.......

"A POSTERIORlERROR ESTIMATES AND ADAPTIVITY IN FINITE ELEHENT METHOD"

J. T. Oden and L. Demkowicz

Texas Institute for Computational HechanicsThe University of Texas at Austin

ABSTRACT

This paper addresses the general topic of adaptive methods for automaticallyenhancing the quality of numerical solutions to linear and nonlinear boundary-value problems in solid and fluid mechanics, and reviews some of the recentwork of the author and his collaborators on this subject.

KEYWORDS

Adaptive Finite Element Methods, A Posteriori. Error Estimates.

INTRODUCTION

The basic objective of an adaptive finite element method is to improve thequality of an initial finite element approximation by automatically changingthe model: refining the mesh, moving mesh nodal points, enriching the localorder of approximation, etc. Thus, all adaptive methods must attempt to re-solve two basic issues: 1) how is the quality of the approximate solution tobe measured? and 2) how does one adapt the model to improve the quality of theapproximation?

The first question is generally resolved by attempting to measure the localapproximation error in some appropriate norm. The error, of course, is thedifference between the exact solution u and a finite element approximation uhof u on a given mesh. Since u is not known, the problem of assessing thequality of an approximation reduces to one of a-posteriori error estimation:the determination of estimates of the error using computed finite element solu-tions. A number of important papers on various schemes for a-posteriori errorestimation has been contributed by Babuska and his collaborators (1978a, 1978b,1984).

Once an estimate of the distribution of the error is available, the difficultquestion of how to best modify the model to improve accuracy arises. Thereare three general approaches:

h-Methods. Here the mesh is refined; the mesh size h is reduced and the numberof elements in the mesh is increased in regions of large error.

p-Methods. Here the mesh is fixed but the degree p of the polynomial shapefunctions is increased over elements in which a high error is indicated.

Moving Mesh Methods. In these methods, the number of nodes and the type offinite element remains constant during the adaptive process and the nodalpoints are moved to regions of high error.

Of course, one can also employ combinations of these strategies. But thecorrect strategy for use of ~ombined methods is apparently a delicate issueand one in which much additional study needs to be done.

We shall describe here three methods for error estimation and show how thesecan be implemented in each of the three adaptive schemes listed above.

A-POSTERIORI ERROR ESTIMATES

We describe three classes of a-posteriori error estimatipn. The first one,based on the computation of element residuals, was introduced by Demkowicz,Oden, and Strouboulis (1984) and includes several of the results of Babuskaand colleagues (1978a, 1978b, 1983). Second method, based on the concept ofan interpolation error estimate, was first advocated by Diaz, Kikuchi, andTaylor (1983) and then investigated by Demkowicz and Oden (1985a, 1986b, 1985c).Finally the last concept, built upon the "extra-superconvergence" result ofsome very special Petior-Galerkin schemes has been recently introduced byDemkowicz and Oden (1985e).

Residual Hethods

Consider the abstract boundary-value problem, find u in V such that

<Au, v> = <f,v> for all v in V

where

(1)

A = a (possibly nonlinear) operator from a reflexive Banach space of admissiblefunctions V into its dual V*

v = an arbitrary test function in Vf = given data in V*<.,.> = duality pairing on V* x V

This problem is equivalent to the abstract problem: Au = f in V*.

A Galerkin approximation of (1) consists of seeking a function uh

in a finitedimensional subspace Vh of V such that

(2)

The ~ehidual rh is the degree ~ith which the approximation uh fails to satisfythe original conditions on the solution:

(3)

Since the residual belongs to the dual space V* and not necessarily V, itsmagnitude must be measured with respect to the norm /I ·11* on V*:

(5)

llrhll*= sup "'rh,v>v G V v

(4)sup<rh,v>

IIvll~lwhere 11·11 is the norm in V.

In some of the error estimators that we have developed, we use in the followingprocedure to approximate the supremum in (4):

• The original finite element approximation uh is computed in a spaceVh = vfi of spanned by low-order (say, linear) piecewise polynomial shapefunctions, resulting in the residual rA.

• The full space V is approximated by a higher-order tinite element spaceV~, p > I, spanned by piecewise polynomials of degree. p. _ ___ _.__

An approximation of the residual r~ is constructed according to

Ilr~ll*~ cllva - v~11 + sup < r~, v~ >Ilvh II ~ I

uhere C is a constant, va is an element of V realizing the sup in (4) and vhis an arbitrary element of Vh' If h is the mesh size (i.e. for a partitionTh of elements K,

h = max ~, hK = diameter (K)

we generally have

so that itlmak~s sense asymptotically (as h + a) to approximate sup <r~, v>by sup <rh, vh>·

As an example of how this procedure can be implemented, consider the modelproblem,

Find u in V I{v G H (n) ; v = a on rl} such that

fnfv dx +fr gv ds for all v in V2

(6)

This is the variational form of the model Poisson problem,

-6u = f in n C JR2

u = a on r can1(7)

auan g on r 2C an

w~th 6 = 62, Hl(g) th~ usual Sobolev space of functions with derivatives in1.. (n), and an = r1 U r2.

We define

Vh= {vhGV: vhlK6 Ql(K) ,n=UK}

Vh(K) = {Vh 6 Pp(K)}

VhO(K) = {Vh G vh (K) ; p > 1, v~ interpolant Vh = a}

(8)

(9)

where Ql(K) is the usual set of bilinear functions defined on a quadrilateralelement K and P (K) is the spaceof polynomials of degree p defined on K.

p

The residual r~ satisfies

1 p { 1 p<rh, vh> = ~ J (-6uh-f)vh dxK K 1

au+ J H~

aK'an dn1

d~ IhJ (--- - g) vP ds} = ~< R , vP>a~r an h K K h

2

where u~ is the (coarse-grid-initial) ihnite element approximation of udetermined by using1the space Vh and RK is the functional on V~ defined asindicated, with dU*h/dn an approximat~on of au/an obtained from an adjacentelement to K.

It is not enough to simply calculate ~h as an indicator of the error inelement K. In general, we wish to have an indicator ~K of the error whichwould contribute to a global error estimate from above as well as from belowand which would converge to zero at the same rate as the actual error, e.g.

(10)

Such an error indicator is obtained as a solution of the auxiliary problem,

JK

V4J • VvP dx =K h <~h , for all vh in V~O (K) (11).-

We generally compute the solution of (11) using the concept of hierarchicelements in which the stiffness matrices are only modified by the additionof a row and column with the addition of each degree of freedom (see, e.g.,Carey and Oden, 1981 for details). Using (11), (9), and (5), we have (towithin terms of O(hP»

{C L JK K

(12)

where C is a (hopefully) known constant. Through this estimate is global, weuse II ~Klll K as an estimate of the local error over Tach element K. Ingeneral, reducing II~Klll K implies a reduction in Ilrl"* which (particularlyfor linear self-adjoint problems on Hilbert spaces) iJplies a reduction inIIu-u~ II·

Interpolation Error Estimates

It is well known (see, e.g. Oden and Carey, 1983) that for linear ellipticproblems the approximation error Ilehll~=llr-ufi11vcan be bounded above by theso-called interpolaton error,

(13)

For the model problem (6), for example,

(14)

If u is smooth enough;a local int~polation ~o~ ehtimate can be derived ofthe type (for Ql-elements)

1 .lu-vhll,K ~ C hK1u12,K

where

2Iu b, K = f K W dx

a2u a2uW dx = (--z + --z) dx dy

aXl aX2

(15)

(16)

(17)

The basic problem we face when attempting to make use of any of these estimates .~is that we must calculate the higher order derivatives of the unknown solutionusing only available information, i.e., through use of the currently availablefinite element solution ub. There are numerous a priori techniques forestimating the second der~vatives u , u or u , but many are somewhat

xx xy yyintuitive and not all are based on rigorous estimates. Exceptions are thetechniques based on so-called "extraction formulas" introduced by Babuskaand Miller (1984a, 1984b). Following their idea one can prove that, if u isregular enough, then the second derivatives at an arbirtrary point (xo'YO)satisfy

- Cl - - aufn (~ +~) f dxdy - fan u an(~ +~) ds + fan (~ + ~) an ds

Here ~ = ~ cO~2e where (r, e) are the polar coordinates centered at the point

(xO' YO) und~r consideration and l is an arbitrary, regular function. By the

proper choice of $, one can eliminate the boundary terms in (17). Of course, uon the right-hand side of (17) remains still unknown, but when replaced by itselement approximations ufiresults in a formula for approximation of second-derivatives at (xO' YO) of the same order of accuracy as the L2-error in the

approximation of u by u~. For example for the first order ap~roximation we can"extract" the difference of second order derivatives with O(h ) order of

convergence! Formula (17), when combined with equation (7), allows us tocalculate each of the derivatives separately. Also, by choosing

1 sin2e. h f 1 " "h· d d' .~= ~ 2 ~n t e same ormu a, we can extract t e m~xe er~vat~ver

a2u--- (xO' YO)'axayOne method we have used successfully in applying the estimate (15) is toconstruct the function ~ using a bivariate blending function of Gordon and Hall(1971, 1973) type.

Note that we still have a global estimate although we "apply it" to K

(18)

An a Posteriori-Local Error Estimate Based on a "Extra-Superconvergence" Result

In the subsequent discussion for the sake of clarity we restrict ourselves toI-D (with respect to space variable) situaton, although the method is by nomeans restricted to this case only. Solving a convection-dominated lineardiffusion problem by method of characteristics, we arrive naturally at thefollowing "model elliptic problem":

= f in (0,1) (19)

u(O) = a, u(l) = b

Viscosity constant E is implicitly assumed to be small and right-hand side fincludes both source term as well as a convected solution from the previoustime step. For small E and "shock-type" behavior of the solution an applicationof standard Bukuov-Galerkin method to (19) results in spurious, oscillatoryresults around a place of irregular behavior. This is not however the case iffor a given mesh

(20)

with standard trial h~t-functions ~i being employed we introduce the followingnew test functions ~., i = 1...,N

~

-E~ . + ~. = a in (x. l' x.) i = 1, ...,N..l.xx l. ~- ~

i(x.) = 6 .. j=O, ...,N+lJ 1J

(21)

Using Petrov-Galerkin approach to (19) we arrive at the following approximateproblem.

N+lF' d i . u (0) = a, u (1) = b~n uh = ~ uh ~~,

i=O u u

1 1 1OJuh~· + EOJuh~' = oj f~. V ~. (22)

J x Jx J J

One of the crucial results is what we call the extra-superconvergence result:

U = Un at nodes xi i = 1, ...,N (23)

that is the exact solution coincides with is approximate substitute at nodes.In this way the global problem (19) breaks up naturally into a series of localproblems allowing us particularly to calculate truly - local error estimates,for instance of the following form:

~

h3 ~!lIu- uhIlLQ)(xx ) 2. £/h2 + £1T2) 1/ f-uhllL2(x x ) (24)

k, k+l \ k' k+l

with h = xk+ -xh and computable -L2-residual on the right-hand side of (24).For all the details and much more information concerning the subject we referto Demkowicz and Oden [1985e).

MESH REFINEMENT STRATEGIES BASED ON THE A POSTERIORI ERROR ESTIMATES

While many issues remain open in the area of reliable a-posteriori error esti-mation, still further complications exist in designing efficient adaptivealgorithms based on these estimates. The basic problem can assume the form ofan optimal control problem in which one has to attain a discrete approximationwhich is optimal in some sense determined by the error measures and the strategyused to reduce error. The entire problem is further complicated by the factthat our a-posteriori estimates are global in nature(particularly the residual-type estimates discussed earlier) even though they are used locally as a basisfor enrichments of the solution.

In this section we describe three methods developed by Oden and colleagues,(1985), Demkowicz and Oden (1985), Demkowicz, Oden, and Strouboulis(1984), andDemkowicz, Oden, and Devloo (1985).

An h-Method

Consider a quadratic mesh and the associated error estimate (18).a function h(x,y) specifying a "density" of the mesh by:

h(x,y) = ~ if (x,y) G K

the error estimate (18) may be written in the form of a functionalJ(h) = In h2W dxdy,

where W is given by (16).

This leads to a natural minimization problem:

If we define

(23)

(24)

Find h = h minimizing the functional (24) and subjected to the constraint:opt

Ifn h dxdy = N (25)

where N is simply the number of elements. The solution of this problem is

2hK

fK W dxdy = const. for every element K (26)

which results in a very simple iterative - mesh refinement scheme, provided wecan estimate the function W. For additional details we refer to Demkowicz,Oden, and Devloo (1985).

Perhaps the most natural is to apply the h-method in the case of local error

estimate (24). Given a threshold TOL we accept the solution whenever theestimate (24) gives a smaller value otherwise subdividing the given element intotwo smaller and resolving local problem only,resulting in a new exactvalue of the solution. Each of such local problems consists of one singleequation only, which results in a very powerful and cheap solver to the problem(19). For details see Demkowicz and Oden (1985e).

A p-Hethod (see Oden and others, 1985; Demkowicz, Oden, Strouboulis, 1985)

It is difficult to predict the rate of convergence of local interpolation errorin the case of the p-version since it depends only on the unknown regularity ofthe solution (Babuska and all, 1981), and it is almost impossible to say any-thing about a local order of convergence of the error or the residual-typeestimate. One way that we have used estimates of the'type (12) effectively isto employ the following steps:

1. Solve local problem (11) and determine local contribution I~KII K to theglobal error estimate. '

2. Normalize the local indicators I~Kll,K by subdividing by the largest one.

3. If a < I~Kll,K < 01 the first order approximation is retained.

° < I~Kll,K < 02 a second-order approximation is used.

02 < I~Kll,K < 1 a third-order approximation is used.

Here the numbers 01 and 02 are chosen rather arbitrarily and ~~ corresponds tofirst-order approx~mation on a uniform mesh.

Moving Mesh Strategy (see Demkowicz and Oden, 1985a)

A popular moving finite element method. due to Miller, is based on L2-residualestimate in its basic form which finds justification in only the one-dimensionalcase. In Demkowicz and Oden (1985) we present a moving mesh strategy based onthe interpolation error estimate that can be considered as a generalization ofthe Brackbill and Staltzman strategy (1982).

We consider a fixed mesh of elements K that is mapped onto a distorted mesh insuch a way as to minimize the interpolation error. We may penalize the inter-polation error functional further by appending to it additional terms to smooththe mesh and prevent the Jacobian of the map from vanishing.

In the method discussed in Demkowicz and Oden (1985a) the optimal mesh resultsfrom minimizing the functional

J = 10 + aI + BJ

where

I = ~J (x2 + y2 + x2 + y2)d~dna A Ann ~ ~K K

1I = ~ J - l(x2 + y2)u2 + (x2 + y2)U2 ] d~dnK K J ~ ~ nn n n ~~

1J = ~f (x~y + x y)2 j d~dnA Ann n

K K

(27)

(28)

(29)

(30)

Here a and 8 are constant parameters, 10 is the Dirichlet integral which treatsas a constraint the invertibility of the transformation defining the mesh. Inthe absence of other terms, its minimization yields a conformal mesh and thenthe method is merely a generalization of the well-known conformal-map meshgenerators. The functional K forces the mesh lines to be orthogonal and thisorthogonality allows us to simplify the form of the error functional I.

NUHERICAL EXPERIMENTS

In this section we present some representative examples.

A Heat Conduction Problem With a Moving Domain

We first considerdomain n = (0, 4and initIal data.correspond with a

a linear transient heat conduction problem defined on a moving+ O.lt) x (0, 3) with purely homogeneous Dirichlet boundaryWe choose the data on the right-hand side of the equation to

prescribed exact solution:

u = 10 e-5(x-l-0.2t)2 x (4+0.1t-x) y (3-y) C. Where

C = rt for a < t < 0.5'Ll for t > 0.5

This problem has been solved for a mesh with 24 elements (see Fig 1). Thetime step has been chosen 6t = 0.1 and the solution was computed for 20 timesteps. The constants 01 and 02 described in the previous chapter have beenchosen as follows:

01 = 1/20, 02 = 1/2.

Figure 1 shows the mesh enrichment for terms t = 0.5, 1.0, 1.5 and 2.0 whileFigures 2 and 3 present the computed first-order and enriched solutions on thesection AA (see Fig. 1) compared to the exact solution. For additional detailswe refere to Demkowicz, Oden, and Strouboulis (1984).

A Stokes Problem

The second problem we consider deals with the applicaton of the moving meshstrategy described earlier rather than the penalty-formulaton. In this examplen is the rectangle (0,2) x (0, 1) and we define on this domain an initial uniform16 x 8 mesh of rectangular elements. Dirichlet boundary conditions are usedwith the velocity ~ prescribed as uo = (1,0) along the top edge and ~ = (0, 0)along the remaining sides (driven cavity problem).

The optimalN

J = 10 + ~i=l

mesh results from a minimization of the functional

<x.1. +f3J + yK~ ~

Where ai' i = 1, 2, f3 and y stand for positive (given) real numbers, I. isdefined by (25) with u replaced by the i-th component of the velocity lfield,JO is a similar term for the hydrostatic pressure given previously. Figure 4presents a computed mesh for <Xl= <X2= 4, 8 = 4, y = 0.1.

.A Model Elliptic Problem

As an example of the application of the h-method combined with the interpolationerror estimate, we have solved the model elliptic problem (7) with homogeneousDirichlet boundary conditions. The right-hand side of the equations correspondto the following (exact) solution defined in n = (0, 1)2:

u(x,

Ijl(x)

y) = ljl(x)ljl(y), where

-e: 1= e + Ax + B

e(x-x )2 + e:a

and A and B chosen in that way that 1jl(0)= .(1) = O. With xo = 0.55, yO = 0.5, -

e: = 0.02, and e: = 0.05; after 19 mesh refinements we have obtained a meshx ypresented in Figure 5. The error together with its exact and computationalerror estimates are shown in Figure 6.

--·Petrov-Galerkin Method for Model Elliptic Problem

A~ an example of the application of the h method combined with the specialPetrov-Galerkin scheme we have solved problem (19) with a right-hande side fand boundary conditions corresponding to the following, exact solution:

u(x) = -exp(-x)-exp [(l+£)(x-l)I£] + (A-B)x - A

with constants A and B chosen so that u(O) = u(l) = O. For small £, the solutionpossesses a thin boundary-layer near x = 1. The numerical data have been chosenas follows:

£ = O. OlLoo - error tolerance TOL = 0.0001

Figure 7 presents both numerical and exact solutions (with the scale used theyare undistinguishable) and the resulting optimal finite element mesh of 50 elements.

ACKNOWLEDGEHENT

This work was supported by the U.S. Office of Naval Research under ContractNOOOl4-84-K-0409. Helpful discussions of the work reported here with Hessrs.

T. Strouboulis and Ph. Devloo are gratefully acknowledged.

REFERENCES

1. Babuska, I. and W. C. Rheinboldt (1978). Error estimates for adaptive finiteelement computations. SIAM J. Numer. Anal., 15 (4).

2. Babuska, I. and W. C. Reinboldt (1978). A-posteriori error estimates for thefinite element method. International Journal for Numerical Methods inEngineering, ~, 1597-1615.

3. Babuska, I., A. Miller, and H. Vogelius (1983). Adaptive methods and errorestimation for elliptic problems of structural mechanics. Inst. for Phys.Science and Technology, University of Haryland, Lab. for Numerical Analysis,Technical Note BN-I009.

4. Babuska, I., B. A. Szabo, and I. N. Katz (1981);· ·Tne p-version of the finiteelement method. SIAM J. Numer. Anal., ~ (3), 515-545.

5. Babuska, I. and A. Millerelement method - Part 1:higher derivatives of the1085-1109.

(1984). The post-processing approach in the finitecalculations of displacements, stresses and otherdisplacements. Int. J. Numer. Methods Eng., 20,

6. Babuska, I. and A. Miller (1984). The post-processing approach in the finiteelement method - Part 2: the calculation of stress intensity factors.Int. J. Numer. Hethods Eng., 20, 1111-1129.

7. Brackbill, J. V. and J. Saltzman (1°82). Adaptive zoning for singularproblems in two dimensions. J. Computational Physics, 46, 342-368.

8. Carey, G. F. and J. T. Oden (1983). Finite Elements: A Second Course,Prentice Hall, Englewood Cliffs, N.J.

9. Demkowicz, L., J. T. Oden, and T. Strouboulis (1984). Adaptive methods forflow problems with moving boundaries. I. Variational principles anda-posteriori estimates. Computer Methods in Applied Mechanics and Engineer-~, 46, 217-251.

10. Demkowicz, L. and J. T. Oden (to appear). On a mesh optimizaton method basedon a minimization of interpolation error. Int. J. Eng. Sci.

11. Demkowicz, L., J. T. Oden and Ph. Devloo (to appear). On an h-type meshrefinement strategy based on minimization of interpolation errors. ComputerMethods in Applied Mechanics and Engineering.

12. Demkowicz, L. and J. T. Oden (to appear). Extraction methods for second-derivative in finite element approximations of linear elasticity problems.Communications in Applied Numerical Hethods.

l3. Demkowicz, L. and J. T. Oden. An adaptive characteristic Petrov-Galerkinfinite element methods for convection-dominated linear and non-linearparabolic problems in one space variable. TICOH Report 85-3, April 1985,The University of Texas at Austin, TX 78712, USA.

l4. Diaz, A. R., N. Kikuchi, and J. E. Tayloroptimization for finite element methods.Mechanics and Engineering,~l, 29-45.

(1983). A method of gridComputer Methods in Applied

IJ. ~ordon, W. J. (1971). Blending function methods of bivariate and multi-variate interpolation and approximation. SIAM J. Num. Anal., 8, 157-177.

16. Gordon, W. J. and C. A. Hall (1973). Transfinite element methods: blendingfunction interpolation over arbitrary curved element domains. Numer. Math.,21, 109-129.

17. Oden, J. T. and G. F. Carey (1981). Finite Elements: Mathematical Aspects,Printice-Hall, Englewood Cliffs, N.J.

18. Oden, J. T., L. Demkowicz, T. Strouboulis, and Ph. Devloo (to appear). Adap-tive methods for problems in solid and fluid mechanics. In I. Babuska andO. C. Zienkiewicz (eds.), Adaptive Methods and Error Refinement in FiniteElement Computation, John Wiley and Sons, Ltd., London.

FIGURE 4. Driven cavity problem. Optimal mesh after 8 FErecalculations; a = 4., e = 4., y = 0.1.

Ln " error II

\\

f1' computational error~ ..\ estimate based on

.....\ the extraction formula

.....\\\".\

'..\.....\

.'.:.\...\

.'.\-..:,.....~

:FIGURE 5. Refined mesh after19 FE recalculations.

1.0

o

-1.0

-2.0

1.0 2.0" Ln(N1/2)\.\ .'.

error estimate~·'

-semi norm error

FIGURE 6. Convergence of the interpo-lation-error estimate based on the ex-traction formula.

~. A A.

t=0.5 t=1.0

A.

t=1.5 t =2.0

~1o~rder A) 20 order

FIGURE 1. Hesh enrichment for the heat conduction problem

4.003.002.004.003.002.001.000.00

~l 1\ 00

TIME =0.5 gl / I ""TIME =2.00

-- 1st Order -1st Order

----- Enriched0 0q Exact 00 0c-. c-.

0 00 0. .0 0... ..

FIGURE 2. Heat conduction problem.Computed solutions on section AA forp = 1 and adaptive correction fort = 0.5.

FIGURE 3. Heat conduction problem.Computed solutions on section AA forp = I and adaptive correction fort = 2.0.