctc.

On f2 we construct families of hierarchic shape functions organized into three categories:

(1) Vertex (or /lode) functions

~1(g,'1/)=t(J +00 +'1/),

~3(g, '1/)= to - 00- '1/) ,

JJ2(g,'1/)=t(I-g)(1 +1]),

tfr4(g,'1/)=:tO +0(1-1]),(3.1 a)

J. T. Oden. Optimalh-p finite element methods 3]

(2) Edge functions

-(1) (-It(k (~,17)=~(1 +17)Pk(~)'

'(2) (-1/(k (~.17)=~(I-OPk(17),

- (3) 1(k (~.17)=2:(1-17)Pk(~)', (4) 1(k (~, 17)=2:(1 +17)Pk(17).

k =2 .... ,p,

k =2, ... ,p.

k =2, ,p.

k =2, ,p,

(3.tb

Here Pk are the integrated Legendre polynomials.

where Pk-I is the Legendre polynomial of order k - I,

(3) Bubble (or internal) functions

(3.2

(3.3

When mapped to a typical clement flK in the mcsh. these functions become thc local element shapfunctions for the clement and arc denoted

i = 1. 2, 3, 4.

i= 1,2,3.4. k = 2, ... , P ,

i.j=2 ..... p. (3.4

REMARK 3.1. Numerous generalizations of (3.1). (3.3) are possible. For example. polynomials cdiffering degrees p 7' could be used in the definition of each edge function ,~/) and polynomials cdifferent degree ql and q2 could be used in the product functions Pj(OPj(17), i=2, ... ,q\, j:2, .... q2' This would result in 4 + E:= I (p 7' - I) + (q\ - I)( q2 - I) degrees of freedom for an elemen~Also, irregular II-meshes could be used in which interelement constraints are imposed to enforecontinuity of basis functions across interelement boundaries, such as described in [11]. While datstructures that employ this level of sophistication are used in our applications. they are unnecessary fcour present purposes, which is to demonstrate basic properties of hop methods on a simple modiproblem,

An interpolation function defined on fiK as a linear combination of the local shape functions is of thform

4 ( P ) PK _ i K k K(i) ij KVI/(x) - j~ VK"'i (x) + k2;2 VKi(k (x) + i.~2 vKbjj(x).

x E fiK, fiK E !!ll" 1~ K ~ NELEM . (3.:;

Globally. over the connected mesh of finite elements, we demand that the finite clemenrepresentations be continuous. Let ~, Z~) and /J~/' denote the global basis functions associated with th

K Kr) Kshape functions l/J;. (k I and bjj, respectively. i.e"

~(x)lf) = l/J~(x) if node.::1 coincides with node i of element ilK'K

(3.6

314 J. T. Odell. Optimal h-p finite element methods

etc. Then the space of hop finite element functions generated on Ii is

'Vh = H~(il) n span{ {~}, {Z~)}, {B~} ; .1= 1, NNODES ' N = 1, ... ,NEDGES •

Ct = 1, ... ,NELEM; k = 2, , PN' i, j = 2, ... , PK} , (3.7)

and a function vh in "VII is a continuous function of the form

NNOPI'.S NEDOI,S PN NELEM PK

vh(x) = L V4~(X) + L L vZZ~)(x) + L L v~B;j(x).1-1 N-t k=2 a-I i.j-2

andv"laf2 = 0 .

(3.8)

(3.9)

The fine mesh approximation of (2.4) is now characterized by the following problem: Find tih E "V'Isuch that

(3.10)

REMARK 3.2. In (3.8), PN is the highest degree of the polynomial edge shape functions sharing acommon edge on an interelement boundary and PK is the highest order polynomial bubble function forelement ilK E2lh. Thus, (3.8) provides for the use of edge functions ,~i)of different degrees fordifferent edges (recall Remark 3.1) and, therefore, for a nonuniform distribution of spectral orders Pover the mesh.

3.2. A priori error estimmes

The space "Vh possesses standard interpolation properties of h-p finite element methods (e.g" (12]).Thus, for element ilK' if h K = diam(ilK) and P K is the lowest degree of the complete polynomialscontained in the span of shape functions associated with this element, then. for any functionIV E H'(fl/(), r> 0, an interpolant w" of the form (3,8) can be found such that

where s = 0,1 and c is independent of hK, PK and w. Then. globally,

NELEM h2vK

IIw-whll;J)~c2 L 2(~-2)111V11;.J)K. K=l PK

~ c2J.L2(h, p)IIIVII;.11 '

(3.11 )

(3.12)

where J.L=suPK(h~Klp~-2). If r>PK+s for all K and if h=maxhK and p=minKPK for !!lh' then,asymptotically as h -+ 0, P -+ 00, we may assume that

I p+s-r

Ilw - ~v'llls.12 ~ C 'pr-s IIwllr,J)' (3.13 )

Since 'V', C 'V, we can set v = vh in (2.4), subtract (3.10). and obtain the orthogonality condition,

where e"

is the fine mesh approximation error,

eh = U - uh '

(3.14)

(3.15)

J. T. Oden, Optimal hop finite elemeTIImethods

By standard arguments (e.g., [13, 14]), we know that when (2.5) holds

and, therefore, if u E H'(il) nH~(il), the following a priori estimate of the error holds:

31

(3.lt

We note that this error bound, unfortunatcly, depends upon E-I which may be a very large quanti~

3.3. Simplified case

In order to simplify thc discussion and to bcttcr manage the notation, we focus on a simplified speci,case depicted in Fig. 1. Here each element has nine degrees of freedom and is the affine image ofmaster element iJ equipped with the following shape functions:

(1) Vertex

.j,1.2.3.4(~' TJ) = t(1 ± ~)(1 ± TJ) ;

(2) Edge

.J3 2p(s) = Vg(s -1);

(3) Rubble

As a further simplification, we denote

(3.17.

(3.17t

(3.17(

so that functions v defined over these shape functions arc rcprescnted as the linear combinations

8

v(~, TJ) = 2: v(~.j,j(f TJ) + Vbfj(~, TJ)·ja\

For clcmcnt ilK'

8

v:(x) = ~ v~t/Jf(x) + v~b\x);=1

Mr NNElEM

vloE rio : vlo(X) = 2: V~1Ji'N(X)+ 2: V; B,,(x)N=I ,,-I

(3.H

(3.1~

(3.2(

(bK(x) = SKu B,,(x), X E flK), and Viola!} = O. For thc mcsh shown in Fig. 1, Mr = NNODES + NEDGES = 4-12 = 16 and NELEM = 9.

316 J.T, Oden, Optimal hop finite element methods

• • •

• • •

• • •

IXl

T4d1

1

2

2

3~ ~ HI--1-1--1 -..j

Fig. I. A rectangular mesh of nine-nolled rectangular hierarchic clements.

The fine mesh solution of (3.10) is of the form

MI" NELEM

uh(x) = 2: U~1/'N(X) + 2: U;Ba(x).N=\ K-t

(3.21 )

so that upon substitution of (3.21) and (3.22), we arrive at the system of equations

(3.22)

KNa = f!l1(1/'N' Ba) .

K"f3 = f!l1(Ba' Bf3) . Fa = .:£(Ba) .(3.23)

Arranging the vertex and edge degrees of freedom into a vector Vo and the internal degrees of freedominto a vector Vo, we write (3.22) in the matrix form

(3.24)

Then, eliminating VB by 'matrix condensation', we have

(3.25)

J. T. Oden. Oprimalh-p finite element methods

The coefficient matrix of Uo is the Schur complement of Kuu and we refer to the right-hand sivector as the Schur complement of Fu. Denoting these by

(3.2

(3.25) becomes

(3.2'

A similar construction can be made at the element level. Since

OOK(U, v) == 1 (eVu· Vv + c· Vuv + emv) dx ,Ilx

:l'K(V) == 1 fv dx .Ilx

the fine mesh solution satisfies the local equation

Thus.

(3.2:

(3.2'

(no sum on K or b) . (3.3'

where repeated indices i. j are summed from 1 to 8 and

(3.3

Then. in obvious symbolic notations, for each element flK E 2lh•

( K K (kK )-lkK ) ~ K K (kK )-l(fK K)kuu-kub bb bU UOK==Jo +uo -kUb bb b +ub '

K (kK)-I[fK K kK ]U b == bb b + U b - bUUOK ,(3.3

(3.3

4. Optimal hop finite clements: the coarse-mesh approximation

We now consider a 'coarse-mesh' approximation Uti of (2.4) which has the following features:- uti belongs to a subspace 0Zl tI C rh of trial functions. We will design OU tI to have special properti

relative to rh•

- uti will be constructed using an optimal Petrov-Galerkin approximation of (2.4), where the meanilof 'optimal' has to do with the specific design of spaces of trial and test functions and is to be malclear below.

318 J, T. aden. Optimal hop finite element methods

Towards constructing such approximations, we introduce formally two coarse-mesh spaces:

CUh = span{XN}~~1 = space of trial functions,

'WH = span {PN} ~~ J = space of test functions,

CUH C "Vh, 'WH C "V", dim CUll = dim 'WII = Mr.

(4.1 )

If dim "Vh = Mr + No (No being the number of bubble functions), we will denote by "VII theMr-dimensional subspace of "Vh spanned by the functions {1/'N}~~I' The precise definitions of the basisfunctions XN and cJJN are given shortly.

The coarse-mesh Petrov-Galerkin approximation of (2.4) is characterized by the following problem:Find UII E CU II such that

(4.2)

The approximation error ell = II - VII satisfies the orthogonality condition

(4.3)

and, since

(4.4)

thc relative error betwecn thc Iinc mesh and the coarsc mcsh approximation,

(4.5)

satisfies the orthogonality condition

(4.6)

4.1. Optimal Petrov-Galerkill methods

We now design the trial and test functions to be optimal relative to a specified bilinear form on "V. Inparticular, let a : "V X "V -- R be any bilinear form on "V possessing the following properties:

(a.i) a(·,') is symmetric,(a.ii) a(· .' ) is independent of OO(... ); i.e., a( . " ) is independent of E, c and u,(a. iii) a(· " ) is positive definite.

For cxample, one choice of a( . " ) is

a( II, v) = In Vu' Vv dx. u. v E "V .

The norm associated with a( . , ' ) is denoted

(4.7)

(4.8)

For example, if a( ... ) is given by (4.7), then IIv IIa = Iv I J' Also, restrictions of a( . , . ) to elements ilKare denoted aK(',' ); i.e., a(u, v) = EK aK(u, v).

In the discussion to follow, a(· " ) will be a bilinear form satisfying conditions (a.i), (a.ii) and (a.iii).

J. T. Odell, Optimal h-p fillite element methods 3

DEFINITION 4.1: a-Optimal trial functions. A set of linearly independent trial functions {XN}~~Isaid to be a-optimal relative to rh if and only if

(4.~

Let {X N} ~~ I be such a set of a-optimal trial functions and let .stJ denote the triple, (a( . " ). 9B(. , .{XN} ).

DEFINITION 4.2: a-Optimal test functions. For any triple .stJ, a set of linearly independent te:functions {CPN} ~~ I is said to be a-optimal relative to .s4 and rh if and only if

(4.H

Henceforth, we refer to sets of functions {XN}, {CPN} satisfying (4.4) and (4.5) as simply a-optimal trioand test functions.

For given g/J( .. ' ), a( . ,. ) and rh• such a-optimal trial and test functions always exist. It is seen th.

a-optimal trial functions are a-orthogonal to the functions in the fine mesh space rh that do not belonto rH

. In the case of the simplified examples given in Section 3, the functions XN would be orthogon;to the bubble functions B". Then (4.9) becomes

( 4.11

It is noted that the X N are independent of @( . , . ) and. therefore. do not depend upon e or c or (The a-optimal test functions CPN do indeed depend upon 00(' .' ) and. therefore. on e and c and (.

The construction (4.10) can be regarded as symmetrizing 00(· .' ) (if c ~ 0) or. in any case, renderinthe discrete problem to one independent of E (and. of course. CT).

A number of obvious properties of a-optimal Petrov-Galerkin approximations can be listed, blthere are also several not so obvious properties that are very important. We record these in the ne>section. But we first show how such optimal functions can be constructed.

4.2. Calculation and localization of a-optimal trial and test functions

We now record two theorems which provide concrete formulas for calculating a-optimal trial and te~functions and for the construction of local a-optimal trial and test functions for each element.

THEOREM 4./. (a-Optimal trial functions). Let a :r X r -IR be a bilinear form satisfying condition(a.i). (a.ii) and (a.iii). and let "JI.h = span{ {1/'N}~~I' {Ba}~~\EM}. Then(i) the functions

XN(X) = 1/'N(X) - aNaa"fJ BfJ(x), 1:!S; N:!S; Mr- 1:!S; a. {3:!S; NELEM

are a-optimal trial functions with respect to '}I'h, where

(4.12

(4.13

and repeated indices are summed over their ranges.(ii) The restrictions of the function XN to element ilK E 2lh are the local a-optimal functions.

and satisfy

(4.14

(4.15

320

(iii) The global matrix

J. T. Odell, Optimal hop finite element methods

where aNM = a(1JtN,1Jt.~f) and Y(aNM), is the Schur complement of aNM'

PROOF. (i) A direct substitution of (4.11) reveals that

(4.16)

Thus, XN satisfies (4.11).(ii) Because a"p = (a(B", Bp» and aap are block diagonal, the restriction of XN to DK is precisely the

function defined in (4.14). Local orthogonality condition (4.15) follows by direct substitution.(iii) This also follows by direct substitution of X,~f into a(1JtN,XM). 0

A similar collection of results holds for a-optimal test functions.

THEOREM 4.2. (a-optimal test functions). Let a: r X r -IR be a bilinear form satisfying conditions(a.i), (a.ii) and (a.iii), with 00 : 'Y x 'V -IR given by (2.3a), and let rh = span{ {PN} ~~1' {B" }Z~~EM}.Then(i) the functions

are a-optimal test functions relative to (a(· .. ), 00(· .' ). 'VII), where

(4.17)

and[~(X) = 1JtR(x)- KpRK"P B,,(x)

A Z = Y(KMR)-IY(a,~IN)

(4.18)

(4.19)

and repeated indices are summed throughout their ranges. Here KNll' K"N' KNR and K"p are thestiffness arrays defined in (3.23),

)-1Kap = (K"p ,

and Y(aMN) is the Schur complement of aMN defined in (4,16).(ii) Moreover. the functions CPNsatisfy the orthogonality condition

(iii) For each element nK E IQh' the local shape functions

j summed over its range. where

K'and the constants a j I satisfy

(4.20)

(4.21 )

(4.22)

(4.23)

(4.24 )

(4.25)

J. T. Oden. Optimal hop finite element met/rods

are local a-optimal test functions in the sense that

and

and index N corresponds to index i of ilK'

PROOF. (i) An arbitrary function vh E rh is of the form

Thus. a direct substitution of (4.17) and (4.19) yields

@(vh, CPN) = @(V~'lJIM + V;BA• AZrR)

= A ZV~[@(lJIM' 'frR) - KfJRKafJ@(PM' Ba)]

+ A ZV;[@(BA,lJIR) - KfJRKafJ@(Ba, BA)]

= A ZV~'(KMR - KMaKafJ KfJR) + 0

= V~'A ZY(KMR)

=V~'Y'(aMN) (by (4.19))

= a(vh,XN) (by (4.16».

In view of (4.9). cPN is an a-optimal test function.(ii) It is obvious that

£!J.J(Ba' cJJN) = A Zoo(Ba. lJIR - KfmKAfJBJ

= A Z(K"R - KfJR13~)=0.

3,

(4.2c

(4.2'

(iii) Owing to the block-diagonal structur~ of f!JJ(Ba, BfJ)' it is easily seen that the function 1:of (4.2·is the restriction of FN of (4.18) to ilK' with an appropriate relabeling on indices. The coefficieex:i correspond to element matrices which. when assembled, produce the global matrix A Z(4.19), i.e .. if s is the assembly operator.

The matrices k~ and Y(k~) are semi-definite; therefore, (4.25) holds.Condition (4.26) follows from a direct substitution of (4.23) into @(bK

• cp~). 0

REMARK 4.1. Let A: 'V - L 2(il) be the operator defined by

oo(u, v) = (Ali, v) = (Ii. A*v) •

where A* is the adjoint of A. Since (by virtue of (4.26))

the local optimal test functions are designed so that the local adjoint AK maps them into a spal

322 J.T. Oden, Optimal hop finite elemelll methods

orthogonal to the bubble function bK in L 2 (ilK ). The related Local Adjoint Method and the ELLAMapproach (see [9]) choose test functions in the kernel of A;.

S. Properties of a-optimal methods

Reviewing the notations of the preceding section, we denote- a: 'JI' x 'JI' -+ ~ a bilinear form satisfying conditions (a.i), (a.ii) and (a.iii).- {XN}Z~I a set of a-optimal trial functions relative to a fine-mesh finite element space 'JI'h'- OUH

= span{XN}~~l'- {lPN} ~~ I a set of a-optimal test functions relative to the bilinear form OO(. , . ), the space 'JI'h, and the

a-oftimal trial functions {XN} ~-l'

- 'W = s..ran{<PN}~~I'- uH = UUXN (sum on N) = the coarse mesh Petrov-Galerkin approximation satisfying (4.2).- ehH = uh - Uu = the relative approximation error.

The most obvious property of the a-optimal Petrov-Galerkin approximation is that it is characterizedby a symmetric positive-definite system of equations. We record this in the following proposition.

PROPOSITION 5.1 (Symmetry). If uH(x) = uZXN(x) is the solution of (4.2). then the coefficients aredetermined by the system

(5.1)

where A.\IN is the symmetric positive-definite array

(5.2)

and X Nand (PN are a-optimal trial and test functions, respectively.

PROOF. This follows immediately from the fact that 9lJ(uH, lPN) = ll(ul/' XN) = u~A,\fN' 0

The advantages in computing the coefficients U~ using a symmetric positive-definite matrix are,unfortunately, nullified by the work needed to compute the right-hand side. Indeed, using the notationsof (3.23) and (3.26).

.P(CPN) = AZ.:t(1J'R - KR"KAUBA)

= Y(KRM)-IY(IlMN)(FR - KRuKAa FA)

= [Y(KRM)-I(FR - KR"KAU FA)]Y(aMN)

= U'~Y(aMN) , (5.3)

where U~ are the coefficients of the fine mesh vertex and edge functions. Thus, it appears that onemust essentially solve the original fine-mesh problem to obtain the vector .P((PN). Fortunately, in manycases this needs to be done only once locally. for one typical element, and then used successively foreach element.

PROPOSITION 5.2 (Coincidence of coefficients). Let uh(x) = U~1J'M(X) + U;BA(x) and ul/(x) =u ~XM(X) be the fine mesh and a-optimal coarse mesh approximations of (2.4), respectively. Then

(5.4)

J, T. Oden. Oprimal hop fill ire elemem merhods

PROOF. From (5.1) and (5.3), u~:AMN=/I;:9'(aMN)=.:t(CPN)=U·~Y(a,\'N)' Thus, (5.4) hoibecause Y(aMN) is positive-definite. 0

As an intermediate result, we state the following.

PROPOSITION 5.3 (Independence of coefficients). The relative error ehH satisfies the inequality

(5.

where II . IIa and wf{ are independent of the coefficients £, lei, a.

PROOF. This follows from the definition (4.5) of the a-optimal test functions, the orthogonal,condition (4.3), and the fact that

where W f{ = W ZXN (sum on N) is an arbitrary function in GUIt. 0

A much stronger characterization of ehll can be derived with some simple calculations. Returning(3.8) or to the simplified case (3.20), we recall that any function vh in cyh can be represented asbilinear combination of vertex, edge and bubble functions. Thus. the relative error can be written

(5.

with N summed from I to Mr and a from I to NElEM. The next result establishes that the useoptimal trial and test functions for computing /lit leads to a type of superconvergence of the metho

PROPOSITION 5.4 (Extras/lperconvergence). If /If{ is the a-optimal. coarse mesh, Petrov-GalerAapproximation satisfying (4.2), then the relative error ehlt of (5.3) is such that

(5.

that is, the relative error at the vertex and edge degrees of freedom is zero.

PROOF. Since the a-optimal test functions cPM E W·f{,

0= g?J(ehf{'(J)",)

= g?J(E~1JtN+ E;Ba' cP,\,)

= E~a(1JtN' XM) + E;a(Ba' X",)

= E~9'(aNM)

(by (4.5)

(by (4.4») ,

where the Schur complement Y(aNM) of aNM is necessarily positive-definite. 0

REMARK 5.2. In the one-dimensional case, it is well known that optimal trial and test functions cbe designed to make the total error, ef{ = /I - uH precisely zero at the nodal points (see, e.g., [7]). Tgeneralization to two-dimensional cases. with ef{ replaced by ehll, was introduced in [8].

REMARK 5.3. Condition (5.7) is easily extended to the more general case in which (3.8) holds. Th(if

324 J.T. Oden. Optimal hop finite elemelll methods

e = EA1]r, + EN Z(k) + Eij B~.hH !J k N a IJ'

(5.8)

with obvious sums on repeated indices, we have

E!J = 0, .1= 1, , NNODES '

EZ = 0, N = 1, , NEDGES' k = 2, .... PN .

REMARK 5.4. There is another way to arrive at (5.7). Starting with (5.6), we note that

E~lJtM(X) + E~BA(X) = Uh(X) - UH(X)

= U'~lJtM(X) + U~BA (x) - U ~(lJtM(X) - aMaaaA BA (x))

= (U~ - U~)lJtM(X) + (U~ - U~aMaaaA)BA(X),

Thus, in view of (5.4), E~1= 0 and

Next, we construct a precise local a posteriori formula for the relative error ehJl·

(5.9)

(5.10)

PROPOSITION 5.5 (A posteriori error estimates). Let r~ denote the element residual for elemeflt DK:

Then the relative error ehJl at any point x E iiK is given by

where

and

PROOF. On each element {JK' the fine mesh approximation satisfies the equality

where II is the exact solution. Since uh = UH + ehH, we have

from which (5.11) follows. 0

(5.11 )

(5.12)

(5.13)

(5.14)

J. T. Odell. Optimal hop finite element methods 3.

REMARK 5.5. In the more general case in which Uh is given by (3.6), the elementwise error is easiishown to be of the form

ehH"" (91JK(b:t, b~))-Ir~b:lx) ,

r~ = !tK(b~) - 91JK(Ull' b~). 1 <!iOi, j. k. I <!iOp. x E ilK' (5.1:

Several useful results follow from (5.12). For example:• Directional error indicator. The gradient of ehlt in ilK is, of course, calculated immediately frO!

(5.12), Let

(5.1e

Then the change in ehH in the direction of a unit vector" is

(5.17

A more useful indication of the direction of maximum change of the local relative error may be t,idcntify 2l K of clcments surrounding flK,

(5.18

gives an indiction of the change in ehH as one progresses from ilK to other clements in 2l K.

Local LP-estimates. For l~p<!iOoo and ilKE2lh, we have

(5.19

Numerous other examples (e.g .. negative- or fractional-norm estimates) could be easily obtained fran(5.12).

The next result shows the special role that bubble functions have in stabilizing finite elemenmethods.

PROPOSITION 5.6. Let the a-optimal coarse mesh Petrov-Galerkin approximation UH of problen(2.4) be characterized by

xEfl .

where XN are a-optimal global and local trial functions satisfying (4.9) and (4.15). respectively, ancrepeated indices Nand i are summed throughout their ranges, 1 <!iON <!iOMr, 1 <!iOi <!iO8. Then the coefficient.U~/K are precisely those of the fine mesh approximation obtained through matrix condensation, as if.(3.30).

PROOF. Note that for every element ilK'

326

But

and

J.T. Odell. Optimal hop fillite elemelll methods

(5.20)

(5.21)

(5.22)

Introducing (5.21) and (5.22) into (5.20) gives, equivalently, the first equation in (3.33), multiplied bythe coefllcients a fk defined in (4.25) which was obtained from (3.30) by matrix condensation.

6. Generalizations and discussion

6.1. Interpretation of bubble functions

A review of the orthogonality properties that define a-optimal trial functions (recall (4.9» revealsthat XN need only be a-orthogonal to the functions in r" - 'VII. One could, for example, deflne rll asmerely the space spanned by the nodal basis functions '/'.1 ~. L1= 1. .... NNODES (recall (3.8». Then thea-optimal trial functions X.1 are a-orthogonal to both the internal bubble functions 8" and the edgefunctions Z~). All of our global results still hold, including the extrasuperconvergence properties (5.7)and (5.9). Only the error characterization (5.12) is based on the fact that the functions bK vanish on theelement boundary aflK, but (5.10) is not. Even so, the definition of this boundary on which internalfunctions vanish is somewhat arbitrary. Indeed. if two adjacent elements share edge functions along acommon side (see Fig. 2(a», then these edge functions qualify as interior bubble functions for thecomposite element.

Many generalizations of this idea are possible. If one chooses any patch of elements fl~ C fl in anarbitrary hop mesh. all of the global basis functions with supports contained in the interior of fl~ qualifyas bubble functions. This is illustrated in Fig. 2(b). In an adaptive algorithm setting, a cascadingsequence of patches could be identified with the structure of these internal functions and, therefore, ofthe a-optimal solutions, changing with each adaptation.

6.2. Three-dimensional problems

Generalizations of the theory to three-dimensional cases is straightforward, but, of course, three-dimensional implementations carry a significant increase in complexity and computing effort. In thethree-dimensional case. if f1 = [-1. 1]3. we now have vertex, edge and face shape functions as well asthe internal bubble functions:

(I) Vertex

(2) Edge• (ot)(1.2.3.i~,T/, ()=t(1 ±~)(I ±1])Pk(O,- (kl(S.6.7.8(~,T/. n=t{l ±()(I ±{)Pk(1]) ,

'~~:O.11.I2(~.1]. n = t(I ± 1])( 1 ± nPk( 0 .

J. T. Oden. Optimal IJ-pfinite element methods

(a)

p=1D

p=2[B

p=3.

p=4.

P=5.

(b)

3

Fig. 2. (a) Bubble functions for a composite element dcfincd by adjacent clemcnts sharing a common side: (b) a patch fJ~elemcnts in a non-uniform h-p mcsh for which all intcrior basis functions vanishing on ilfJ~ are effectively bubble functiom

(3) Face8:/(~, TJ, n == t(1 ± g)P,.(TJ)Pj(n .• 3.01 ISij (~.TJ. n==}(l ±TJ)p,.(g)Pj(n.

8~'\~,TJ. n == t(1 ± np,.( g)Pj(TJ) .

(4) Bubble (internal)

328 J. T. Oden. Optimal h-p finite element methods

with 2 ~ i, j, k ~ p and p the integrated Legendre polynomials given by (3.2).The local function u are thus of the fonn

8 12 Pi

vet Tj, 0 = L u~,fr;(~, Tj, 0 + L L U~k,:k)(~, T}, 0;=1 ;=1 ka2

(, Pi

+ 2: 2: u~8;k(~' T}. 0;-2 ;.k=2 '

PI P2 P3

+ L 2: L U~kb;jk(~,Tj. O·;=2 ;=2 k=2

The remaining steps in defining uh' UH, XN, lPN' etc., for a given form a(· " ) parallel those discussedfor the two-dimensional case.

6.3. a-Optimality and preconditioning

The use of a-optimality for preconditioning deserves some discussion. Considering again the modelproblem (2.1) and a bilinear form a(- ,.) with properties (a.i), (a.ii) and (a.iii), we address the issue ofpoor conditioning of the stiffness matrices that arises with increasing the spectral order p of the elementshape functions. What is sought is a preconditioner that stabilizes the problem in the sense that thecondition number grows very slowly with increases in p. We are interested in 'preconditioning by linearelements' (c.g., [13 \) or 'nodal preconditioning'. in which the stiffness matrices corresponding to nodaldegrees of freedom (which generally are well-conditioned with respect to p) are left inlact whendesigning a preconditioner. Other preconditioning methods. such as those in [14], involve relatedcoarse- and fine-grid computations and could conceivably be brought within the framework of thistheory.

In most cases, it is sufficient to focus on a typical clement ilK E 22" or the master clementh = F ~ 1(DK). A main source of ill-conditioning is the interaction between the edge shape functions '1and the bubble functions bK

. This problem can be resolved on the element level by using a-orthogonaledge functions ,~ defined by

(6.1)4

. ~ EI~J~.L. Pr '

r=1

where p; is the largest polynomial degree of a shape function associated with edge r. Then,

(6.2)

This is a legitimate definition of local edge shape functions because the values of ,~ and t~remain thesame on the clement boundary so that the corresponding global basis functions Z'N (or Z~» arccontinuous throughout il.

We now regard the coarse mesh approximation space 'VH as that corresponding to only the nodaldegrees of freedom; the edge functions and bubble functions lie in 'Vh

- 'VH. For element ilK' the local

a-optimal trial functions are of the form

where now

4

1~ Ci ~ 4. 1~j, k ~ L p~ x E DK •,.=1

(6.3)

,.

J. T. Odell. Optimal h-p finite element methods 3:

etc. Thus. for all a. j,

To compute the corresponding a-optimal test functions. we define

andK _ KP" j" K - K K -I K K jk K K~U(x)-aR [t/1p(x)-mK"jp(,,(x)-(kbb) (kbP+kt>"mKfljp)b (x)].

/ K' -K KThen, for arbitrary wil E r I (wh(x) = w~t/1R(x) + wU j (x) + wbb (x». we have

N1OI•HI NELHI

uP( I) - " uP ( K) -" It K"U,(kK)~J Who (~. - L.J •7JK Wh' ~It - L.J WoCl'Jl J R" .K-I K-I

where

and. as usual. repeated indices are summed (except. of course, the bubble label b). Then

where s is the assembly operation and now

(6.'

(6.~

(6.t

(6./

(6.f

(6.~

(6.H

REM ARK 6.1. In (6.5), we assume that k~ is invertible: if this is not the case, the indicated operatioJmust be done globally, on the assembled matrices.

REMARK 6.2. In (6.5)-(6.9), it is understood that the edge matrices kj~' k:;. k\ arc computed usirthe a-orthogonal functions ,~ of (6.1) and thus differ from the matrices described in Sections 3-:Throughout. repeated indices are summed over their ranges. 1 ~ a. {3. v ~ 4: I ~ i. j. k..::; E:=, p~. pbeing the highest degree polynomial edge function on edge r. r = 1.2.3.4.

The operations leading to (6.9) render the matrix problem of the coefficients w~ (or Viindependent of the coefficients E. leI. CT. However. the conditioning of the problem may still grorapidly with increasing p. To resolve this problem. we introduce the form

(6.11

where

(6.12

330 J. T Od('II. Optimal hop finite elemellt methods

Here. the parentheses surrounding (j) indicate that repeated index j is not to be summed within thesummation sign.

The major property of the form c( . ,. ) is embodied in the following proposition which is proved in[13] and which generalizes a similar analysis for h-version methods developed in [141.

PROPOSITION 6.1. For the element shape functions l/J:. l;, bK• lVith l~given by (6.1), positive

constants mi' m2 exists sllch that

(6.13)

where c(· ,. ) is defined by (6.11). Moreover, if a uniform p-version approximation is Ilsed lVith p thedegree of the largest complete polynomial in the space of element shape functions, or, asymptotically asp -'> 00, P denotes the largest degree of element shape fllnctions in a family of meshes with h = constants,then

(6.14)

where c is independent of p and the partition 2},h'

REMARK 6.3. The number m2/ml represents the 'relative condition number' (d. [13]) of the formsa(' .. ) and c(·.·). Suppose c = O. Then a1( ... ) is symmetric-positive definite and the analysis in[13.14] apply to our model problem. In [13]. the preconditioning form is taken to be

(6.15)NCI.E.M

- '" ( 0 f3kK '" (j) (i)kK kK)- L.J u 0 v 0 ofj + L.J u{';V E jj + ub Vb bl> .KtJ i

Then (6. 13) holds with a( ... ) replaced by 98( ... ). In this case the constant c in (6.14) depends upon E

and may be very large. The present results thus generalize this result in the following ways: OO(. , . ) canbe unsymmetrical and c in (6. 14) is independent of E as well as p and 2},h'

REMARK 6.4. Returning to the construction of a-optimal trial functions, let us select for the forma(· " ) the bilinear form c(· .' ) defined in (6.15). Then

(6.16)

so that the usual bilinear shape functions associated with nodal points qualify as a-optimal trialfunctions:

(6.17)

Suppose. again. that c = 0 so that ~(. ,. ) is symmetric and positive definite. Then, the functions lK,: arck KglJ-orthogonal to the bubble functions and kib = kbl = O. [n this case. (6.9) degenerates to

whereY(k:,) = k:, - k:k(k~)-Ikf. - k:b(k:b)-Ik;,.

K= Schur complement of kOI

'

(6.18)

(6.19)

J. T. Odell. Optimallr-p fillite element methods

and the global optimal lest functions are

(~\(x) = 9'(Kolr) -1K"A ~(x) ,

x E fl, 1~ r. .1, A ~ NNODES ' (6.21

where K,.ol is the global stiffness matrix corresponding to nodal degrees of freedom. In other words, thmatrices associated with the problem

with llH = 1l:;X,. = 1l~1f',. will have a condition number that grows like c(1 + log2 p) with c independerof p.

Acknowledgment

The work presented here profited from numerous conversations with Leszek Oemkowicz whose helis gratefully acknowledged. The support of several agencies for work on this subject is also gratefullacknowledged; in particular, my work was supported by ONR under Contract N00014-J-92-1161. AR<under Contract OAAL03-92-0253. NSF under Grant ASC-9111540. and the Aerothermomechani('Laboratory at NASA Langley under Contract NASI-18746. This work was completed under thsupport of DARPA Contract DABT63-92-C-0042. Partial support from IBM through an agreemer.dated January 14. 1991 is also gratefully acknowledged,

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