~ The Amerlcen Society of~ Mechenlcel Engineers

R.pr/nr~ FromRecent Developments In Computational

fluid Dynamics - AMD·Vol. 95Editors: T. E. Tezduyar end T. J. R. Hughes

(Book No. G00455)

PARADIGMATIC ERROR CALCULATIONS FOR ADAPTIVEFINITE ELEMENT APPROXIMATIONS OF

CONVECTION DOMINATED FLOWS

J. T.OdenTICOM. The University of Texas at Austin

Austin, Texas

T. StrouboulisDepartment of Aerospace Engineering

Texas A & M UniversityCollege Station, Texas

J. M. BassThe Computational Mechanics Company, Inc.

Austin, Texas

Abstract: New eTTOre~timation technique.! are de~cribedfor two-dimemional convec-tion-dominated flow~. Ezample~ of appliadiom to linear convection problem" are given.The methocU are ~ed on d~continuow finite-element technique.t. Re~u1~ ~how that con-ventional interpolation methocU may lead to grou undere~timate~ of actual eTTOr and beinadequate for we in adaptive method~.

1 INTRODUCTIONAdaptive finite element methods may have a great impact on computational fluid dynamics;they give credence to the concept of optimal CFD algorithms which can deliver the bestpossible numerical results for the least computational effort, and they have proved to bevery effective in many complex flow simulations (see, e.g., [5, 6, 13, 14, 15J or the survey[12]).

The success of adaptive methods hinges on the availability of a reliable error estimator- a technique for using a computed solution to obtain an a-posteriori estimate of errorin each element in an appropriate norm. When the estimated error exceeds a preassignedtolerance. the mesh is refined (an h-refinement) and/or the polynomial degree is increased(a p-enrichment). Thus, adaptive methods not only provide for a systematic reduction oflocal error but also can yield an estimate of the error itself, which can be a very usefulindication of the overall reliability of the calculation.

To date. several methods of error analysis have been proposed for linear elliptic prob-lems [1,2,3) and some techniques for parabolic problems and incompressible Navier-Stokesequations have been tested [5, 10, 16J, but the overwhelming majority of adaptive codesconcerned with solving practical flow problems employ simple interpolation results (e.g.,[6,8,9, 14, 15]). The behavior of error in convection-dominated flows has apparently notbeen addressed in the literature.

In the present paper, we describe some preliminary results on the estimation of errorin finite element approximations of linear convection problems in two dimensions. The

study is part of a broader study of error estimation for general flow simulations that willappear in a forthcoming paper. The principal issues of interest uncovered or developed inthe present study are:

• The standard interpolation methods [6, 8, 9, 14, 15Jmay be completely inadequate forerror prediction in many situations in convection-dominp.ted flows. They can yieldestimates many orders-of-magnitude less than the actual error and, thus, cannot beused to drive an effective adaptive scheme.

• The discontinuous finite element method of Lesaint and Raviart [7J can be used asa basis for an effective error estimation scheme; such a scheme is developed in thisnote.

• The error predictor developed here can be used effectively in problems with varyingsmoothness in initial data, including problems with quite sharp gradients.

2 ERROR ESTIMATION FOR A MODEL HYPER-BOLIC PROBLEMFor clarity of presentation, we shall confine our attention to the model convection problem:

~+a'Vu

u

u(x,y,O)

f

9

uo(x, y)

in 0 CR?

on 811-

inO

(1)

Here u = u(x, y, t) is a scalar-valued function on 0 x (0,00), a = a(x, y) is a givenbackground velocity field, f = f(x, y, t) a source term, Uo the initial data, and g the inflowdata on the inflow boundary 811- = {(x, y) e 80 : a . n $ ° a.e.}, and we assume aLipschitz boundary 80 = 80- U 80+.

A weak form of (1), valid over the space-time domain

D = 0 x (O,T)

is characterized as the following problem:

Find u e W such that

r (UI + a· Vu - f)v dOdt + rT r la.nl(u - g)v dsdtJD Jo Jeo- (2)

+ fo(u(., 0+) - uoO)v(·,O)dn = 0 VvEW

wherein W is an appropriate space of admissible test functions; e.g., W = {v = v(z, y, t) ELOO(HI(n), (O,T))}.

It is possible to develop a "local" form of (2) valid for Bubdomains of D. Let 1j., denotea partition of 0 into convex, almost disjoint subdomains K (eventually K will denote afinite element) and consider a partition of the time interval into time steps,

° = to < tl < ... < tn < ... < tN = T; IT = U KKET~

148

Then, over the slice Dn = K x (tn, tn+d, the solution of (2) satisfies

1 11"+'l(UI + a.Vu - f)v dndt + la.nKI(u+ - u-)dsdtD" '" 8K- (3)

(4)

+ i(u(.,t~) - u(·, tn))tl(·,t~)df1. = a "Iv E WK

Here oK- is the inflow boundary to K(oK- = {(z, y) E oK; a.nK < Oa.e.}) and nKis the unit exterior normal to the subdomain boundary oK, and WK is a space of testfunctions with domain Dn = K x (tn, tn+1).

Now suppose that a finite element (or finite difference) approximation uA to (2) (or(1) has been obtained by some numerical scheme. Then it is clear that the approximationerror eA = u - uA satisfies the equality

r A A l'n+1lJI (e, + a· Ve )v df1.dt + Ja· nl(u+ - u_)v dsdtD" I" 8K-

+ 1(u(-,t~) - u(·.tn))v(·,t~)df1. = r rAvdf1.dtK JD"

where rA = f - u~ - a· VuA is the local element residual. The test functions v in (4) maypossess jumps at oK.

The plan is to now construct an approximate solution EA of (4) over each space-timeelement. Thus, if EA = EA(x, y, t) is an approximation of the error on D and E7< is itsrestriction over K, we write

with l{Ji, t/Ji polynomial shape functions in t and (x, y), we introduce this function into (4)in place of eA, and replace v by VA = l{Ji( t)t/Ji( X, y) to obtain a system of local equations forEb. With the approximation to the error thus determined, the local error can be computedin any norm.

3 THE STEADY-STATE CASEIn the steady-state case, (4) reduces to the local discontinuous finite element approximationof Lesaint and Raviart [7J. The discrete approximation of the local equation for errorreduces to

r a.VE'vAdf1.+ r la'nKI(~-E~)vAds=<rn,vA>JK 18K-where now

(5)

EA = EEJ'Mx,y) (7)j

and Wk is a space of piecewise polynomials defined on K which are generally discontinuousacross oK.

The solution of (5) element-by-element is straightforward. We assign an ordering ofthe elements in nA, following [7J, so as to trace the propagation of error along stream linesthroughout the mesh. We compute the approximate inflow condition uA 180- = gA and thecorresponding inflow error eA 180- = g - gA. This defines E~ on all elements intersectingthe inflow boundary &f1.-. Element residuals in the inflow boundary elements are thencomputed and (5) is solved for each of these elements. With E~ then determined for the

149

next set of adjoining elements, the solution can be advanced to these elements. Repeatingthis process advances the calculation of E" throughout the mesh.

4 SAMPLE RESULTSThe approximate error E" computed using the above procedure is generally computedusing polynomial shape functions vll = ,pj of higher order than those used to calculate u\but this may be unnecessary.It is interesting to compare error estimates computed in this way with those obtained

using interpolation estimates. The so-called interpolation estimates are based on finiteelement inteIl>olation theory; since

II ell IIm.q,K$1I u - iill IIm,q.K + II ull - iill IIm.q,Kwhere iih is the local interpolant of the solution u and II . IIm,q.K denotes the Wm·q(K)-Sobolev norm, and since (see Ciarlet [4Jor Oden and Carey [11]) for quasi uniform refine-ments,

C = constant independent of u or the mesh size hK

/'!p+l,r,K = WJ>+l,r(K) seminorm

1$ q,r $ 00

then, if II uh - uh IIm,q.K is of higher order in hK than the interpolation bound, the localerror in the wm.9(K) norm is bounded by the local error indicator,

~-~+P+l-m "OK = ChK lu lp+l.r.Kprovided lulllp+l.r,K = lulp+l.r.K+O(hi<), CT > 1. For example, for m = 0, q = r = 2, P = 1,this reduces to the familiar estimate [8, 9, 15J,

II ell IIV(K}$ OK = Chklulll~.~,KThe constant C can be estimated in a straightforward calculation of the "master" elementK used to generate the mesh.

Let us now compare the quality of this crude error estimator with that derived in thepreceding sections.

We let a = (2,2), n = [O,64J2 and by imposing different types of inflow boundaryconditions 9 or right-hand side forcing functions f, we obtain steady-state solutions withvarying degree of smoothness.

Example 1. Smooth Sinusoidal SolutionWe first consider the case in which the steady-1ltate solution is a smooth sinusoidal

wave, namely:

u(x, y) = sin (;¥)with il = y-x

We let the inflow conditions g coincide with the exact solution and let f = O. An approxi-mate steady-state solution was obtained on a uniform 32 X 32 grid by integrating a set ofinitial data using a one--step finite--element Lax-Wendroff algorithm. The steady-state er-ror was estimated using interpolation-type estimates, using both bilinear and biquadraticlocal approximations of the equation of the error respectively. We use the notations:

150

DG(I) bilinear error estimator

DG(2) = biquadratic error estimatorInorder to compare the various estimates we constructed cumulative curves of the form

M

e(M) = L'7~i' M = 1,2···. NELEM;.1

where '7~. stands for the square of the L2-norm of the error or its estimator computed,over the nith element. Here {nil denotes the special ordering which is used to solve theerror equation with the discontinuous Galerkin. Figure 1 gives the cumulative curves forthe square of the L2-norm of the error and its various estimators for tbe problem withsmooth sinUllOidalsolution. We note that while the interpolation-type estimate appears toprovide a sharp bound for the exact error the DG(2)-approximation of the error practicallycoincides 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 steady-state solution:

u(x,y) = sin (;r) tanhG(Jl- 32)}

{

y- xJl-

lI-z+64

, ifll-x~O

, ifll-Z<O

Figure 2 gives the cumulative error curves for the square of the L2-norm of the error andthe various estimators. We note that the DG-approximations are very close to the exacterror (the DG(2)-approximation coincides with the exact error) but the interpolation-1'!rrorestimate differs from the exact error by several orders of magnitude. Figure 3 presents thefinite-eIement grid, the contours of the error and the contours of the DG(I)-approximationof the error. The contours of the DG(2)-approximation of th error are indistinguishablefrom those of the exact error.

Example 3. "Discontinuous" SolutionIn order to show the collapse of the interpolation-type estimates for non-smooth solu-

tion, we consider the following solution:

1 , Jl ~ 8

1-~ 8<-< 12, -y-u(x, II) =

0 Jl ~ 12

Jl=Y-x

Figure 4 contains the adaptive finite-element grid and the contours of the steady-state numerical solution. Figure 5 shows the contours of the exact error and the DG(2)-approximation. Figure 6 gives the contours of the DG(l)-approximation of the error andthe interpolation error conturs. Note that the contours of the interpolation-bound areseveral orders of magnitude less than those of the exact error or the DG-approximations.

151

Figure 1.

Figure 2.

EXACT 0INTERPOLATION 0·DG(I) METHOD b.DG(2) METHOD ¢

ELEMENT COORDINATE NO. xl0-3 mID

Smooth Sinusoidal Solution. Cumulative curves for the square of the £2-norm of the exact error and its estimates.

EXAcr & 00(1)lNI'ERPOLAnON00(1) METHOD

ELEMENT COORDINATE NO. x10-'> mID

Almost Discontinuous Solution. Cumulative curves for the square of the£2-norm of the exact error and its estimates.

152

(b)

(a)

(c)

Figure 3. Almost Discontinuous Solution: (a) Finite-element grid; (b) contours of theexact error and (c) contours of the DO (I)-approximation of the error.

153

Figure 4.

(a)

"In ~r-. l1li ••• I.""" _. '.1_"" 18'TIII'f&-'.1."

(b)

Discontinuous Solution: (a) Adaptive finite-element grid and (b) contours of thesteady-state numerical solution.

154

Figure 5.

~------~~-_._-!

II"'; c:.. ,_. 1111...... -12 MJ ••• ,.... IIftP ... • •• .- ....

(a)

(b)

Discontinuous Solution. (a) Contours of the exact error and (b) its DG(2)-approximation.

165

Figure 6.

l

(a)

II'" ~ 1.1••. tI ..... -.J. I. "II"" UT'D'IIL1o .........

(b)

Discontinuous Solution. (a) Contours of the DG(l)-approximation of theerror and (b) contours of the interpolation-type error estimate.

156

Moreover, the interpolation~rror bound decreases in the streamline-direction, which isopposite to the behavior of the exact error. Figure 7 presents the cumulative error curvesforthe steady-state solutions; again the DG-approximations are very closewith the exact errorwhile the values of the interpolation-bound are too small (in the figure the interpolation-bound curve coincides with the x-axis).

The magnitude of the squares of the global L'-error norms and the corresponding esti-mates are given in the table below.

Table 1. Error Estimate Comparisons

Exact Error Interpolation- DG(l) error DG(2) errorSolution (global L'-error2) type estimate estimate estimateSmoothSinusoidal 0.119612 0.120957 0.112623 0.119596SolutionAlmost-Discontinuous 0.315754 0.011202 0.302882 0.315670Solution"Discontinuous"Solution 4.446456 0.007325 3.861574 4.404879

Example 4. Steady-Convection of a Cosine-HillWe also consider the non-homogeneous linear hyperbolic equation:

Ul + a .Vu = f in nu = 0 on an-u(·,O) = U(·)

Again we let n = [0,64]', a = (2,2) and we chose f = a . VU where U is a "cosine-hill"given by:

{

A sin (1rr)U(r, y) ~ 0 '[II ,'" R

, r> R

Here r is the polar radius from the center of the cosine-hill,

r = v(x - x.)' + (11 - 11.)'

and we let Xc = Yc = 32,R = 16 and A = 100. (The initial grid was taken to be a uniform8 x 8 grid while the interpolant of the exact solution on the initial grid was used as theinitial condition for the calculation.) Figure 8 shows the time-evolution of the exact error,its DG(2)-approximation and the interpolation-type estimate as the solution is integratedforward in time. Figure 9 gives 2D and 3D views of the final solution and grid. Notethat the grid was adapted using the error indicators based on the DG(2)-approximationof the error. Finally, Figure 10 presents a comparison of the contours of the exact andapproximate error function at the final time.

157

EXACfDOl APPROXIMATION

_ 002 APPROXIMATIONINTERPOLA nON ESTIMAlE

ELEMENT COORDINA lE

Figure 1. Discontinuous Solution. Cumulative curves for the square of the L1-normof the exact error and its estimates.

EXACfDG2 APPROXIMATIONINTERPOLATION ESTIMAlE

TIME (sec x10-1)

Figure 8. Steady-Convection of a Cosine-Hill. Time-evolution of the square of theL2-norm of the error, its DG(2)-approximation and of the illterpolation-type error bound.

158

Figure 9.

(a)

(b)

Steady-Convection of a Cosine-Hill. (a) Finite--eIement grid, contours and(b) 3D view of the approximate solution at the final time.

159

(a)

(b)

Figure 10. Steady-Convection of a Cosine-Hill. (a) Contours of the exact and (b) theDG(2)-approximation of the error at the final time step.

160

5 CONCLUSIONSIn this note, a simple procedure for a-posteriori error analysis is given which can beused effectively for estimating local errors in finite element approximations of convection-dominated flows. The method is based on a discontinuous Galerkin approximation of theevolution of error over individual elements. .

The results of this study also show that popular interpolatIon methods can produceestimates of error in adaptive finite element computations that are grossly in error. How-ever, the accumulated error over a mesh obtained using the crude interpolation estimatescan be estimated quite accurately with the discontinuous Galerkin technique developedhere. This suggests that these methods may be useful as post-processing techniques thatcan enhance adaptive results generated using a fast but low accurate error estimator. Re-sults on nonlinear problems and Navier-Stokes equations are to be reported in a lengthiercompanion paper.

Acknowledgement: The support of this work by ONR under Contract N00014-84-K-04009, and by NASA Langley Research Center under Contract NAS 1-18570 is gratefullyacknowledged.

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