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COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 78 (1990) 201-242NORTH-HOLLAND

A POSTERIORI ERROR ESTIMATION OF FINITE ELEMENTAPPROXIMA TIONS IN FLUID MECHANICS

T. STROUBOULISDepartment of Aerospace Engineering. Texas A. & M. University. Texas. U.S.A.

J.T. ODENTexas institute for Complllational Mechanics. The University of Texas lit Austin. Texas. U.S.A.

Received 18 April 1989

Several techniques for a postenon error estimation of finite clement approximations of time-dependent problems are developed and tested. These include a discontinuous Galerkin method forlinear convection problems. a residual-type method for diffusion problems and an operator splittingmethod for convection-diffusion problems. Some extensions to certain classes of nonlinear hyperbolicprohlems are also presented.

I. Introduction

While the idea of estimating the numerical error inherent in computational methods hasbeen an important issue in numerical analysis since the primitive beginnings of the subject. thegeneral area of a posteriori error estimation has taken on special significance only in recentyears due to its importance in adaptive computations. Some feel that the most promising andchallenging ideas in modern computational fluid dynamics involve the use of adaptive meshstrategies wherein the structure of the flow solver and of the computational grid changedynamically during a calculation in an attempt to optimize the solution procedure: to producebest results in some sense for the least computational effort. The quality of the results isjudged by some computable estimate of the error or, at least. by an indication of trends in theerror throughout the computational grid. Then error estimates become more than just amathematical embellishment in proofs of convergence of various methods; they are elevatedto a practical and crucial component of the method itself. Moreover, a very useful byproductof adaptive methods that employ good a posteriori error estimates (and these are somewhatrare) is that they automatically contain an indication of the reliability of the numericalsolution, a computable estimate of the error elementwise in some appropriate norm, through-out the mesh.

The subject of a posterori error estimation and adaptivity in finite element computationsemerged from the work of Babuska and Rheinboldt [1-3J and Babuska and coworkers [4-81.The earliest of these works focused primarily on linear elliptic two-point boundary-valueproblems ancl on linear elliptic problems in the plane. Adaptive methods and error estimationtechniques for linear parabolic problems in one space-dimension were studied by Bieterman

0045-7825/YO/$3.50 © 1990. Elsevier Science Publishers BV (North-Holland)

202 T. Strouboulis, J. T. Oden, A posteriori error estimation of FE approximlltions

ancl Babuska [6-8], while Coyle ancl Flaherty [10] presented rcsults for linear and nonlinearparabolic problems in one and two dimensions and Johnson et al. [11] studiecl error estimatesancl timc-step control for diffusion problems in one dimension. Moreover, the case ofsingularly perturbed elliptic and parabolic problems which is of great interest because of itsrelation to viscous high-speed aerodynamics has been studied by Babuska and Szymczak[5.9. 12] and Oemkowicz and Oden [13, 14]. Alternative approaches to error estimation inelliptic and parabolic boundary-value problems were proposed by Oden et al. [15]. Bank 116].Bank and Weiser [17]. and Strouboulis and Oden [18]; recently Oden et al. [191 constructederror estimates in a very general framework of an hop adaptivc strategy. A survey of thelitcrature on adaptive methods and error estimation was compiled by Oden and Oemkowicz[20] and a broad indication of the state of the subject is givcn in the proceedings volumes121-23].

The development of a computable and reliable estimator of the error in approximatesolutions of the Euler equations of gasdynamics or of the Navier-Stokes equations is anextremely complex and difficult undertaking that will likely be the subject of research formany years. Indeed. the subject of a posteriori error estimation in modem numerical analysisliterature has scarcely been advanced beyond the simplest linear elliptic or parabolic prob-lems. Nevertheless. we demonstrate in the present work several new approaches to errorestimation that give very good results for elliptic, parabolic and hyperbolic initial boundary-value problems in many dimcnsions and which have good potential for extension to thecomplex problems encountercd in computational fluid mcchanics.

Thc prcsent work addresscs thc general issues of a postcriori estimation of thc error inadaptive finite element calculations in many spatial dimensions; the examples that accompanythis paper focus on transient and stcady-state problems in two dimensions but the proceduresapply without modifications to the three-dimensional case. Several new approaches to errorestimation for hand hop adaptive schemes have been developed and implemented innumerous test cases elsewhere [24].

The basic objective is to develop an error estimation package that is capable of processingany approximate solution uh supplied by a user. The particular algorithm used to calculate theapproximate solution lih is, in general, not the same as the one employed to compute theerror. The error estimator package can be used throughout a transient calculation as a tool tocontrol the evolution of error or to drive an adaptive method, or it could be employed purelyas a post-processor to determine the reliability of a computed solution. Obviously, since theapproximate solution li" serves as data in any error estimation technique, the quality of u" mayor may not affect the quality of the estimator of the error.

The present study is divided into four major parts:I. Convection problems: hyperbolic problems in two dimensions.2. Diffusion problems: parabolic problems in two dimensions.3. Convection-diffusion problems: convection or diffusion dominated parabolic problems

in two dimensions.4. Nonlinear problems: extensions to nonlinear convection problems.The methods developed here for convection problems arc new and are based on thc

discontinuous Galerkin method [25]. The a posteriori error cstimation techniques employedfor diffusion problems are a natural extension of the residual methods introduced for ellipticproblems [15J. An operator-splitting technique [9] is employed for convection-diffusion

T. Strouboulis. J. T. Odell, A posteriori error estimation of FE approximations 203

problems so that the methods developed for parabolic and hyperbolic problems can becombined into an effective approach valid for the entire range of the convection and diffusionparameters. The studies of nonlinear problems are preliminary in character, but theydemonstrate how the error estimation techniques can be extended to problems with variousnonlinearities.

2. Preliminaries

2.1. Notations and model problems

We begin by considering a model class of linear convection-diffusion problems of the form

2 a ( au)u, + a ·Vu -.2: -;- aij -a =I in n x (0, T),I.J=I uXi Xj

2

min [0, (a' n)]u +.2: niaij :Ii = g on r; x [0, T] ,I.J=I uXj

U = 0 on r,x [0, T] ,

(2.1a)

with initial conditions

u(x, 0) = uo(x), x En. (2.1b)

=aiii

Here the following notations and conventions are usecl:

u = u(x, 1) is a scalar-valued function of the position vector x = (x I' X2) E n k 1R2 andtime I, 0 ~ t ~ T,auat '

= a(x, t) = (al (x, t), a2(x, I» = the convective now vector,aij(x) = diffusion coefficients, with

2 2

2: aij(x)titj;::: aD 2: t; V~ = (gl' tz) E IRZ,

i.j=1 i=1(2.2)

ni =

I =g =~.r;

components of a unit exterior normal to the domain boundary an,I(x, I) = source term,g(x, I) = flux boundary data,disjoint partition of an, an = ~ U tz, ~ n r2 = 0.

For compactness in notation. we introduce the following bilinear and linear forms:

1 z au avB: V x v~ IR, B(u, v) = 2: (lij -;- -;-:- dx ,

11i.j-I UXi Ox'j(2.3)

L :V ~ IR , L( v) = r Iv dx + r gv ds ,In Jr2(2.4 )

204 T. Strollbolllis, J. T. Oden, A posteriori error estimation of FE approximations

where dx = dx) dx2, and V is the space of admissible functions,

V= {v = vex) Ivir. = 0, Illvlll < +oo}, Illvlll = energy norm of v (!~\/B(v, v).I

(2.5)Then, if (u, v)d::~/fJ} UV dx, (2.1) can be written in the equivalent weak form,

(2.6)(u( . ,0), v) = (uo' v)find u = u(x, t) E V, such that(u

l' v) + (a· Vu, v) + B(u, v) = L(v) ,

\fv E V \ft E [0, T] .

A semi discrete Galerkin approximation II of the solution to (2.6) can be constructed usingfinite element methods in the standard way: a family Th of partitions of [l into finite elementsconsidered over which the test functions VII are piecewise polynomials. The rcsulting space V"of finite element tcst functions mayor may not bc assumed to be a subspacc of V. Over atypical element K E Th' the approximation uh of u is of the form

N

uh(x, t) = 2: u;(t)t/1;(x) , t/1;(xk) = &~, I~j, k ~ N ,;=1

. . kwhere uf(t) is the value of u" at node Xf of element K, &; is the Kronecker symbol and t/1 arethe conventional polynomial shape functions of the Lagrangian family. Many variants andgeneralizations of this simple notion are possible, and, in fact, some of thc developments givenlater apply to the much more general setting of hop finite element approximations studied in124]. When V C V", we can formulate the finite element model of (2.5) dircctly:

Find u" = 1I,,(x, t) E V", such that

(Uh, VII) + (a ·Vu", v,.) + B(u", v,.) = L(v,,),,\fv" E Vi, 'Vt E [0, T] .

(2.7)

A key to the analysis of problems on unstructured meshes is that all of the terms appearingin (2.6) can be expressed as the sum of contributions from each element K in the mesh. Inparticular, we can write

(u, v) = 2: r uv dx = 2: (u, v) K ,K JK K

B(u. v) = 2: BK(u, v) , L(v) = 2: LK(v) ,K K

1.2 dU dV

BK(u, v) = ,2: aj; -;- -;- dx ,K j,;=1 uXj l1X;

LK(v) = (f, V)K + tK aK(u)v ds ,

\fu, v E V . (2.8)

T. StrolibOlllis. J. T. Oden, A posteriori error estimation of FE approximations 205

,~ J\ au.

where U'K(U) = ~ 11; ({;j ax. IS the flux across the elemcnt boundary aK.1./ I

solution U of (2.6) satisfies over each element the 'weak conservation law'.

(Il" V)K + (a 'Vu, V)K + BK(u, v) = (t. V)K + tK U'K(U)V ds,

(u(· .0), V)K = (uo' V)K '

't;ft E (0. T] 't;fK and 't;fv E V(K) ,

Thcn the exact

(2.9)

(2.10)

where V(K) is the restriction of functions in V to element K and it is understood thatU'K(Ii) = g on aK n I~.

Similarly, the finite element approximation satisfies the semidiscrete weak statement

Ii" E V\K). t E (0. T] .

(II"." V',)K + (a ·Vuh. vhh + BK(lih· v,.) = (f. VI.)K + tK O"K(Il)Vh ds,

(11,,(,,0), VI,) = (110, v,,) ,

't;fUh E V"(K) .

The semidiscrete weak statement (2.10) is the starting point for the construction of discretestatements for II". Several alternatives are possible; one may discrelize the time-derivativeusing a Taylor-series expansion in time to obtain a Taylor-Galerkin algorithm, or one maychoose appropriate spaces of test-functions which lead to a Petrov-Galerkin algorithm [12]. Inthe examples that follow the approximate solution Ii" was computed using algorithms of theTaylor-Galerkin type.

2.2. Evolution of the error

Given an approximate solution u" of (2.9) we define the approximation error function,

(2.11)

Introducing Ii = e}, + Ii" into (2.8), we arrive at the following equation for the evolution of theerror within element K:

e"EV(K), tElO, T],

(e"" U)K + (0 ·Ve", U)K + BK(e", v) = ("" v) K + tK O"K(U)U ds.

(el,(' ,0), vh = (uo - IIh(' ,0), uh .

't;fu E V(K) ,

whcrc ,." is the rcsidual.

(2.12)

(2.13)

206 T. StrollbOlllis, .f. T. Oden. A posteriori error estimlltion of FE approximlltiof/S

The basic strategy of the various methods of error estimation that we shall consider is this:approximate the solution ell of the error equation (2.12) on some appropriate polynomialsubspace V\K) C V(K). For the transient cases. we refer to such schemes as EOE (evolutionof error) schemes. The structure of these schemes depends upon the values of the coefficientswhich enter in the definition of the operators in (2.1).

2.3. Special cases

Because the effectiveness of different schemes for error estimation may depend on theelliptic or hyperbolic character of the initial boundary-value problem satisfied by the errorfunction. it is instructive to break down our description of a posteriori error estimationtechniques into independent studies of special cases. We will then combine the results toconstruct a method for the general case (2.1).

1. ConvectionWe first consider the first-order hyperbolic problem (B(u, u) == 0)

u, + a . 'VII = f in n ,II = lion an- ,

with initial condition (2.1b). Here aIr is the inflow boundary

an- = {xeanla(x, I)' 1l(x)<O}.

For this problem, II (and ell) can suffer jumps

(2.14)

(2.15)

across space-time surfaces of discontinuity. The estimation of thc crror in approximatesolutions of (2.13) is discussed in Section 3.

2. DiffusionHere we set, for simplicity,

a = 0, B(u, u) = In EVU ·Vu dx, (2.16)

with E a small positive constant. In this case, (2.1) reduces to the classical non-homogeneousdiffusion problem

11, - Edit = f in n , (2.17)

where d = a2/ax~ + iJ2/iJx; is the Laplacian operator and the initial condition (2.lb) is alsoimposed. The a posteriori estimation of the error in approximate solutions of (2.17) isdiscussed in Section 4.

T. Strollbolllis, J. T. Odell, A posteriori error estimation of FE approxillllllions

3. Convection-DiffusionNext, we combine (2.14) and (2.16) into a model convection-diffusion problem

UI+ a .Vu - e tJ.u = f in n .

207

(2.18)

The a posteriori error analysis in this case involves a splitting algorithm which combines themethods introduced in Sections 3 and 4.

4. NonlinearFinally, we consider extensions of the methods presented earlier to a problem with

quadratic nonlinearities. We demonstrate that the schemes presented in Sections 1 through4 can be naturally extended to estimate the error in approximate solutions of nonlinearproblems.

3. Linear hyperbolic problems

3.1. The disconti/luous-Galerkin algorithm

We now focus our attention to the model convcction problem:

ul + a' Vu = f in n ,U=ll inan-, (3.1)

u(x, 0) = un(x) in n .

A weak form of (3.1), valid over the space-time domain

DK:::: K x (0, T) .

can be constructed as follows:

Find u E W(K), such that

iT (UI + a ·Vu - f. V)K dt + iT LK-Ia. nl(u - u-)v ds dt

+ «(u(·. 0+) - Uo(')), V)K = 0 'Iv E W(K).

Here aK- denotes the inflow boundary of clement K,

and thc sct of admissible functions W(K) is defincd by

W(K) ::::L ""(H"(K); (0, T)) ,

(3.2)

(3.3)

(3.4)

208 T. Strollholllis, J. T. Oden, A posteriori error estimation of FE approximations

where Ht/(K) is thc sct of functions with square-intcgrable clerivatives in the a direction.namely

(3.5)

Here u( . , T~) = limHT-,- u(· , t) and u -(x, t) = limu->o+ u(x + UIlK• t).Let uh be a finite element or finite volume approximation of u computed on a given mesh Th

and for a discrete set of time-instants

0= to < t) < ... < ttl < t"+1 < ... < tN = T.

Then over the domain D~ = K X (t", t,,+I) the error-function satisfies the following weakstatement:

Find ell E W(K) , such that

= f'"·J (f, V)K dt - f',,+1 r ,_Ia '1llJhv ds dt - (I~.V)K 'Vv E W"(K).'II I" )a/\

Here W"(K) = (L 2(t". (11+ I)' Ht/(K»,

del}. nr = - (( - a' v((h , h

is the space-time residual.

denotes the jump at the intlow boundary and

denotes the jump at the time-instant ttl'We then seek a local approximation of the error

(klleh KEPk(K), tE(t",t"+I)'such that

f',,+1 (e(kl + a ·Ve(k), V(k)) . dt + f',,+1 f la· IlI(e(k) - e(kl-)v(k) ds cit'" h, h h K 'n ilK- h h "

«(kJ( + ) (k)( -) (kl)+ eh " til - eh " ttl ,vh K

f'"'' f',,··1 f= (rh• V~,k» K cit -._ la· 1lIJhV~,k) ds dt - U;:. V~,k)KI" Itl ill<.

'Vvl.k) E W"(K) .

(3.6)

(3.7)

T. Strollbolliis. J. T. Odell, A posteriori error estimatioll of FE approximmioflS 209

Here Pk(K) denotes the local space of polynomial of complete order k definecl over elementK.

The discrete weak statement (3.6) may be employed to construct kth-order space-timepolynomial approximations of the error function in element K or, in other words, EOE(k)schemes. In the following we present some details about the discrete problem involved in theEOE(I)-algorithm. Let us consider first a trilinear space-time element as shown in Fig. 3.1.Using linear interpolation in time we obtain

n+ 1() "()

( ) _ e" x - ell xell" x, ( - /::.t '

1- t" t"+ I - IvlI(x, t) = v;:+ I(x) 1ft" + v~(x)

Here we denote e~+I(x) = ell (x, t,,+I)' V=+I(X) = v,,(x, t"+I)' Assuming that thecoordinates are fixed in time we can integrate the time-variable analytically to obtain- Acceleration term:

- Convection term:

1'··· /::.('" (0 ·Ve", V/.)K dl ="6 ([20 'Ve~+1 + a 'Ve~], V~+I)K

+ III ([ V ,,+1 2 V "] ")"6 a' e" + 0' e" ,V" K'

Integrating the rest of the time-integrals in (3.4) we obtain

(3.8)

nodal

(3.9)

(3.10)

5

6

2

Fig. 3.1. Typical trilinear space-time element with the local enumeration of its nodes.

210 T. Stroubou/is, J. T. Oden, A posteriori error estimlllion of FE approximatio1lS

EOE(J)-disc,ete equations:Find e~+I, e~ E Q l(K), such that

/I 11+1) A11+1 11 Vh+VII u.t n+l n 11+1

(eh - eh' 2 K + 6 ([2a ·Veh + a ·Veh], v" )K

at([ n"+1 2 n1l] 11+1)+6 a'yell + a·ye" ,v K

+ ~t LK-1a' nll2(e;;+1 - e~+I,-) + (e~ - e~'- )jv;:+l ds

at 1 1 1[( 11+1 11+1.-) + 2( 11 11.-)] 11 d + ("+0+ 11-0+ ")+ 6 aK- a' n ell - ell ell - ell v" s e" - e" , v" K

at ([2 11+ 1 11I 11+ I ) at ([ 11 2 11+ II /I)= 6 'II + '" ' vh K + 6 'h + '" 'Vh K

+ at 1 la· nl(2}n+1 + }n+l)vn+1 ds6 aK- ,. h h

+ a6t1 la· nl(J~+1 + 2}~)v;: ds - (I~, V;;)K 'VVh E QI(K) .

iJK-(3.11)

Here Q I(K) denotes the space of bilinear functions defined over the element K. In thederivation of (3.11) we assumed that the residual varies linearly in time, namely

(3.12)

The discrete equation (3.6) leads to an 8 x 8 matrix problem for the nodal values of the errorin element K; this problem is solved using Gauss-elimination.

We note that the complete definition of the discrete problem (3.6) requires the knowledgeof the 'upsteam' values of the error e~'- on the inflow boundary aK- of element K. Lesaintand Raviart [25] showed that in the case of a constant convection field a it is possible to solvethe discrete problems (3.6) by processing the elements sequentially according to an orderingwhich allows us to know the upstream values of the solution when we reach element K. Thisordering is described as follows:

_ N

n = U K~, {KI' K2' ... , Kt, ... , KN} ,~=I

each side of iJK,- is either a subset of an-or a subset of aK:' for some m < 1 .

Here aK+ denotes the outflow boundary of the clement K dcfincd by

aK+ = {x E iJKla(x, t)· n(x) > O} .

(3.13)

(3.14)

T. Smmboulis. J. T. Odell, A posteriori error estimation of FE approximations 211

An example of a finite element grid ordered according to (3.13) is given in Fig. 3.2. We notethat the orclering (3.13) may be defined for non-constant convection vectors a = a(x, t)provided that the convection-field does not have closed streamlines.

In the steady-state case (3.6) reduces to the local discontinuous finite element approxima-tion of Lesaint and Raviart [25]. The discrete approximation of the error satisfies the discreteproblem:

Find ehkllK E Pk(K), such that

( V (k) (kl) f I I( (kl (kl-) (k) d - ( (kl) f I /J (k) da' e" , Vii K + iJK- a' n eil - eh Vii s - ril, vh K + ale a' n "v" s(3.15)

'VV},kl E PiI(K) .

In our steady-state numerical experiments we employ bilinear and biquadratic approximationsof the error; the size of the corresponding linear algebra problem is 4 x 4 for the bilinear and9 x 9 for the biquadratic approximation.

i < j

i < 1

Hj

31 35

22 36

26 32

19 24 28 298 9 33 34

15 16 18 27

12 13 17 233 4 25 30

6 11 14 20

" 10

1 21

2 7

Fig. 3.2. Upstream partial ordering. (a) Schematic definition. (b) A finite element grid ordered with respect to thevector a.

212 T. Strouboulis. J. T. Oden, A posteriori error estimation of FE approximations

(3.16)

3.2. Numerical examples

3.2.1. Steady-state sollllions; comparisons with interpolation-type estimatesWe consider several model problems with steady-state solutions and we compute bilinear

and biquadratic approximations of the error in approximate solutions obtained using Taylor-Galerkin algorithms. At the same time we compute the estimates of the error usinginterpolation-type estimates; these estimates are very popular in the application of adaptivefinite element and finite volume methods in compressible gas-dynamics. Our results indicatethat these estimates can be incorrect by several orders of magnitude for 'rough' solutions. Wenote that the majority of the solutions in compressible gas-dynamics contain singular surfacesand for these solutions the interpolation-type estimates are invalid.

The so-called interpolation-type estimates are based on finite element interpolation theory;let lih, rih denotes a finite element approximation and the corresponding finite elementinterpolant of II. Then the triangle inequality gives

Ilehllm.q.K ~ lIu - ilhll",.q,K + Iluh - il"II,,,.q.K·

Here" . 11",.q.K denotes the W",·q(K)-Sobolev-norm, and since (see Oden and Carey [26]), forquasi-uniforms refinements,

II - II :< Ch2lq-2Ir+k+ 1-", II IIU - uh ",.q.K -= U k+l.r,K'

C = a constant independent of U and the mesh size hK '

j'lk+ 1.r.K = Wk+l.r(K) seminorm ,

l~q,r~x.

(3.17)

then under very restrictive assumptions one can argue that an error indicator eK of element Kis such that

(-9. = Ch2lq-2Ir+p+l-"'1 IK li" p+l.r.K . (3.18)

Here luhlk+l.r.K denotes an 'extractecl' value of the semi norm and k denotes the completepolynomial order of the finite element basis employed in the approximation.

We will now compare the quality of this crude (but popular) estimator of the error and theestimates based on piecewise polynomial approximations to the error computed using (3.10).Let

a=(2,2), f=O.

and consider steady-state solutions with varying degrees of smoothness by choosing appropri-ate inflow-data. The steady-state solution is computed by integrating a set of initial-data usingthe Lax-Wendroff algorithm and the steady-state error is cstimatcd using intcrpolation-typeestimates and from bilinear and biquadratic approximations of the error computed as solutionsto (3.10). We use the notations:

T. Stroubolliis. J. T. Odell, A posteriori error estimation of FE approximatiolls

DG( 1) = bilinear error estimator.00(2) = biquadratic error estimator.

We now proceed to the description of the examples:

Example 1. Smooth sinusoidal solutionWe consider a smooth sinusoidal solution of the form

213

u(X .. x2) = sin (;;), with i = x2 - Xl .

In Table 3.1 we present comparisons between the exact and the estimated global L 2 -norms ofthe error for an approximate solution computed on a 32 x 32 uniform grid; because of thesmoothness of the solution all three estimators (interpolation, DO(I) and DG(2)) are in goodagreement with the exact error.

Example 2. Almost-discontinuous solutionWe next construct a smooth solution with a sharp transition layer by choosing the

boundary-conditions to correspond to the steacly-state solution

u(X1, x2) = sin ~: tanh {~ (i - 32)} ,

if x2 - Xl ~O,if x2 - XI <0.

The results listed in Table 3.1 show that while the DG-approximations deliver very accurateestimates of global L 2-error norms the interpolation-type estimate differs from the exact errornorm by several orders of magnitude. Figure 3.3 presents the finite element grid, the contoursof the error and the contours of the DG( 1)-approximation of the error. The contours of theoG(2)-approximation of the error are indistinguishable from those of the exact error.

Example 3. 'Discontinuous' solutionIn order to demonstrate the collapse of the interpolation-type estimates for non-smooth

solutions, let

Table 3.1Error estimate comparisons

Exact global Interpolation DG(I) error DG(2) errorSolution L ~-error norm type estimate estimate estimate

Smooth sinusoidal 0.119612 0.120957 0.112623 0.119596solution

Almost discontinuous 0.315754 0.011202 0.302882 0.315670solution

Discontinuous 4.446456 0.007325 3.861574 4.40879solution

Curvilinear convection 66.150523 8.043176 61.101277 65.606250with rough inflow data

214 T. Strollbou/is. J. T. Oden. A posteriori errol' estimation of FE approximations

(a) (b)

(e)

Fig. 3.3. Almost-discontinuous solution. (a) Finite element grid. (b) Contours of the exact error. (c) Contours ofthe DG(1 )-approximation of the error.

{

l. .i~8,u(xp x2) = 1- Hi - 8), 8 ~.i ~ 12,

0, .i ~ 12 ,with .i = x2 - Xl'

Figure 3.4 shows the adaptive finite element grid and the contours of the computedsteady-state solution. Figure 3.5 gives the contours of the exact-error and its DO(2)-approximation, while Fig. 3.6 presents the contours of the 00(1 )-approximation of the errorand the interpolation-error contours. Note that the contours of the interpolation-error boundare several orders of magnitude less than those of the exact error or the DO-approximations.

T. Strouboulis, 1. T. Odell, A posteriori error estimation of FE approximations 215

(a)

DOlIn' CJJlflUU "IN = O.IOClt-G1 MJ" O.IOR.DI IWTfllr"L- 0.1_.00

(b)

Fig. 3.4. Discontinuous solution. (a) Adaptive finiteelement grid. (b) Contours of the steady-state approx-imate solution computed using the two-step Lax-Wendroff algorithm.

(a)

11II1II1 C./IIT1URI MIM. a.SOOf-Ol MAl. D.1OI(..oo IHTU'tIlL= D.t5Gl-G1

(b)

Fig. 3.5. Discontinuous solution. (a) Contours of theexact error. (b) Contours of the DG(2)-approximationof the error.

Example 4. Non-constant convectionIn order to test our error estimation procedures for non-constant convection fields we now

consider a curvilinear convection field given by

a(xp x2) = (x2 + 16, -XI)'

The solution domain is the same as in the previous examples namely, n. = [0, 64fWe choose initial, boundary conditions compatible with the following exact solution:

216 T. StrollbOlllis. J. T. Odell. A posteriori error estimation of FE approximations

/

!....

""'AI CllCf.UItS 11111" 0.toal-02 ItAX. a.2oshua UIlUy .... O.atOl.OI

1(1:11". CIMlIUJtl MIN;. a.tln-os MAl. O.1IIl-D2 INTOYIft..;0.".-05

(b)

(b)

Fig. 3.6. Discontinuous solution. (a) Contours of theDG(l)-approximatioll of the error. (b) Contours ofthe interpolation-type error bound.

Fig. 3.7. Curvilinear convection with rough-inflowdata. (a) Adaptive finite element grid. (b) Contours ofthe steady-state approximate solution computed usingthe two-step Lax-Wendroff algorithm.

100,100-5(1'-32) ,80,80 - ¥ (r - 64) ,to,

r~32,32 ~ 1'~36,36 ~ 1'~64,64~ 1'~68.68~,. .

Here r = Vxi + (Xl + 16) is the polar radius measured from the center of curvilinear convec-tion (0, -16). Figure 3.7 contains the adaptive finite element grid and the contours of the

T. Stro/lbo/llis, J. T. Oden, A posteriori error estimation of FE approximations 217

steady-state solution which was computed using the Lax-Wendroff algorithm. Figure 3.8 givesthe contours of the exact error and its DG(2)-approximation and Fig. 3.9 shows the contoursof the DG( 1)-approximation of the error and the interpolation-error contours. Again theinterpolation error bound is several orders of magnitude less than the actual error. Moreoverthe spatial distribution of the interpolation-error does not resemble at all the spatialdistribution of the exact error.

3.2.2. Solutions in space and time; evolution and control of the errorWe now consider model problems with space-time solutions and we compute trilinear

approximations of the error in approximate solutions computed using the Lax-Wendroff

SICMI ''"'1Ll1S ,qll I (1.\00 -t, ••.• r.UI05 .l)! 1~1(11\'1Il' 0.'':10 -00

(b)

Fig. 3.8. Curvilinear convection with rough-inllowdata. (a) Contours of the exact error. (b) Contours ofthe DG(2)-approximatilln of the error.

COl)

(b)

Fig. 3.9. Curvilinear convection with rough-inflowdata. (a) Contours of the DG( I )-approximation of theerror. (b) Contours of the interpolation-type errorbound.

218 T. Strollbolllis, J. T. Odefl. A posteriori error estimatio/l of FE approximlltiO/lS

algorithm. Our cstimates of the error arc obtained by solving thc discrete cquations (3.11) ineach element.

We may control the error in the Lax-Wendroff algorithm with respcct to the oG(1)-algorithm (used to estimate the error) by restarting the solution procedure at a set of discrctetimes T = {1"), 1"2'1"J' ... }. We let

(3.19a)

(3.19b)

Herc 11k denotcs thc L 2-projection operator from the discontinuous flnite element spacewhere the error is defincd into the continuous finite element space where the approximatesolution is computed, namely:

N,

11k : n Pk(K<)~ H"(fl).<~I

(3.20)

The discrete-times 1"(< 1"2< ... < 1"" can be set by the user or they may be selected tocorrespond to the time-instant at which the error norm exceeds a user-specifled tolerance.

We now proceed to the examples.

Example 1: Rectilinear convection of a cosine hill: calclliation of the evolution of errorWe now consider the homogeneous linear-convection problem

au + au = 0 in fl .at ax(

We let fl = (0, 170) x (0,50) and we choose boundary and initial conditions compatible withthe 'cosine-hill' exact solution:

{

A . (7T'r)ll(X., x

2' I) = Sill 2R ' r:;;;' R,

0, r>R.

where

are the coordinates of the centcr of the cosine-hill at the initial time and R = 15 is the baseradius of the cosine-hill. We computcd three solutions to this problem, a uniform grid solutioncomputed on a coarse 34 x 10 uniform grid, and two adaptive grid solutions by allowingadaptation of the grid by one or two levels over the original coarse grid. We integrated thesolutions forward in time up to thc final time T = 120. Wc did not employ control of the errorin this examplc. Figures 3.10, 3.11 and 3.12 display the contours and three-dimcnsional vicwsof the computed solutions at the final time. Figure 3.13 shows the discrepancy between theexact and the estimated global L 2-norm of thc error for 0, 1 and 2 levels of refinement. We

T. Stro/lbo/llis. J. T. Odell. A posteriori error estimation of FE approximations

DENSI', CBNT6URS "IN 0.505E-01 HAx:;: O.100f"03 lNTfRVAl.: 0.500E-OI

219

Fig. 3.10. Rectilinear convection of a cosine-hill. Solution contours and three-dimensional view of the approximatesolution at the final time obtained on a uniform grid of level O.

DENSITY C8NTBURS "IN = 0.50SE.01 HAX' 0.100['03 INTERVAL' 0.500[.01

Fig. 3.11. Rectilinear convection of a cosine-hill. Solution contours and three-dimensional view of the approximatesolution at the final time obtained on an adaptive grid of level 1.

220 T. Srro/lbo/lli.l'. J. T. Oden, A posteriori error estimation of FE approximations

OENSITl C8NT8URS KIN:: 0.505E-Ol ~AX 0.lOOE-03 (NTERVRl= 0.500£-01

Fig. 3.12. Rectilinear convection of a cosine-hill. Solution contours and three-dimensional view of the approximatesolution at the tinal time obtained on an adaptive grid of level 2.

H

a

l:l:: ..0 ..l:l::l:l::UJ ..~ ..~i:!:l ~g ..0 EXACf "V-~ 00(1) - LEVEL 2

8:11 0.'0 1.11 .... a." "" 1.11 I.U

TIMEx 10.2

Fig. 3.13. Rectilinear convection of a cosine hill. History of the exact global L1-norm of the error and itsEOE-estimate for grids of level O. I and 2. respectively.

T. Stroll ball lis . 1.T. Oden, A posteriori error estimation of FE approximations 221

note that as the Icvel of refinement is increased the discrepancy betwecn exact and approxi-mate error is reduced.

Example 2: Linear convection of a cosine-hill along the diagonal of a square domain:estimation and colltrol of the evolution of error

We now consider the linear-convection problem:

Wc let n = (0,64)2 and we choose boundary and initial conditions compatible with a'cosine-hill' solution which convects along the diagonal of the square domain. We chose theradius of the support of the cosine-hill R = 16, wc placed the center of the hill at the initialtime at (16, 16) and we integrated the solution forward in time until the center of the hillcoincides with thc point (48,48). Wc computed the solution and thc evolution of the globalL 2 -norm of the error using uniform and adaptive grids with rcfinements up to four levels ovcra coarse 8 x 8 uniform initial grid. Figures 3.14 through 3.26 present the approximate solutionswhich were computed on adaptive grid with two. three and four levels of refinement with orwithout control of error and the time-histories of the global L 2 -norm of the error and itsestimate. In the calculation with error control we employed (3.19a) to correct the solution andreset the error every 3 time-steps. We experimented in resetting the error in two ways at thetime instants that the solution is corrected, namely:

(i) the new value of the error is set according to (3.l9b),(ii) the new value of the error is set equal to zero.

Figure 3.25 shows that (i) leads to better control of the error.We note that when the error control algorithm is employed the EOE( 1)-estimate cannot

provide an accurate valuc of the total error; instead it gives an indication of the additionalerror between solution resets. In order to obtain an estimate of the total error in conjunctionwith error control. an EOE(2) estimate must be used.

Fig. 3.14. Linear convection of a cosine-hill along the diagonal of a square domain. Finite element grid with 4levels of refinement and solution at the initial time.

222 T. Strollholllis, 1.T. Oden. A posteriori error estimation of FE approximmio/ls

(al

(bl

Fig. 3.15. Linear convection of a cosine-hill along thediagonal of a square domain. Finite element grid with2 levels of refinement and solution at the final time.(a) Without error control. (b) With error control.

(al

(b)

Fig. 3.16. Linear convection of a cosine-hill along thediagonal of a square domain. Finite element grid with3 levels of refinement and solution at the final time.(a) Without error control. (b) With error control.

T. Strouboulis. 1.T. Odell, A posteriori error estimation of FE approximations 223

(a)

(b)

Fig. 3.17. Linear convection of a cosine-hill along thediagonal of a square domain. Finite element grid with4 levels of refinement and solution at the final time.(a) Without error control. (b) With error control.

(a)

o

(b)

Fig. 3.18. Linear convection of a cosine-hill along thediagonal of a square domain. Contours of the approxi-mate solution at the final time on an adaptive grid with2 levels of refinement. (a) Without error control. (b)With error control.

224 T. Strouboulis. J. T. Odell, A posteriori error estimation of FE approximations

(a)

(bl

Fig. 3.19. Linear conveelion of a cosine-hill along thediagonal of a square domain. Contours of the approxi-mate solution atlhe final time on an adaptive grid with3 levels of refinement. (a) Without error control. (h)With error control.

(a)

(b)

Fig. 3.20. Linear convection of a cosine-hill along thediagonal of a square domain. Contours of the approxi-mate solution at the final time on an adaptive grid with4 levels of refinement. (a) Without error control. (b)With error control.

T. Strollboulis, J. T. Oden, A posteriori error estimation of FE approximations

"c"'"'"

i I // '..J I / ~ 3

I /\ /\ /\ ~/\ /'J\...---2~

TIMETIME

225

Fig. 3.21. Linear convection of a cosine-hill along thediagonal of a square domain. Time evolution of theglobal L 2 -norms of the error and their 'estimates' forsolutions computed on adaptive grids with 4 levels ofrefinement with and without error control. Curve 1:Exact error without error control. Curve 2: EOE( 1)-error estimate without error control. Curve 3: Exacterror with error control. Curve 4: EOE( 1)-error esti-mate with error control.

TIME

Fig. 3.23. Linear convection of a cosine-hill along thediagonal of a square domain. Time evolution of theglobal L 2-norms of the error and their EOE( I )-esti-mates computed on adaptive grids with 2. 3 and 4levels of refinement without error control. Curve 1:Exact error; adaptive grid of level 2. Curve 2:EOE( I )-estimate of Curve I. Curve 3: Exact error;adaptive grid of level 3. Curve 4: EOE(1 )-estimate ofCurve 3. Curve 5: Exact error; adaptive grid of lcvel4. Curve 6: EOE( 1)-estimate of Curve 5.

Fig. 3.22. Linear convection of a cosine-hill along thediagonal of a square domain. Time evolution of theglohal L 2 -norms of the error for solutions computedon adaptive grids with 2. 3 and 4 levels of adaptiverefinement with and without error control. Curve 1:Level 2: without crror control. Curve 2: Levcl 2: witherror control. Curve 3: Level 3; without error control.Curve 4: Level 3: with error control. Curve 5: Level 4:without error control. Curve 6: Level 4: with errorcontrol.

TIME

Fig. 3.24. Linear convection of a cosine-hill along thediagonal of a square domain. Time-evolution of theglobal L 2-norms of the error and their EOE( 1)-esti-mates computed on adaptive grids with 2. 3 and 4levels of refinement with error control. Curve 1: exacterror: adaptive grid of level 2. Curve 2: EOE( 1)-estimate of Curvc I. Curve 3: Exact error; adaptivegrid of level 3. Curve 4: EOE( 1)-estimate of Curve 3.Curve 5: Exact error: adaptive grid of level 4. Curve6: EOE( 1)-estimate of Curve 5.

226 T. StrOllbolllis. J. T. Oden, A posteriori error estimation of FE approxim(l(iolls

~,/vvvvvv V'"0'"'"w

':.J'':'

6

TIME

Fig. 3.25. Linear convection of a cosine-hill along thediagonal of a squarc domain. Time-evolution of theexact global L 2 - norms of the error computed on adap-tive grids with 2. 3 and 4 levels of refinement and witherror control. Curve I: Adaptive grid of level 2; theerror is reset to zero after the solution is corrected.Curve 2: Adaptive grid of level 2; the error is resetaccording to (3.19h) after the solution is corrected.Curve 3: Adaptive grid of level 3; the error is reset tozero after the solution is corrccted. Curve 4: Adaptivegrid of level 3; the error is reset according to (3 .19b)after the solution is corrected. Curve 5: Adaptive gridof level 4: the error is reset to zero after the solution iscorrected. Curve 6: Adaptive grid of level 4; the erroris reset according to (3.1911) after the solution iscorrected.

4. Linear diffusion problems

TIME

Fig. 3.26. Linear convection of a cosine-hill along thediagonal of a square domain. Time-evolution of theexact global L 2-norms of the error computed on un-iform and adaptive grids with 2 and 3 levels of refine-ment. Curve 1; Exact error vs. time for an adaptivegrid solution with 2 levels of refinement. Curve 2:Exact error vs. time for a uniform grid solution with 2levels of refinement. Curve 3; Exact error vs. time foran adaptive grid solution with 3 levels of refinement.Curve 4: exact error vs. time for a uniform gridsolution with 3 levels of refinement.

4.1. A posteriori error estimates for finite element approximations of parabolic boundary,initial value problems

We now consider the parabolic boundary-value problem:

Find u = u(x, t), which satisfies

ul - .± -aa. (ajj aau,) = f in n x (0, T) ,/.,'"I x, x,

2 CJuj.~1 /ljaij (h

j= g on r2 x [0. T] ,

u = u on ~ x [0. T] ,

(4.1)

T. Strol/holllis, J. T. Odell. A posteriori error estimation of FE approximations

with the initial condition

u(X, 0) = II(I(X) .

227

Here ii = ii(x, t) denotes the given data on the kinematic part of the boundary and the rest ofthe notation is described in Section 2.1.

There are several ways to construct finite element approximations for time-dependentboundary-value problems, namely:

1. One can define a weak space-time formulation of (4.1) and can design space-time finiteelement approximations by employing polynomial test and trial spaces in space and time(such a procedure is followed in [15]).

2. One can discretize the space variables only (semidiscretization) using the finite elementmethod; the resulting system of ordinary differential equations is integrated in time usingan O.O.E. solver. This procedure is often called the method of lines example (see16-8]).

3. One can discretize the time variable using a Taylor-series expansion in time while leavingthe space-variahle continuous. This leads to a Taylor'-weak statement which can befurther discretizcd in space using the Galerkin finite element method. Such schemes arereferred by the name Taylor-Galerkin methods.

Bieterman and Babuska 16-8J developed a posteriori estimates for a method of linesprocedure for parabolic problems. The method employs local problems with bubble-typesolutions to compute local approximations of the error and rests on the major assumption thatthe time-component of the error is controlled by an adaptive time-stepping algorithm. Herewe follow very similar lines with Babuska and Bieterman and we assume that II" = ult(x. t)denotes an approximate solution obtained by using a first-order Taylor-Galerkin algorithm,namely:

G· n () n+1 fIvcn IIh = IIh " tn ' compute lilt rom

(1I~+I, v,.) + Ilt B(II~, vh) = L(v,.) 'Vvit E Vit. (4.2)

We then assume that we can control the time-step Ilt so that we can estimate theacceleration 117+1 = 11,(', tn+1) with desired accuracy using a first-order difference in time, i.e.,

Given tol choose Ill, such that

(4.3)

The tolerance tol may be selected according to the recommendations of Bieterman andBabuska 16J, i.e., it is intimately connected to the spatial discretization error. Then, in orderto estimate the error. wc define local spatial problems in each element K as follows:

Find e;;+1 E H1(K), such that

B ( n+ I ) (' ) (n+1 ) \.J VK ell ,v = r". v K - II, ,v vV E . (4.4)

228 T. Strollbolllis, J. T. Odell. A posteriori error estimation of FE approximations

Hcrc (r;" v)" dcnotes thc action of the spatial residual on thc test functions and is defined by

(4.5)

We seek error indicators e~+1 E Mh(K), where Mh(K) denotes the space of bubble-functionsover the element K, namely

M,,(K) = {W(x) E Tl H '(K) I w(xi) = 0, where xi is a node of the finite elemcntKET,.

basis employed in the calculation of IIII} .These error indicators are obtained from (4.4) after the time-derivative aul at and thenormal-derivative a;i aul ax; are approximated based on the approximate solution u".

4.2. Numerical examples

Example 1. Propagating exponential pulse solurion of the diffusion equationWe solved the linear diffusion equation

lit - All = f(x, t), x En = (0,64)2 ,

with the forcing function f, the boundary and initial conditions chosen to correspond to theexact solution

Hcre we selected A = 100, (l = 0.10, b = 0.15, V = 0.15, x~ = 10 and x~ = 15. Figures 4.1through 4.5 show the approximatc solution at the initial and the final time and the histories of

(a) (b)

Fig. 4.1. Propagating exponential wave solution of the diffusion equation. (a) Grid and solution at the initial time.(b) Grid and solution at the final time.

T. StrollbOlllis. 1. T. Oden, A posteriori error estimation of FE approximations 229

iEXACT ERROR

P"Ak$ (OrtMpl)nd to

tl ... time!. o( grid ,tdi.\plio!l

ESTIMATED EllllOR

T1~'E

Fig. 4.2. Propagating exponential pulse solution of thediffusion equation. History of the exact and estimatedglobal energy norm of the error. Time-step M·, 240time-steps. At· is equal by the stability limit of theforward-Euler time integration. The grid was adaptedevery 10 steps.

TIME

Fig. 4.3. Propagating exponential pulse solution of thediffusion equation. History of the exact and estimatedglobal energy norm of the error. Time-step! At·, 480time-steps. At· is equal by the stability limit of theforward-Euler time integration. The grid was adaptedevery 10 steps.

~."T"'m"mOH

. ~~V EXACT EHHOH "

'"o'"'"uJ

TIME TIME

Fig. 4.4. Propagating exponential pulse solution of thediffusion equation. History of the exact and the esti-mated global energy norm of the error. Time-step1At·. 960 time-steps. At· is equal to the stability limitof the forward-Euler time integration. The grid wasadapted every 10 steps.

Fig. 4.5. Propagating exponential pulse solution of thediffusion equation. History of the exact and estimatedglobal energy norms for solution obtained using time-step size At*. !At*. !At*. Note that reduction of thetime-step size leads to reduction in the global error.At· is equal to the stahility limit of the forward-Eulertime integration. The grid was adapted every 10 steps.

the global energy-norm of the error and its computcd estimatc for approximate solutioncomputed using timcslcps ilt*, !ilt*. 1ilt*. whcrc ilt* denotes the stability limit for thetime-step employcd in the forward Euler time-integration scheme. We note that in all casesour error estimation technique provides a very effective error bound which does notdeteriorate as the solution is integrated forward in time.

230 T. Stroubolllis, J. T. Odell. A posteriori error estimation of FE approximl/tions

Example 2. Rotating Gaussian hill so/wion of the diffusion equation

We solved the diffusion equation with initial, boundary conditions and forcing functionwhich correspond to the exact solution of a rotating Gaussian-hill given by

Here n = [0,64]2, A = 100 and (x~, x~) is the position vector for the center of the Gaussian-hill and we let x~ = 16 + 32 sin (11't), x~ = 16 + 32 cos (11't). By taking values of the time-variablet E [0, !] the center of the Gaussian-hill moves from (16,48) to (48, 16) as the Gaussian-hillrotates around the center of rotation (16, 16). Figures 4.6 through 4.12 show the solution andthe time-evolution of the energy-norm and its estimate. The sudden drop of the error near theinitial time observed in Figs. 4.8-4.12 for the level 3, 4 calculation is due to the fact that theinitial data was interpolated on a coarse level 2 grid which resulted in high-error in the initialconditions.

Fig. 4.6. Rotating Gaussian hill solution of the diffusion equation. Three-dimensional view of the solution at theinitial, at intermediate and the final time obtained from an adaptive finite element calculation.

T. Slrollbolllis, 1.T. Oden. A posteriori error estimation of FE approximations

Fig. 4.7. Grid, contours and 3-dimensional view of theapproximate solution (computed with an adaptive gridwith 4 levels of refinemem) at the final time.

231

..-_.~-

TIME

Fig. 4.8. Rotating Gaussian hill solution of the diffu-sion equation. Time-history of the global energy-normof the error and its estimate for the adaptive gridsolution with three levels of refinement. Curve 1:Estimate of the error. Curve 2: Exact error.

I

1i~

nn

5. Convection-diffusion problems

(5.1)

5.1. A posteriori error estimates for finite element approximations of convection-diffusionproblems

We now consider the parabolic convection-diffusion problem:

Find u = u(x, t), which satisfies

2 a ( au) .tit + a .Vll -.~ -;- {lij -;- = f tIl.n X (0, T) .

I.J= I uXi uXj2

i.tl lliaij :'~j + min [0, (a' 1I)Ju = g on 1; x fO, T] ,

u = II on I~ x [0, T] ,with the initial condition

u(X, 0) = uo(x) .

232

I. /

T. Strollbolllis, J. T. Oden. A posteriori error estimation of FE approximations

I-~~~._--

5b

\I'L.---------. --....

II _\ --- ------ .........-..-_.1--

TIME

Fig. 4.9. Rotating Gaussian hill solution of the diffu-sion equation. Time-history of the global energy-normof the error and its estimate for adaptive grid solutionwith refinemcnts of level 2 and 3 respectively. Curve1: Estimatc of the error on thc adaptive grid with 2levels of refinement. Curve 2: Exact error on theadaptive grid with 2 levels of refinement. Curve 3:Estimate of the error on the adaptive grid with 3 levelsof refinement. Curve 4: Exact error on the adaptivegrid with 3 levels of refinement.

;!; h ..-r---.---r-- __;;:.[ ~-.ffi~~-~ ,

TIME

Fig. 4.11. Rotating Gaussian hill solution of the diffu-sion equation. Time history of the global energy normof the error and its estimate for adaptive grid solutionswith 3 and 4 levels of refinements. Curve 1: Estimateof the error (level 3 grid). Curve 2: Exact error (level3 grid). Curve 3: Estimate of the error (level 4 grid).Curve 4: Exact error (level 4 grid).

TIME

Fig. 4.10. Rotating gaussian hill solution of the diffu·sion cquation. Time-history of the global energy·normof the error and its estimate for the adaptive gridsolution with 4 levels of relinement. Curve I: estimateof the error. Curve 2: Exact error.

L = --t 4

TIME

Fig. 4.12. Rotating Gaussian hill solution of the diffu-sion equation. Time-history of the global energy-normof the error and its estimate for adaptive grid solutionwith 2. 3 and 4 levels of refinement. Curve 1: Estimateof the error (level 2 grid). Curve 2: Exact error (level2 grid). Curve 3: Estimate of the error (level 3 grid).Curve 4: Exaet error (level 3 grid). Curve 5: Estimaleof the error (level 4 grid). Curve 6: Exacl error (level4 grid).

T. Strouboulis. 1.T. Odell. A posTeriori error estimation of FE approximations 233

We construct a finite element approximation of (5.1) using a combination of the Lax-Wendroff algorithm for the convective term with the forward-Euler being employed for thediscretization of the second-order terms. This form of time-stepping is called LW-FE (Lax-Wendrofflforward-Euler) and for a typical time-step it reads as follows:

O· 1/ () n+1 fIven u" = ll" ., tl/' compute lI}, rom

(lI;:+I, vh) + ~tB(ll~, Vh) + ut(a, VlI~) - !~t2(a ·V(a ·Vu~), Vh) = L(vh) 'fIvh E Vh.

(5.2)

Our objective is to ohtain convergent estimates of the error in the finite element solution ultwhich remain valid for the entire range of the parameter

(5.3)

Here lal = Va~+ a;, laiil = mini.i=I.2la;J h is the mesh size of the computational cell forwhich the parameter Pc is evaluated. The parameter Pc is called the Pec1et number andcharacterizes the behavior of the discrete approximations of (5.1). When Pc ~ 1 takes largevalues we say that the discrete problem (5.2) is convection-dominated, where if Pe ~ 1 we saythat the discrete problem is diffusion-dominated. Whenever an adaptive-grid is employed it isconceivable that in the very fine parts of the grid the discrete problem can be diffusion-dominated while at the same time it can be convection-dominated in the coarse parts of thegrid. We refer to this possibility to emphasize the fact that we must construct an errorindicator which remains effective for Pc E [O,:xl) in order to be of meaningful use in anadaptive environment.

The error e" satisfies the following local problem in each element K and for the fixedtime-instant tl/+)'

find e~+1 E H1(K), such that

( 1/+1 ) (V n+1 ) B ("+1 ) ( )ell ,V K + a' lilt . V + K elt , v = 1'", V KI

'fIv E V . (5.4)

Here I'll denotes the convection-diffusion residual dellned by (2.13). In order to measure themagnitude of the error in an element K. we employ the following norm:

(5.5)

Here we remember that Illeltlll~ = BK(e". ell) is the energy-norm which corresponds to thebilinear form B and we define

(5.6)

The norm of the error 11I1 e~+ 11I11 is estimated by employing a combination of the errorestimation procedures used in Sections 3 and 4. In particular we have

234 T. Strollbolllis, J. T. Oden, A posreriori error estimation of FE approximarions

I1II 11+111112 = III c.II+11112 + I d,"+112ell K ell K ell a.ilK· (5.7)

Here IIl1e~+ 'IIII~denotes the estimate of the corresponding error norm and e~,11+ I, e1.·11+1 arethe convective and diffusive components of the error which are obtained using the followingtime-splitting procedure:

1. Convection-Step:

find

e~·n+1 E W(K), such that

JI.,+ I JI,,+ 1 f

(e~ + a' Ve~, V)K dt + _Ia· 1l1(e~- e~'-)v ds dt1'1 I In aK

+ (e~(', t:) - e~(-, t~), V)K

J~+l J~+lf= (I", V)K cit - __ Ia· "IJ"v ds dt - (/7" V)KI" t 11 (1/\

VEW~.(5.8)

Here r denotes the convective-component of the source f and the remaining of the termswere defined in Section 3.2. DifJusion-Step:

Find e1.·I1+ I E H '(K), such that

BK(e~·n+l, v) = (r~, v) K Vv E V. (5.9)

Here (r~:, v) K denotes the action of the diffusive part of the spatial residual on the testfunctions and is defined by

(r~, v) K = (Jd, V)K + BK(U~+J, v) + iK aij ::. njv ds. (5.10)J

Here Jd is the diffusive component of the source-term; clearly J= I" + Jd.In our applications we employed trilinear approximations in space and time for e~ as

described in Section 3 and biquadratic bubble function in space for the approximation ofd.Il+1eh •

The following example demonstrates that the outlined procedure leads to very sharpestimates of IIl1e;;+lIIIIK for all values of the Peclet number.

5.2. Numerical examples

Example 1. Forced convection-diffusion of a Gaussian hillWe determine boundary, initial condition and the source term which correspond to the

exact solution

u(x, t) = A exp {- k[(x - xo) - at]2} . (5.11)

We let A=IOO, K=O.03, a=a(1,I), aij=eoij; we let !2=lo,64f and xo=(16,16). The

T. Stroubouli.\·, J. T. Oden, A posteriori error estimation of FE approximations 235

approximate solutions are obtained on an adaptive grid with refinements up to level 3 withminimum mesh size h = 1. We considered the cases described in Table 5.l.

In each case we calculate the solution, the exact and approximate error integrating thesolution 136 time-steps. Figures 5.1 and 5.2 show the solution at the initial time and thecomputed solutions at the final times for various choices of parameters. In each case the

Table 5.1Minimum Peclet number in the calculation

Convection Diffusion Pecletspeed lal constant E number

V2 0.01 70.71V2 0.1 7.071V2 1 0.7071Y.02 1 0.2236yOm 1 0.0707

(a)

(b)

Fig. 5.1. Forccd convection-diffusion of a Gaussian-hill. (a) Solution at the initial-time. (b) Solution ob-tained after 136 steps· for a = (1, I) and e = 0.01.

• The time-step was detcrmined from the stabilitycondition of the explicit algorithm.

(a)

(b)

Fig. 5.2. Forced convection-diffusion of a Gaussian-hill. (a) Solution obtained after 136 time steps fora = (1. 1) and /; = 1. (b) Solution obtained after 136steps for a = (0.01. 0.(1) and E = 1.

236 T. Strouboulis, J. T. aden, A posteriori error estimation of FE approximatiollS

time-step was obtained for the stability condition for the LW-FE algorithm. Figures 5.3, 5.4and 5.5 show the evolution of the global error norms of the error and its estimate for the casessummarized in Table 5.1 It is clear that the error estimate remains sharp for the entire rangeof the parameters.

TIME

Fig. 5.3. Forced convection-diffusion of a Gaussian-hill. time history of the global IllIellll-norm of theerror and its estimate obtained for a = (1. 1) and£ = 0.01. Curve 1: Exact error. Curve 2: Computedestimate.

a = (0.1.0.1).< = 1

TIME

Fig. 5.4. Forced convection-diffusion of a Gaussian-hill. Time history of the global IIllellll-norm of theerror and its estimate obtained for (a) a = (1. 1).£ = I. (b) a = (0.1. 0.1), £ = 1. Curve I: Exact error(a = (1,1). £ = 1). Curve 2: Error estimate (a = (I. 1).£ = I). Curve 3: Error estimate (a = (0.1. D.]), £ = 1)Curve 4: Exact error (a = (D.I. 0.1), £ = 1).

D.E

Fig. 5.5. Forced convection-diffusion at a Gaussian-hill. Time history of the globallillellil-norm of the error andits estimate for 5 different sets of parameters. Set A: a = (I. I). £ = 0.01. Set B: a = (1. 1), E = 0.1. Set C:a=(1,l), £=1. Set D: a=(O.1.0.l), £=1. Set E: a=(O.OI,O.OI). £=1.

T. Strollbolllis, J. T. Odell. A posteriori error estimation of FE approximations

6. Nonlinear transport problems

237

Here we examine the extension of some of the error estimation procedures for nonlinearconvection problems with quadratic nonlinearities. We now report some numerical experi-ments which demonstrate that the error estimation methods give meaningful results fornonlinear problems.

6.1. Burgers' equation

We consider the nonlinear, scalar hyperbolic conservation law:

au a (t/) au 0 . n- + - - + - = In JI.at ax 2 ay ,u = u on an,u = Uo at t = to .

Again we let fl = = (0, 64)2 and we choose the inflow boundary iJfl-, the initial andboundary conditions compatible with the exact solution

{

-I,1 ,

u(x, y) = 32 - x ,32- Y

x > 32 and x > 64 - Y ,x < y and x < 32 ,

x < 64 - Y and x > y and y < 32 .

We solved the above problem on adaptive grids of maximum level 3 and 4 using thetwo-step Lax-Wendroff algorithm. The grid was adapted using the interpolation-estimategiven before.

We computed two estimates of the error ell and e" as the approximations of the solutions ofthe following conservation laws:

1. Exact error equation:

Find ell E L 2(fl) , such that

aell iJell (aUh iJUI.)Il-+-=- U-+- .ax ay ax ay

2. Approximate error equation:

Find eh E L 2(fl) , such that

aeh aeh ( aUh auh)Il -+-=- U -+-

II ax ay h ax ay'

The definition of these first-order boundary value problems for eh and ell is completed bynoting that the inflow values of the error are equal to the interpolation-error on the boundary.

238 T. Strollboulis, J. T. aden, A posteriori error estimation of FE approximations

Q(K"lt C•• flU'" 1.111 = ·.1$0(-00 ",n '=' 0.150('00 JKTUWM.= 0.100r.-00

Fig. 6.1. Burgers' Equation. Exact solution interpo-lated on a uniform grid of level 3.

Fig. 6.2. Burgers' Equation. Approximate solutionobtained with the Lax-Wendroff algorithm with anadaptive grid of maximum level 3 shown in the figure.

Figures 6.1 through 6.9 depict the exact solutions, the approximate solutions and thecorresponding grids, the contours of the exact error and its estimates. The slight asymmetryobserved in the contours of the estimated error may be explained from the fact that theordering employed in the discontinuous Galerkin calculation is not symmetric. We now list theexact and the estimated values for the global L 2-norm of the error:

Adaptive grid of level 3:

lIehII1.2(/1) (exact) = 11.336.

IIeh II1.2(/1) (computed) = 11.119 ,

lIehIlL2(n) (computed) = 8.009.

T. StrollbOlllis, 1.T. Oden, A posteriori error estimation of FE approximations 239

SUiMA. ClNllu" liN = 0.200(-01 KAI:I O.lOU.Ol IHTUwAL=- 0.769[-01

Fig. 6.3. Burgers' Equation. Contours of the exacterror for the approximate solution obtained on thegrid of maximum level 3. lIehllL2(11) (exact) = 11.336.

Fig. 6.5. Burgers' Equation. Contours of the esti-mated error for the approximate solution obtained onthe grid of maximum level 3. The error was estimatedby solving an approximate conservation-law satisfiedby the error. IIl\ IIL2(U) (estimated) = 8.009.

Fig. 6.4. Burgers' Equation. Contours of the esti-mated error for the approximate solution obtained onthe grid of maximum level 3. The error was estimatedby solving the exact conservation-law satisfied by theerror. lIeh II L2(Ul (estimated) = 11.119.

Fig. 6.6. Burgers' Equation. Approximate solutionobtained using the Lax-Wendroff algorithm with anadaptive grid of maximum level 4 shown in the figure.

240 T. StrollbOlllis, J. T. Odell, A posteriori error estimation of FE approximations

Fig. 6.7. Burgers' Equation. Contours of the exacterror for the approximate solution obtained on thegrid of maximum level 4. lIehllt.2(fl) (exact) = 11.243.

SI'''''I CllIIuun "'"' o.toor~a. _I' O.IGl[·OI u.tU, .... t. '7ttf·~1

Fig. 6.8. Burgers' Equation. Contours of the esti-mated error for the approximate solution obtained onthe grid of maximum level 4. The error was estimatedby solving the exact conservation-law satisfied by theerror. lIehIlL2(U) (estimated) = 11.048.

Fig. 6.9. Burgers' Equation. Contours fo the estimated error for the approximate solution obtained on the grid ofmaximum level 4. The error was estimated by solving an approximate conservation-law satisfied by the error. lIell(estimated) = 7.274.

Adaptive grid of level 4:

IlehIlL2(fl) (exact) = 11.243,

Ile"IIr.2(fl) (computed) = 11.048,

lIe"IIL2(fl) (computed) = 7.274.

We note the following:

T. StrollbOlllis. J. T. Odell. A posteriori error estimation of FE approximCllions 241

(a) The effectiveness of the estimates based on eh is inhibited by error involved in thedefinition of the conservation law satisfied by eh. This fact suggests an interaction procedurefor the calculation of the error in the solution of nonlinear problems, namely: after thecalculation of the error eh an improved solution uh = II" + e" is obtained. This solution may beemployed to correct the coefficients of the conservation-law satisfied by the error in order toobtain a better error estimate. This iterative procedure may be repeated until there is nosigniflcant change in the computed error.

(b) We observe that the global L 2-error norm is practically the same for the grids of levels 3and 4. This is an indication of the fact that the adaptive procedure employed does not have theadaptivity property with respect to the L 2-norm.

Acknowledgement

Support of this work by NASA Marshall Space Flight Center and portions of the study bythe Office of Naval Research is gratefully acknowledged. The first author was supported bygrant 32122-70120 ATG of the Texas Advanced Technology Program.

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