•

JOURNAL OF COMPUTATIONAL PHYSICS 1../6. 491-519 (1998)

ARTICLE NO. CP986032

A Discontinuous hp Finite ElementMethod for Diffusion Problems

J. Tinsley Oden.* Ivo Rabuska,t and Carlos Erik Raumannt

Texas IlIsritule/or Computational and Applied Mathematin. The Un;renityof Texils at Aus/in. Allstin. Texas 78712

F.-m,d I: "oclcn(cft icalll.lllcxas.cclll. thahllska@licam.lIlcxus.cdll,aud tcarIosb(mticum.lItcxas.cclll

Received Nuvember 13.1997: rcviscd April 23. 1498

We present an extension of the discontinuous Galerkin method which is applica-blc to the numerical solution of diffusion problems. The mcthod involves a weakimposition of coni inuity condit ions on the solution values and onlluxes across inter-element boundaries. \Vithin each element, arbitrary spectral approximations can beconstructed with different orders p in each element. We demonstrate that the methodis elemenlwise conservative. a property uncharacteristic of high-order finite clements.

for clarity. we focus on a model class of linear second-order boundary valueproblems. and we develop a priori error estimates. convergence proofs. and stabilityestimates. The results of numerical experiments on It- and I'-convergence rates forrepresentative two-dimensional problems suggest that the method is robust and ea-pable of delivering exponential rates of convergence. :c I'NX ,v.•d:·no;" p",,,

Key llords: discontinuous galcrkin: tinitc elcments.

I. INTRODLCTIO:\

This paper presents a new type of discontinuous Galerkin method (DGM) that is appli-cable to a broad class of partial di frerential equations. In particular. this paper addressesthe treatment of diffusion operators by tinite element techniqucs in which both the approxi-mate solution and the approximate fluxes can experience discontinuitics across inrerelcmcntboundaries. Among features of the method and aspects of the study presented here are thefollowing:

• 1I priori error estimates are derived so that the parameters affecting the rate of conver-gence and limitations of the method are established;

• the method is suited for adaptive control of error ami can deliver high-order accuracywhere the solution is smooth:

• the method is robust and exhibits clementwise conservative approximations:

491

11021-999 tf')X $2~.()0Copyright I:.£) lINg by Academii.' Prl'ss

All rights of rc.prodUl.:tioll in any foml n:sl:rvl.."d.

492 ODEN. BABUSKA. AND BAUMANN

• adaptive versions of the method allow for near optimal meshes to be generated;• the cost of solution and implementation is acceptable.

The use of dis<:ontinuous tinite element methods for se<:ond- and fourth-order elliptic prob-

lems dates back to the early 1960s. when hybrid methods were developed by Pian an hiscollaborators. The mathematical analysis of hybrid methods was done by Babuska [7].Bahuska et al. [t). 101. and others. In 1971, Nitsche [38] introduced the concept of replacingthe houndary multipliers with the normal fluxes and added stabilization lerms to produceoptimal convergenL:e rates. Similar approaches can be traced back to Ihe work of Percell andWheeler 140[, Wheeler [421, and Arnold [3]. A different approach was Ihe /I-formulationof Delves and Hall [251. who developed the so-called global elemenl method (GEM);applications of the laller were presented by Hendry and Delves in 1261. The GEM consists

essentially in the classical hybrid formulation for a Poisson problem with the Lagrangemultiplier eliminated in terms uf the dependent variables; namely. the Lagrange multiplieris replaced hy the average Ilux across interelement boundaries. A major disadvantage ofthe GEM is that the matrix associated with space discretizalions of diffusion operatorsis indefinite. and thus the method is unable to solve time-dependent diffusion problems:

heing indetinite. the linear systems associated with steady stale diffusion problems need

special iterative schemes. Moreover. the conditions under which the method is stable andconvergent are not known. The interior penalty formulations of Wheeler [421 and Arnold[3] utilize the hilinear 1'01111 of the GEM augmented with a penalty term which includesthe jumps of the solution across clements. The disadvantages of the last approach indudethe dependence of stabilily and convergence rates on the penalty parameter. the loss of theconservation property at clement level. and a bad conditioning of the matrices. The DGMfor diffusion operators developed in Ihis study is a modification of the GEM. which is freeof the above deficiencies. More details on these formulations and Ihe relative merits of each

one will he presented in Section 3.1.The lirst study of discontinuous finite element methods for linear hyperbolic problems in

two dimensions was presented by Lesaint and Ravial1 in 1974 [311 (see also [30.32]). whoderived a priori error estimates for special cases. Johnson and PitkUranta 1281 and Johnson[27] presented optimal error estimates using mesh-dependent n0I111S. Among the applica-tions of the discontinuous Galerkin method to nonlinear first-order systems of equations,Cockbul11 and Shu [ 19-22] developed the TYB Runge-Kurta projection applied to generalconservation laws. Allmaras [11 solved the Euler equations using piecewise constant andpiecewise linear representations of the field variables. Lowrie [371 developed space-timediscontinuous Galerk in methods for nonlinear acoustic waves. Bey and aden 1151 presentedsolutions to the Euler equations. and Atkins and Shu [41 presented a quadrature-free imple-

mentation for the Euler equations. Other applications of discontinuous Galerkin methodsto first-order systems can be found in [I n. 291.

The underlying reason for developing a method based on discontinuous approximationsfor diffusion operalors is to solve convection-diffusion problems. Solutions to convection-diffusion systems of equations using discontinuous Galerkin approximalions have beenobtained with mixed formulations. introducing auxiliary variables to cast the governingequations as a first-order system of equations. A disadvantage of Ihis approach is thatfor a problem in IRe}. for each variable subject to a second-order di ITercntial operator. dmore variables and equat ions are introduced. This methodology was used by Dawson 124J.Arbogast ami Wheeler 12]. and also by Bassi and Rebay [12, 13J for the solution of the

DISCONTINUOUS GALEI{KIN ,"lETHOD 493

Navier-Stokes equalions, Lomtev et a/.I.33-36] and Warburton et at. [41] solved the Navier-Stokes equations discretizing the Euler Iluxes with the DGM and using a mixed fonllulationfor the viscous fluxes. A similar approach was followed by Cockburn and Shu in thedevelopmenl of the local discontinuous Galerkin method [231; see also short-course notesby Cockbull1 r 18].

In the present investigation. a discontinuous Galerkin method for second-order systemsof partial differential equations is presented in which the solution and its derivatives are dis-continuous across element bnundaries. The resulting scheme is elementwise conservative,a property not common to tinite element methods. pm1icularly high-order methods. Theformulation supports 11-. p-, olld lip-version approximations and can produce sequencesof approximate solutions that are exponentially l:onvergent in standard nonns. We explorethe stabi Iity of thc method for one- and two-dimensional model problems and we presenta priori error estimates. Optimal order 11- and fJ-convergenn~ in the lit norm is observedin one- and two-dimensional applications.

Following this Introduction, various mathematical preliminmies and notations are pre-sented in Section 2. Section 3 presents a variational formulation of a general linear di ITusionproblem in a functional selling that admits disl:ontinuities in ftuxes and values of the solu-tion across subdomains. Here properties of this weak fonllulation are presented, laying thegroundwork for discontinuous Galerkin approximations that are taken up in Section 4. InSection 4, the discolltinuous Galerkin method for diffusion problems is presented. A studyof the stability of the method is presented and a priori error estimates are derived. Thesetheoretical results are then conlirmed with llullIerical experiments. The results of severalapplications of the method to two-dilllcnsionallllodel problems are recorded and discussed.It is shown. among other features. that exponential rates of cnnvergence can be attained.

2. PRELlMII'\ARIES A~n :\'OTATIONS

2.1. Model Problems

Our goal in this investigation is to develop and analyze one of the main components ofa new family of computational methods for a hroad class of now simulations. In this paperwe analyze the diffusion operator. Model problems in this class are those characterizingdiffusion phenomena of a scalar-valued lield /I: the classical equation governing such steadystate phenomena in a bounded Lipscllitl. domain Q c R'. d = I, 2, or 1, is

-V'·(AV'II)=S inQ,

where S E L2(Q). and A E (L OC(Q))"'" is a dilTusivity matrix characterized as

A(x) = AI (xl.

O'taTa:::: afA(x)a:::: O'oaT a. at:::: 0'0> O. Va E ffi.".

(I)

a.e. x in Q.The boundary aQ consists of two disjoint paris. rn on which Dirichlet conditions are

imposed. and rt\ on which Neumann conditions arc imposed: I'D n r.'1= Vl.fD U rN = aQ,

494 ODEN. IlAIlLJSKA. AND BAUMANN

and meas fD > O. whereas boundary conditions are

It = f on 1'0(AVu),n=g onl'N.

(3)

Unfortunately. the classical statement ( 1 )-(2) of these model problems rarely makes sensefrom a mathematical point ofviev. ...In realistic domains Q and for general data (A. S. f. g).the regularity of the solution It may be too low to allow a pointwise interpretation of thesolution of these equations. For this reason. weak forms of the model problem must beconsidered in an appropriate functional setting.

2.2. Functional Settings

As noted previously, Q C IRd• d = I, 2. or 3,denotes a bounded Lipschitz domain. For

classes of functions defined on Q, we shall employ standard Sobolev space notations; thus,HII/ (Q) is the Hilhert space of functions defined on Q with generalized derivatives of order~ 111 in L2(Q). The standard norm on HII/(Q) is denoted lilt II H"'lQ I or simply 111111/11' and thesemin0l111on H'" (Q) is denoted III 111/' Spaces H S (Q), for .I' > a not an integer, are definedby interpolation. The closure of Cii"(Q) in H/II(Q) is Nfl'(Q). and H-II/(Q) denotes thedual of HO'(Q).

2,3. Standard Weak Formulation and (;alerkin Approximation

A classical weak formulation of problem (I )-(3) is stated as follows:Find It E V (Q) such that

B(It. 0) = L(v) VI' E V (Q);

here

\/(Q) = {v E H1(Q) : YllvlrD = O} .

B(u.v)= lVv.AVudx. L(v)= /vSdx+ ': vgds-B(ii.v)../0. ./Q./t:<

(4)

and ii E HI (Q) is such lhat yoii II'll = .t. Yobeing the trace operator. A weak solution ofproblem (I )-(3) is It + ii.

The existence of so]utions to (4) can be established using the classical generalized Lax-Milgram theorem IS. 391.

A Galerkin approximation of (4) consists of constructing families of closed (gener-ally finite-dimensional) subspaces. \I" C V, and seeking solutions It" E V" to the followingproblems:Find u" E VII such that

(S)

Let us assume that the conditions of the generalized Lax-Milgram theorem hold. It followsthat (5) is solvable if there exist y" > 0 such that

inf sup IB(u. tJll ~ YI,.lIEl'1. l'eh.

'11/1"',,=1 1I1'1I1',,~1

(6)

and

DISCONTINUOUS GALER KIN METHOD

supIB(I/. vll > o. v =1= O. v E V./lEV"

495

(7)

A straightforward t:alculation reveals that the enor U - Uh in the approximation (5) of (4)satisfies the estimate

(8)

where M is the continuity constant defined a~

B(I/, v) ~ Mllullvlll'lk. 'Vu. v E V.

Proofs of the generalized Lax-Milgram Iheorem and of the estimate (X) can be found in

[5,39].

2.4. Families of Regular Partitiuns

Since our discontinuous approximations are to be ultimately defined on partitions of thedomain a, we now establish notations and conventions for families of regular partitions

[17]. Let P = (Ph (Q) lh>o be a family of regular partitions of a c IRd into N = N (Ph)subdomains ar (see Fig. 1). such that for Ph E P.

NIP;.)

Q = Un". and a" n aj = VIfor e =1= f·c=t

Let us detine the interelemelll houndary by

fim = U umf n (jar)'Q,.Q,EP,

On fim. we define Il= "l' on (iJae n (jaIl C fin! for indices e. f such that e> f·

(9)

(10)

2-D Discontinuous Approximation

FIG. 1. NOI,lIiou: slIbdolllains and houndaries after dis~retizatiou.

496 ODEN. BABUSKA. A~D BAUMANN

2,5. Broken Spaces

We define the slH.:alled hroken spaces on the partition P,,(Q).

(ll)

if v E Hili (0.,,), the extension of 11 to the boundary ilQ,,, indicated by the trace opera-tion Yol}. is such that YOF E 1I111

-lj1(aQ,), 11/ > 1/2. The trace of the normal derivative

Yt v E H'"-1/2(iJQ,.). III > 3/2. which will be written as VI." nliJn,. is interpreted as a gen-eralized flux at the clement boundary aQ,.

With this notation. for v In,.E 1I3i2+E (0.,) and v In! E H 3/2-, (0. f). we introduce thejumpoperator [·1 defined on r,'f = QI' n QI '# til as

( 12)

and the average operator (-) for the nonllal flux is defined for (A V v) . n E L2( r,r) as

(AVI') ·n} = ~ (((AVu). n)IiJ!'2,~I"'r+ ((AVv) .n)hnfni·,.:). (' > f. (13)

where A is the dilTusivity. Note that n represents the outward normal from the elcmcnt withhigher index.

2.6. Polynomial Approximations on Partitions

For future reference. we record a local approximal ion property of polynomial finiteelemeIll approximations. Let Q be a regular master element in IR". and let IFn ..J be a familyof invertiblc maps from Q onto 0.,. (see Fig. 2).

For every clement 0.,. E Ph. the tinite-dimensional space of real-valued shape functionsPc Hili (0.) is taken 10 be the space PPc (0.) of polynomials of degree ~ p" defined on Q.

CxFI(;.2. Mappings Q -> 11, and discontinuous approximation.

DISCONTINL:OliS GALERKIN t"fETHOD

Then we dellne

Using the spaces PI',(0.e). we can define

,v,P.,)

\ip(P,,) = IIp/>.(0.,.).1'=1

497

(14)

(15)

N (Pit) being the number of elements in partit ion Pt,.The approximation properties of "1'(Ph) will be estimated using standard local approxi-

mation cstimatcs (see 16]). Let /I E H·'(0.,.): therc exist a constant C depending on s and on

the angle condition of 0.", but independent of II. he = diam(0.,.). and P", and a polynomial

"I' of degree P", such that for any 0::: /' ::: s the following estimate holds:

(16)

here II· lid'!. denotes the usual Sobo\ev nOllll. and IL = min(p" + I . .1').

The following local inverse ine4ualities for a generic element 0.e arc valid for U E P P., (0.e)

and Pe > 0 (sec 18. 14]):

") -I"''''lulii:m, ::: Ch,. p~lIlIlliiiiQ..'1 '.~

I~u 1- ::: CI1,-:1 p;lull.n,·<Ill lI.ap.,

(17)

J. A WEAK FORMULATION OF DIFFUSIOJI\ PROBLEMSIN BROKEN SOBOLEV SPACES

We focus on a model linear diffusion problem in a bounded domain; given data (0., a0..51. f.g). we wish to lind a funct ion II such that

-V· (AVII)=51lI=f

(A VII)·n = g

in0.on ff)on fill,

( 18)

where A E (L oc(0.»dxd is a diffusivity matrix satisfying the conditions stated in (2).

The weak formulation of (\ X) that forms the basis of our discontinuous Galerkin method

is defined on a broken space II (Pit). Pit being a member of a family of regular partitions of0.. In particular. V(Ph) is a Hilbert space on the partition Pit. which is the completion of/I.I/2+f('Ph). f' > 0, with respect to the mesh-dependentnOJ'm

where

, , ,IIvlli = lull!'. + Ivliil';.,.. ( 19)

(20)

49H

and

ODEN, BABUSKA AND BAUMANN

(21)

In (21). the value of II is h,./(2a[) on fj), and the average (he+hfl/(2al) on that partof rint shared by two generic elements st, and st f' the constant a J being defined in (2).A complete characterization of the space Ii(Ph) for the one-dimensional case is describedin 1]41.

We should note that the terms h±a. with a = ]/2. are introduced to minimize the mesh-dependence of an otherwise strongly mesh-dependent nonn. The inner product associatedwith 11·11 v is the symmetric bilinear form

(It. v)v := B/r(It, v) + l (h-211yov YOII + h211(AVv)· n (AVI/)' n) cis.IrD

+ I (h2a«AVv).n)«AVII).n)+h-2a[yov][you])ds. (22)rlill

Next, we introduce the bilinear form B±: V (P/r) x V(Ph) -+ lit detined by

B±(u, v) = B,,(u. v) + l (±(A V'v)· n u - v(AVI/)' n) ds.ft.1>

+ l (±«AVv),ul[u]-«AVu)·n)lvl)ds, (23)• J'lfLI

and the linear ronn L±: Ii(Ph) -+ lit defined by

L±(v) = L {l l'S dX} ± r (AV~')· nf ds + ~ vq ds. (24)rl,.EP;, Jrl,. Jrll ./rN

In (23). we denote by (A Vv) . n} the limits of the sequcnces of averaged l1uxes (A Vvd . n}on rim. With these conventions and notations in place. we consider the following weak orvariational boundary-value problem:Find U E V (Ph) such that

(25)

That (25) indeed corresponds to our model diffusion problem is veri tied in the followingtheorem:

THEORE~13.1. Suppose S, f. and g art! smooth (co/ltinl/ous) a/ld that the .I'o/Illion u to( I H) exists a/ld (A Vu) E H J ('Ph). Then II is a SO/III ion of (25). COIlI'ersely, any slIfficientlysmooth soilltio/l of (25) is also a so11lliO/l iif (i8).

Proof This follo ....o'sfrom standard arguments and use of Green's romlUla. For details,see 114J .•

DISCONTINUOCS GALERKIN METIIOD

Remark 3.1. Let us note that

499

(26)

the above inequality only indicates that the bilinear form is positive semiclellnite. As shownlater. the variational formulation is stable. i.e .. it has no zero eigenvalues: therefore B+(· .. )is positive definite. It is the skew-symmetric part of B+(· .. ) that renders it positive detlnite.

3.1. The Global Element Method and Interior Penalty Formulations

The global elcment method [25. 261 consists in the classical hybrid fom1Ulation for a

Poissoll problem with the Lagrange multiplier eliminated in tet1l1S of the dependent vari-ables: namely. the Lagrange multiplier is replaced by the average flux across interelementboundaries. The GEM can be written as follows:Find It E V (Ph) such that

/L(II. v) = L_(t:) "VI' E V(Ph). (27)

A signilicant disadvantage of this method is that the matrix associated with the above bilin-

ear form is indefinite (the real parts of the eigenvalues are not all positive), which preventsthe solution of time-dependent di I'l'usion problems and also the utilization of many iterativeschemes for thc solution of steady problems. The reason is lhat for advancing in timc thcsolution to timc-dependent diffusion problems. the real part of the eigcnvalues of the space

discretization needs to have the appropriate sign; otherwise the problem is unconditionally

unstable.Given that the goal of this investigation is to obtain a solver for convection-diffusion

equations within the usual CFD settings. it is of paramount importance to generate a nu-merical technique whieh can handle these equations using pseudo-time-marching schemes.

The main disadvantage of the GEM is its inability to generate systems of equations whichare amenable to the aforementioned solution techniques. Other than this disadvantage theGEM performs better (han the technique that \ve advocate when the error is measurcd in theL 2 norm (optimal for any p): this is not the case with the technique that wc advocatc. whichloses one order for even powers p when the error is measured in the L 2 nonn (optimal for

any p in the H t norm). The GEM has the advantage of producing symmetric systems of

equations. but this advantagc is not of intcrest for convection-diffusion problems. whichare intrinsically asymmetric (non-self-adjoint) problems.

The method of Nitsche [3HI and the interior pcnalty formulations of Wheeler [421 andArnold [31 are very similar; Ihey arc b:lsed onthc solution of the following problem:

Find It E Ii (Ph) such that

here (J = K 11-1 is the penalty function. TIle valuc of K is critical becausc if it is too smallthis technique is the same as the GEM. which has the problem of indefinite systems. Inour experience. the parameter K is problcm dependem and has to be chosen very carefully;otherwise the rate of convcrgence is not optimal. Other disadvantages include the loss of theconservation properly. and a bad conditioning oflhe matrices. This technique also produces

500 ODEI\. BABUSKA. AND BAU"vIANN

symmetric systems of equations, but as explained above. this in not an advamagc for thetype of problems that we al'c planning to solve.

The DGM developed in this snl(iy does not present the deficiencies pointed out before.and we believe it is much belter suited to the solution of CFD problems.

4. THE I>ISCONTI'\'l!OUS GALERKIN METIIOI> FOR U1FFUSION PROBLEMS

The variational formulation (25) will now be uscd as a basis for the construction of dis-continuous Galcrkin approximations of the model diffusion problem. In the discontinuousGalerkinmcthod, the partitions of the solution domain. on which problem (18) is posed. arcnow finite clements. across common boundaries of which the test functions can cxperienccjumps. The Galerkin approximation is thus defined on a subspace VI'(P,,) of thc Hilbertspace II (P,,) introduced in Section 2. Thus. the general setting of the approximation result(X) is applicable and can be used to derive £I priori error estimates for the method.

Let us now consider the finite-dimensional subspace Vp(P,,) C V (P,,) defined in Sec-tion 2.6 as

N(P.,)

Vp(P,,) = IIPp,(O.,,).,,=1

Our discontinuous Galerkin approximation (25) in VI'(P,,) isFind Itll E VI'(P,,) such that

(29)

(30)

where B+(·.·) and L+(·) are defined in (23) and (24). rcspectively.An immediate observation is thar the discrete scheme dellncd by (30) is conservative.

This is the subjcct of the following section.

4. L Strong Conservation at Element Level

A solution is said to be globally cOl/servative if the balance of the species (the solution II)

is satisfied on the solution domain as a whole, and locally COI/.I'l'l'l'ative if a partition ofthc solution domain exists such that thc balancc is salisllcd within each subdomain of thispartition.

Considering a generic clement r2" E P", whcn the test function v" is a pieccwise con-stant (unit) function with local SllppOrl on clemem r2". from (30)-(23)-(24) we obtain theapproximation

- l (AVII,,).nds-l. ,«(AVII1,)·n,,)ds= / Sdx+ r gds, (31)Im ..nrll ,m.,nl,,,, .In,. .Ian.nrN

which means Ihat conservation at element level is ensured whcn the flux across iJr2" isdefined as the average nux «(A VL'h)' n,.).

4.2. Numerical Evaluation of the Inf-Sup Condition

In order to apply the generalized Lax-Milgram theorem 15. 391 and the estimate (8).we must know the conrinuity and inf-sup parameters. The continuity condition holds with

DISCO"T1NUOUS GALERKIN METHOD 501

I~ t

M=l.i.e ..

1<'1(;,3. Sllhdolllain for slIccessive glohal rctinclllcnts.

(32)

where 11·11 v is the associated norm deli ned in (19). To verify this. we multiply the integrandsappearing on fim and fn in the definition of B+(' •. ) hy Ita h-f1

, and use standard inequalities114] to show that

Thus. lIsing (19), we obtain

(33)

The behavior of the discrete inf-sup parameter y" appearing in (8) as a function of meshparameters hand p is of paramount importllnce in determining the stability of the DGM. Astraightforward caleula! ion of y" for representative cases can be done using the eigenvalueproblem described below.

THEORE1\'! 4.1. Let 1ih be an n-c/imemional Hilhert space, and CN" E 1R" X}t" thesymmetric positive definite matrix associated 11'itltthe inner product in 1i". Let B: 1i" x1i" --+ IR bc a gcncric bilinearjorm. amI BE jR" X IR" the associlltcd matrix: t!ten thc inlsllJlcondition associated with B(-, .) cml hc cl'alullted usillg the cigcllrallll' prohlc/11

(34)

/i'Olll which we obtaill the illl-wp constallt y" =J;'.mill'

Proof Given that CN. is symmetric and positive definite. it can be factored into lower!upper triangular f0I111 eN" = u~"UN,.' Then. thc norm in 1i" is related to the Euclidennnorm 11·11 in R" as

502 ODEN. BABUSKA. AND BAUtvlANN

FIG.4. I.-shaped domain-nomenclatllre.

Angle 90, aspect ratio 1:1

Angle 90, aspect ratio 1:2

Angle 90, aspect ratio 1:4

Angle 90, aspect ratio 1 :8

Angle 30, aspect ratio 1:1

Angle 30, aspect ratio 1:2

Angle 30, aspect ratio 1:4

Angle 30, aspect ratio 1:8

FUi. S. Two-dimensional stability analysis: duma ins.

0.::l'i'

.50.1

DISCONTINLJOLJS GALERKIN \.fETIIOD

.;-.:,---:. .....

::~::-:-:---..... ". -". . . '. :~~~:~:~.~~~-

. '.--''-"::'~:""'"

"':

90 - 1190 - 1:290 - 1:490 - 1:830 -1:130 - 1230 - 1:430 - 1:8

.......

503

0.012 3 4

p5 6 7 8 9 10

0.1

90·1190 - 1:290 - 1:490 - 1:830 - 1:130 - 1:230 - 1:4 ..30·1:8 ..

0.012 3 4

p5 6 7 8 9 10

FUi.6. Inf-sllp vatucs for a mcsh with 64 c1emcnts. 'HlP: llniform p. Boltonl: random dislrihutinl1 of p.

504 ODE1\. BARtJSKA. AND BAUMANN

1.·10

o;;a:

~a~(;() -------

16"11

32 64o

8 16lill

32 64

FIG. 7. \i-norm of the en-or and convcrgcnce fatc with unifomlmcshes: -i/2u fil.\" = S. S = (4rr)2 sin(4rrx) .

.J 11>05

~

1.. 10 ~l______16 32 64 32 64

FIG. II. V -norm of the error ancl convergencc ratc wilh nonunifuIlll mcshcs: -iI2I/fi/x' =S. S =(4;r)2sin(4rrx). ~h = ±20%.

.............

11>05

g

1.. '0 ~~=~-H ----------

81!/1

16 32 161111

32

"'(;.9. L '-nonn of the error and convergeuce rate with L1niformlllcshes: -iJ'lIfil.r' = S. S = (2rr)' 5in(2;7.\').

, .. 05

[)[SCONTINUOCS GALER KIN METHOD

_ ~: .

505

-~---1.·10

101/h 10

lih70

FIG. 10. U-norlll of thc crror and convergence rate with nonuniform Im:shcs: -iJ'III"x' = S, S =(6;r):sin(6;rx I, 8h = ±20'i;,.

25

~~=-4~!•..I 20~=

~ 6 ;S! · 15. •a: 5 a:

i! ·".4

0;

" "~ ~ 100 3 8u

16 32o

1 4 5Degreep

FIG. II. Ir and p convergence rates in the HI seminorm: -iJ'lI/ax' = s. S = (371')' sin(3JTx).

10

1.. 05

1.. ,0

:::::----.- ~ _. ---------.........

19-13 ---.- ..---.16

Hih

1'1<;. 12. [)islo!1CUdomain: -/).1/1 = s. S = -L';(cxp(-(.r: + .1"»)).

16

506 ODEN. BABUSKA. AND BAUMANN

I

I

I- -- .-1

i; -F

IIl~. _

h-padapt

Degree

1312111096765432 .

iJh-p

adapt

Degree

13121110967654321o

FIG. D. L-shnped domain. Top: ~'ksh and polynomial hasis aftcr h-" ndaplation. Bottom: closc-up viewx 20 at thc corner.

DISCONT[l\UOUS GALERKIN :\1ETHOD

h-padapt

Error

O.827E-02

O.763E·02

O.700E-02

O.636E-02

O.573E-02

O.509E-02I

0.445E-02 iO.382E-02

iI:::::::::I O.191E-02

I0127E-02

o 636E-03 i'-- J O.767E-l~J

FIG. 14. Pointwise error: close-up view (x20 al the corner).

507

508 ODEN. BABUSKA. AND BAUMANN

Let us now consider the equalities

IB(u. vll I(v, Bull I(w, U;/Uu)1 -Tmax = max = max ., = IIU Bull:!'EN/, IluIIN;. \'EJR" IUH"vl WER" Ilwll HI,

using this result. we can write the discrete inf-sup constant y" as

(35)

and y" can be evaluated by solving the eigenvalue problem (34) .•

4.2.1. Olle-Dill/{'/ISiolwJ Study

We consider thc simple Dirichlet problem -II" = f on (0, I). with homogeneous bound-ary conditions. The following analytical result is proven in 1141:

THEOREM 4.2. JIIthe space \l1'(P,,), ifPe :::-:3. I ::::e ::::N (P,,), the hiJillearform B+ (II, v)defined ill (23) satisfies

. f IB+(II, v)1 (I - 4i1.o)til sup :::-: . . .

IIEI" l!EVp IIl1l1vllvllv (I +~)

where 1.23 < ),,,< 1.24 independelltly of the discretization parameters Pt' and h('.

Remark. Theorem 4.2 was proven for polynomial basis functions of degree :::3. Numer-ical experiments indicate. however, Ihat the method is stable and convergcnt for polynomialbasis functions of degree :::-:2.Note that the method is not stable for PI' :::: I.

Using (34), we evaluate y" for various meshes p" and uniform p. Pt' = p. For uniformmeshes. y" = I13 for p = 2. and it changes from 0.503] to 0.5025 when p changes from 3to 8. For nonuniform meshes, both for the case of an arbitrary distribution of nodes in whichthe worst aspect ratio between adjacent elements is 114 and for a geometric distribution ofmesh points such thaI hminl hmax = 10-7, the values of y" are very close to those obtainedwith uniform meshes. the difference being of the order of 1%.

In conclusion. the calculated y" is independent of the mesh size and is independelll of pfor P::: 3.

The asymptotic values obtained are higher than the analytic value because the laller isonly a lower bOllnd of the exact inf-sup constant.

The a priori error estimate for this case is given by the following theorem:

THEOREM 4.3. Let the soJutionll to (18) E H' (Ph (0.». with s > 3/2. !!the approxima-tion estimate (1 (1) hold for the spaces VI' (p,,), then the error t!! the approximate soll/tionl/ DC can he hounded as

(36)

Ivhere /1 r = min (p,. + I, S). E ~ 0+. and the constant C depends on s and on the anglecondition ol0., .. hilt it is independent of II. h", and Pc.

DISCONTINUOUS GALERKIN METHOD

Proof' See proof of Theorem 4.4 with K = O.•

509

4.2.2. Two-Dimensional Analysis

A two-dimensional study of the behavior of the inf-sup parameter y" for the discrete caseis carried out for meshes with different numbers N(P,,) of elements, different degrees of

skewness, aspect ratios, and for uniform and nonuniform /le. These experiments are donefor a model Dirichlet problems. for Laplace's e4uation on a parallelogram. Figure 5 showssome of the suhdomains on which the value of the inf-sup parameter y" is computed.

First the inf-sup paramcter y" is evaluated for meshes with uniform p, p,. = p. 1 :::::e :::::N (Ph). The numerical estimates indicate that the inf-sup parameter y" is asymptoticallyindependcnt of 11, but depcnds on p. Figure 6 (top) shows the inf-sup dependence on /l for

mcshes with M clements: uniform degree p: aspect ratios 1: 1, I :2, 1:4. 1:8: and skewnessof 9lY and 30". The asymptotic dependence on p (within the range of intercst) is '" p-t forall the meshes with skewness of 90". and'" p-l.5 lor those with ske\\lness 0130".

Next, the inf-sup parameter y" is evaluated for meshes with a random distribution of PC'The distribution starts wit h,)" = Pmax for element number 1, and for the remaining elementsthe order is chosen randomly between 2 and Pmax. The numerical estimates indicate thatthe inf-sup parameter y" is not very sensitive to abrupt changes in p. In fact, for onhogonalmeshes. the inf-sup parameter y" is approximately 5% larger for most of the meshes where

Mleast one element has Pr < Pmax. For skewed meshes in a few cases the inf-sup parametcrYh is less than lO/C;smaller than the corresponding value when Pc = Pmax for allthe elements.and in many cases it is larger. Figure 6 (bottom) shows the inf-sup dependence on p formeshes with 64 elements; random distribution of p(': aspect ralios I: 1. I :2. I :4, I :8; andskewness of 90° and 3lF. The asymptotic dependence on p (within the rangc of intercst) isagain'" p-t for all the meshes with skewness of 90°, and "-'p- t.5 for those with skewness

of 30e.

From the above studies it is clear that the inf-sup parameter Yt, is asymptotically indepen-dent of 11 but depends on p. The dependence on p is not signilicant for practical calculations,

since the loss of 0 (p) accuracy can be offset by the betler approximation achieved (0 (hP»

using high p for cases with high regularity. By assuming that y" = O(p;;;~x)' K 2: O. an es-timate of the global rates of convergence of the DGM can be easily estimated. We have:

THEOREM 4.4. Let the solutionu to (18) E H' (Ph (0.». with s > 3/2. and assume thatthe mlue of the inFstlp parameter is y" = C I'P;;;~x with K 2: O. If tile approximation estimate(16) hold for the spaces VI'(P,,). then the error ot'the approximate solurion IIDG can bebounded a,I'

(hll,.-t-E )2

Ilu - u DeII~, :::::c /I~~,X L:~-3!2-E 11/111,n..n..E'P" pc

(37)

where l.J.e = min(Pe + [. s). € ~ 0+. and the constant C depends on s and on the anglecondition (~l 0.('. but it is independen! (d·t/. II". and p,..

Proof Let us first bound III/II v as

lIull~ :::::C L (Iulrn, + h-t lIullf/2+d2, + hlll/lI~/1+d'!J·

Q,.EP,.

(38)

510 ODEN, BABU5KA. AND BAUMANN

h-padapt

Degree

13121110987654321o

h-padapt

Error.15231E-02

.14059E·02

.12888E·02

.11716E·02

.10545E-02

.93730E·03

.82014E·03

.70297E·03

.58581 E-03

.46865E-03

.35149E·03

.23432E-03

.11716E-03

.99156E·10

FIG. 15. Stability Icst: (x 20 at the corncr ) Top: p-enrichmenl al the singularity. [lotlom: Pointwise error andmcsh.

DISCONTINUOUS GALERKIN r\-IETHOD 511

FIG. 16. Top: Mesh and polynomial basis after hop adaptation. Bollom: x20 at the corner.

512 ODEN. BABUSKA. AND BAUMANN

and using the approximation estimate presented in (16). there exists a local polynomialapproximation /I p of /I E H S (Ph (S1») in the n0l11111·11 v such that

Finally. using the continuity and inf-sup parameters. and assuming that the exact solution/I E Il" (P,,), we arrive at the (/ priori error estimate

where s > 3/2. /l" = min(p" + I. s), and C depends on s but is independent of /I. he. andPc, •

Remark 4. I. The error estimate (37) is a bound for the worst possiblc case. including allpossible data. For a wide range of data, howevcr, the crror estimate (37) may be pessimistic.and the actual rate of convergence can be larger than that suggested by the ahove bound.

The value of the parameter K depends on Pe and on d. For d = I, K = 0 regardless of Pe'as long as Pe ::::2, whereas for d = 2 the value depends on the mesh regularity: numericalevidence suggests that for the cases considered K ~ 1.0 - 1.5, again for Pc ::::2, as otherwisethe method is unstable.

5. Nm:1ERICAL EXPERIMENTS

We shall now examine experimentally the perfollllance of the DGM for several repre-sentative examples.

5.1. Two-Point BV Problems

We will first analyze test cases of the Iype

{

d211--" = S on [0. I]

d.\'"/I (x) = 0 at x = 0 and x = I.

(39)

First we consider problem (39) with S = (471')2sin(4;rx), for which the exact solutionis /lerae/(x) = sin(411'x). Figure 7 shows error in the nOllll 11·111/ and II convergence ratefor uniform meshes. Figure 8 shows error and h convergence rate for nonuniform meshesobtained by successive retlnements of an initial grid with a subsequent random displacemelllof value ±O.2011 to each interior node. These figures show an asymptotic convergence rateof order 0 (hI') in agreemcnt with Theorem 4.3. Note: The II convergence rate is given by

CR'1

= log(e2h/c,,)log(2) eh = 1111" -/1,·,111/·

The next test cases measure the error in the L "-norm. Problem (39) is solved with S = (211')2sin(2Jrx) on uniform meshes; Fig. 9 shows error in the L"-norm and h convergence rate.These figures indicate an asymptotic convergence rate of order () (hl>+ I) for p odd and

D1SCO",TINUOUS GALERKIN METHOD 513

o (hI') for p even. This test did not involve p > 7 because after the first mesh refinement the

elTor was lIell < 10-1'. Next, problem (39) is solvcd on a nonuniform grid (±O.20h) withS = (6rr)2 sin(6;rx). Figurc 10 shows etTor and h convergence rate; it is clear from these

figures that the asymptotic convergence rate does not deteriorate for nonuniform grids. The

numerical convergence rates agree with the upper bound of order O(hp) obtained in [III.The following test case deals with hand p convergence rates in the HI seminonn. This

test case is the solution to problem (39) with S = (3;rf sin(3rrx). for which the exactsolution is lie",,! (x) = I + sin(3;rx).

We deline p-convergence rate (CR 1') as

CRp

= 10g(eplcl'+d10g(I+l/p)' el'=llI-uexll·P;::2,

Figure II shows hand p convergence rates in the H t seminorm. Thesc figurcs indicatethat the h convergence rate is optimal, i.e .. 0 (hl'). The behavior of the p-convergcnce ralccan be estimated hy considering that if the optimal error in 1·11 is e p ~ hI' I(p !), then CRp ~

p loge (p + 1)1 h). which is the same as the asymptotic convergence rate shown in Fig. II.

5.2. 2-U Experiments

The lirsttest case is the solution to the Poisson problem

-/j,u = 4( I - x2 - y2) exp( _(x2 + y2)) in n

lI(x. y) = exp(-(x2 + /») on iln.

where n is the subdomain shown in Fig, 3. Thc h convergence rate is evaluated by successive

global refinements of the domain.Figure 12 shows the L2-norm of the error and the convergence rate. It is clear from these

figures that a convergence rate of order 0 (hl'+l) is obtained for p odd. For p even. however.results indicate that for low p the h convergence rate tends to O(hp). but for high P it tendsto O(hl'+t).

The second test case involves hop adaptation for a case with low regularity, which is aDirichlet problem defined on the L-shaped domain shown in Fig. 4. with boundary valuesgiven by II = r20 sin(28 13). which is a solution to Laplace's equation.

The p adaptation process is implemented as follows: for cvery element Qe that is refined.

the values of the indicators II(A V lI) • n lIa.an, and 1[lIllo,an, are stored after II is computed. Ifthe characteristic size of the element h "cw in the current adaptation cycle is smaller than theprevious value h"ld (inherited fromthc parent elcment), and if the following equations hold

10g((l[(AVII)' nllo.iln)old/m(AVII)' nlln.iJn)"ew) < cG~?(f)c) - 1/2).log(holdl hncw) -

log«[[ullo.i1n,}old/(l[ullll.;)n)new) < c(~i~(Pe) + 1/2),log(ho1d/h"ew) e=1

wherc C is a tolerance or value 0.85. then the order p is reduccd. The above equationsare based on the optimal cOllvergence ratc for norms Oil the boundary of elcments withpolynomial degrce p". Neig refers to all the ncighbors of the clement under cOllsideration.

514 ODEN. BABUSKA, AND BAUMANN

-.- - .... - ... -- -- - -

Ii i u-r h--p

I ill--~:~-~~~ -.I I '\ I I br~1

I~~

l I 1.1 .. i I .~;. 2 M'-.'.12l I ' .,-,..l r i ~'Ij j I,. 26.l1:·~l.~ Q~~n! ' 'L __ ~ I .•-<.""

I,! ~I" :;)'FI"">?

! ._:.. ';; Il'_i~E-.'-.U:?

I IlolCu;,>

, :. '.' i~'1i;IEIJ::1 W;.<.o.i .'.i I L,;"-l 1",.1-"'"

J 1"4

11-- J ~\: ..:.t.'-f.'_..;.

I" f I 1 :~~~ ~::~ :~,iul'u.~

, !; ~~bt.l·-'l-J:t

FIG. 17. Top: Pointwise error. Bollom: x20 at the comer.

p=2p=3p=4p=5p=6p=7p=8p=9

hop conyhop sqrt

0.01

j~ 0.001

0.0001

DISCONTINUOUS GALERK[:\, METHOD 515

100 1000N

10000

0.1

0.01 ~OB5

034~.1

p=2p=3p=4p=5p=6p=7p=8p=9 ...

hop conyhop sqrt

100 1000N

10000

FIG. Ill. L-Shape domain: Convergence rates using glohat aud adaptive refinemcnrs. Top: L' norm. Bottom:11 ) seminoml.

516 ODEN. AAI:lUSKA. AND I:lAUMANN

The error indicators II(AVu .nllo.;/n..!meas(aS1,.) and Ilu]lo,;/n..!meas(aS1,.) are usedfor II refinement. The max and min values of these indicators over lhe partition Pj,{S1) are

computed. and if the value of any of the two indicators associated with an element is withina given (speci lied) percentage (usually 30% of the max value, then the element is refined.

The above procedure to reline the mesh and to dccrease the order of the polynomialapproximation is used until the values of the error indicators are below a prespe<.:itied value

for every element in the mesh.In the following test case the II-p adaptive process is initiated with a mesh consisting

or three square elements of size I x I and polynomial basis of order 9. The purpose ofthis test is to show that the singularity at the corner is detected by the II-convergence rateinformation. Figure 13 shows the resulting p distribution and h refinemenl after live cycles.

Note that the low regularity of the solution is detected hecause the order p is minimum (i.e ..p = 2 for stahi Iity reasons) for all the elements close to the corner singularity. Figure 14shows a close-up view of pointwise error at thc corner.

In the following test we attempt to evaluate the sensitivity (loss of stability) of themethod to p-enrichment in /.Ones with low regularity. In this case. the p distribution isobtained hy forcing p-enrichment where low regularity is detected (opposed to the usualprocedure). Figure 15 shows a close-up view at the corner (x 20) of p distribution (lOp)

and the <.:orresponding pointwise error (hottom). From this experiment it appears that themethod <.:anaccommodate p-enrichments even in zones with low regularity without stahility

problems.The next numerical experiment does not include automatic setting of the polynomial

approximation. With the help of analytical studies, we prespecify the polynomial orderafter II relinements as p" = 12 +7(r} /2)".11. where r,. represents the radius of the element'sbarieenter, and [.j is the integer part. Figure 16 shows fJ distrihution and II relinement. andFig. 17 shows pointwise error. The convergence rate for this case is exponential. as shownin Fig. 18, in the curve labeled hop sqn.

Finally. we compare convergence rates using uniform and adaptive refinements.Figure 18 shows the convergence rate using uniform p with global uniform relinements

(curves labeled p = Il), and using lip-adaptation (labeled II-p com'). The convergence rateof un if01111II refinements is exactly equal to the theoretical value N -1j.1 in the H I seminorm.and the convergence rates of the II-adaptation is close to the theoretical maximum whichis N -I in the HI semi norm. The exponential convergence rates shown (as 11-p sqrt) areohtained by setting the polynomial order as described hefore.

In summary, numerical experiments confirm the rohustness of the method under many dif-

ferent conditions. For the class of problems considered. the method appears to be stable evenfor arhitrary distributions of spectral orders and very di fl'erent element sizes and aspect ratios.

6. CONCLUDING COMMENTS

As a brief summary of the major observations of this study. we list the following:

• Diffusion dominated problems can be solved using piecewise discontinuous basisfunctions, without using auxiliary variahles such as fluxes in mixed methods. The discon-tinuous Galerkin method developed herein involves imposing weak continuity re4uirementson interclement houndaries: hoth solution values and fluxes are discontinuous across theseintert~1ces.

mSCONTINUOUS GALER KIN ~1ETHOD 517

• The method resembles hybrid and interior penalty methods. but no Lagrange multiplieror penalty parameter appears in the formulation.

• The method is robust. exhibiting only a small loss of accuracy in H t in which stretch-ing and distortion of elements results in a suboptimal rate of p-convergence; numericalexperiments suggest that for Pc 2: 2 the inf-sup parameter YI, ~ 0 (p-l) for 2-D cases andYh is constant for I-D problems,

• The method is not stable for p" ~ I.• The behavior of the method in L] is different for odd or even order polynomial approx-

imations: for regular mesh relinements in 2-D. the L ]-rate of convergence is experimentallyfound to be 0 (111'+1) for p odd and 0 (hI') for p even, p 2: 2.

• The conditions under which the method is stable and convergent are studied herein.with corresponding a priori error estimates, and tests contiI'm that the method can exhibithigh rates of 11-. p-, and lip-version convergence.

• Coupled with the classical discontinuous Galerkin formulation for transport dominatedproblems, this formulation is applicable to a wide range of problems, from convection-dominated to diffusion-dominated cases 1141.

• The formulation renders a numerical approximation which is elementwise conservativeand. as such, is. to the best of our knowledge, the first high-order [inite element methodever developed with this property.

• The associated bilinear form renders a positive delinite and well-conditioned mutlix.thus allowing the use of standard iterative methods for high p and dist0l1ed elements.

• This formulation should be particularly convenient for time-dependent problems. be-cause the global mass matrix is h!ock diagonal, with uncoupled blocks (see [14]),

ACKNOWLEDGMENT

The support of this work by the Army Research Oflice under Grant DAAH04-96-00l\2 is b'Tatefullyacknowledged.

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