adiscontinuous hp finite elementmethod ...oden/dr._oden... · per.gamon computers and mathematics...

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PER.GAMON Computers and Mathematics with Applications 37 (1999) 103-122

A Discontinuoushp Finite Element Methodfor Diffusion Problems:

I-D Analysis1. BABUSKA

Texas Institute for Computational and Applied MathematicsThe University of Texas at Austin, Austin, TX 78712, U.S.A.

C. E. BAUMANNTexas Institute for Computational and Applied MathematicsThe University of Texas at Austin, Austin, TX 78712, U.S.A.

andCOMCO, Austin, TX, U.S.A.

J. T. ODENTexas Institute for Computational and Applied MathematicsThe University of Texas at Austin, Austin, TX 78712, U.S.A.

(Received and accepted January 1998)

Abstract-This paper presents the mathematical analysis of a new variant of the discontinuousGalerkin method which is applicable to the numerical solution of diffusion problems, not requiringauxiliary variables such as those used in mixed methods. The focus of this study is on a class of linearsecond-order boundary value problems for which we prove stability and a priori error estimates inboth the finite- and infinite-dimensional spaces. © 1999 Elsevier Science Ltd. All rights reserved.-*',

Keywords-Discontinuous Galerkin, Finite elements, Diffusion operators.

1. INTRODUCTIONThe aim of this work is to present the mathematical foundations of a new type of DiscontinuousGalerkin Method (DGM) applicable to a broad class of partial differential equations. This paperaddresses the treatment of second-order diffusion operators by finite element techniques in whichthe shape functions are discontinuous across interelement boundaries. The method supportshp-approximations on arbitrary meshes, including nonmatching grids. The main features of themethod presented here are the following:

• the method does not need auxiliary variables such as those used in hybrid or mixedmethods;

• the method is robust and exhibits elementwise conservative approximations;

The support of this work by the Army Research Office under Grant DAAH04-96-0062 is gratefully acknowledged.

0898-1221/99/$ - see front matter © 1999 Elsevier Science Ltd. All rights reservedPII: 80898-1221(99)00117·0

Typeset by AJVtS- 'fEX

104 I. BABUSKA et al.

• coupled with the classical discontinuous Galerkin formulation for transport dominatedproblems, this formulation is applicable to a wide range of problems, from convectiondominated to diffusion dominated cases;

• the associated bilinear form renders a positive definite and well-conditioned matrix, thusallowing the use of standard iterative methods for high p and for distorted elements;

• the formulation is particularly convenient for time-dependent problems, because the globalmass matrix is block diagonal, with uncoupled blocks;

• a priori error estimates are derived so that the parameters affecting the rate of convergenceand limitations of the method are established;

• the method is suited for adaptive control of error, and can deliver high-order accuracywhere the solution is smooth; and

• the cost of solution and implementation is acceptable.

The development of discontinuous finite element methods for second-order elliptic problemsdates back to the early 1970sj Nitsche [1] introduced the concept of replacing the boundarymultipliers with the normal fluxes, and added stabilization terms to produce optimal convergencerates. A very similar approach is that of Percell and Wheeler [2], Wheeler [3], and Arnold [4].A slightly different method was the p-formulation of Delves and Hall [5], they developed theso-called Global Element Method (GEM); applications can be found in [6]. The GEM consistsessentially in the classical hybrid formulation for a Poisson problem with the Lagrange multipliereliminated in terms of the dependent variables; namely, the Lagrange multiplier is replaced bythe average flux across interelement boundaries. A major disadvantage of the GEM is that thematrix associated with space discretizations of diffusion operators is indefinite, thus the methodis unable to solve time dependent diffusion problems; and being indefinite, the linear systemsassociated with steady state diffusion problems needs special iterative schemes. The interiorpenalty formulations of Wheeler [3] and Arnold [4] utilize the bilinear form of the GEM augmentedwith a penalty term which includes the jumps of the solution across elements. Disadvantages ofthis approach include the dependence of stability and convergence rates on the penalty parameter,the loss of the conservation property at element level, and a bad conditioning of the matrices.The DG~ for diffusion operators developed in this study is a modification of the GEM, which isfree from the above deficiencies. More details on these formulations, and the relative merits ofeach one are presented in [7,8].

The underlying reason for developing a method based on discontinuous approximations fordiffusion operators is to solve convection-diffusion problems. Solutions to convection-diffusionsystems of equations using discontinuous Galerkin approximations have been obtained with mixedformulations, introducing auxiliary variables to cast the governing equations as a first-ordersystem of equations. A disadvantage of this approach is that for a problem in IRd, for each variablesubject to a second-order differential operator, d more variables and equations are introduced.This methodology was used by Dawson [9], and Arbogast and Wheeler in [10], also by Bassi andRebay [11,12] for the solution of the Navier-Stokes equations, Lomtev, Quillen and Karniadakisin [13-16] and Warburton, Lomtev, Kirby and Karniadakis in [17] solved the Navier-Stokesequations discretizing the Euler fluxes with the DG method and using a mixed formulation forthe viscous fluxes. A similar approach was followed by Cockburn and Shu in the development ofthe Local Discontinuous Galerkin method [18], see also a short-course notes by Cockburn [19].

The discontinuous Galerkin method analyzed in this paper supports h-, p-, and hp-versionapproximations and can produce sequences of approximate solutions that are exponentially con-vergent in standard norms. We explore the stability of the method for one-dimensional diffusionproblems and we present a priori error estimates. Optimal order h- and p-convergence in the HInorm is observed in one-dimensional applications.

Following this Introduction, Section 2 introduces the model problem and the variational formu-lation in an infinite-dimensional setting, Section 3 presents the analysis for a finite-dimensional

-

Diffusion Problems 105

space of basis functions, Section 4 introduces an a priori error estimate, and finally Section 5presents numerical experiments which confirm the theoretical results from previous sections.

2. INFINITE-DIMENSIONAL CASEIn this section, we focus on two-point diffusion problems in one-dimensional domains

(d = 1). The class of problems considered is first solved using infinite-dimensional spaces ofbasis functions, the objective is to identify a complete space of functions which will make possi-ble an approximation to the exact solution within any tolerance, however small it might be.

2.1. Model Problem and Approximation

We consider a model second-order boundary value problem characterized as follows.Let the domain D. c JR. be subject to a partition 'Ph consisting in N(Ph) elements D.e as shown

in Figure 1,

N(Ph)

n = U ne, D.e = (xe,Xe+1),OcEPh(O)

Given data (Do C JR., S, j, g), find a function u such that

d2udx2 = S,

u= j,dudn = g,

on Do,

at fD,(1)

where f D and f N are the sets of points where Dirichlet and Newmann boundary conditions areimposed, i.e.,

fDnfN = 0, fDUfN = {Xl,XN+d. and fD =I- 0. Thus, we consider Dirichlet problems (fN = 0)or mixed boundary conditions with u specified at Xl or XN+ll and ~~ at the opposite end, i.e.,du _ du· f f - {} d du - du .f f - { }dn - - dx 1 N - Xl , an dn - dx 1 N - XN+l .

The set of interior points is fint = {X2"'" X N }; and n = {n} = {±1} is the unit outwardvector to D. at the end points f D U f N, and at Xi E fint, it is arbitrarily chosen pointing to theelement with lower index.

Let V(Ph) be the space of discontinuous basis functions which we characterize in Section 2.2.The solution to problem (1) can be obtained as follows:

where

I find u E V(Ph) such that B(u,v) = L(v), 'r/v E V(Ph) I, (2)

(3)

(4)


" " " " " "- - - - - -• • • - -- • • •X1

e=1 x e=2 x x e=N-1 x e=N x2 3 N-1 N N+1

"e-e

~I

"e-

;,

I' I , ,

Figure 1. Domain and discontinuous approximation.

du dudn (Xi) = n dx (Xi),

[U](Xi) = u(xi)ln, - u(xi)lnj , e> f,

/ du ) 1 (dU du )\ dn (Xi) = '2 dn (xi)lne + dn (xi)lnj ,

A norm associated with (3) can be defined as follows:

lIulI~ = L luitOe + L (hilU

2 + hi (~~ r) (Xi)neE'P" xiErv

+ Xi~nt (hil [UJ2 + hi (~~) 2) (Xi),

where

(5)

(6)

2 1 du dulullo = - -dx," dx dxneand hi is a boundary function used to homogenize the norm, defined as follows:

{

he a-, XiE nenrD,2

hi = (he + h,) (7)2 ,XiE(anenan,)Crint, l:::;e, f:::;N(Ph)'

The motivation for the preceding definition of hi will become clear during the proof of stabilityfor the finite-dimensional case (Section 3.1). The space of functions

H3/2+«(Ph) = {v E L2(n) : vine E H3/2+fne, Vne E Ph(r!)}

is the biggest functional space among all H5(Ph) on which 1I.llv remains finite.

(8)

Diffusion Problems

For 'u, v E V(Ph), the bilinear form can be bounded as follows:

B(u, v) ~ lIullvllvllv,

where 11.llv denotes the above defined mesh-dependent norm.

107

(9)

2.2. Characterization of the Space V(Ph)

When the domain 0 is subject to a family of regular partitions {Ph}, the norm 1I.llv is welldefined and we can discuss the completion of H3/2+f(Ph) under the norm 11.llv in the followingtheorem.

THEOREM 2.1. The completion of the space H3/Hf(Ph) under the norm 11.llv is the productspace

where(10)

and

with norm

IIwllx = v'lIvll~ + IIqll~,IIqll~= L

w = (v, q) E x,(hiq?) ,

(11)

(12)

IIvllb = L Ivltoe + L (h;lV2) (Xi) + L (h;1[V]2) (Xi)'O,EP" x;Erv Xierint

(13)

PROOF. The space X furnished with the norm 1I.lIx is complete. Given u E V(Ph), let us associatethe element (u,<'i) so that u = u, '\fOe E Ph(O), and

Xi E rD,

then(14)

Let {un} E V(Ph) be Cauchy under 1I.lIv. Then (14) implies that (un, <'in) is Cauchy in X, andgiven that X is complete, (un, <'in) ---+ (u, p) E X·

Now let (u,p) E X, then there exist {u~,} E H20e, Oe E Ph, such that

L Ilu~ - ull~IOe ~ e~,fl, EP"

en ---+ 0,

where C depends on Ph but is independent of u and n.

(15)

108 1. BABUSKA et al.

Given that X is complete, we can construct the sequence {U~} E H2(Oe), VOe E Ph(O), sothat

L llii~- u~117:[I(n,) ~ €~,

n"EP"

du~dn = Pi,

Hence, (u~, p) - (u,p) in X and {un} E V(Ph) is Cauchy under 1I·llv.

2.3. The Variational Formulation in the Space X

The variational formulation in the product space X is the following:

I find A = (u,p) E X such that BH = (A,W) = LH(W), Vw = (v,q) E X I, (16)

where

and

BH(A,W) = L {I ~~~~dX}n,EP" n,

+ L (qiU - PiV) + L (q;[u] - p;[v]) (Xi)Xi ErD Xi Erlul

(17)

(18)

2.3.1. Stability analysis in the space X

Using (11)-(13), the bilinear form (17) can be bounded as follows:

To prove stability, we present the following theorem.

THEOREM 2.2. The stability of the variational formulation (16) is given by the following estimate:

and

. f IBH(A, w)1 > 1III sup -),.Ex ",Ex IIAllxllwlix - 2J6)"¢D ",¢D

inf sup IBH(A, w)1 > O.wEx AExw¢D )"¢D

(19)

(20)

PROOF. Taking an arbitrary A = (u, p) and w = 77(U-Y,P+z), where y E U(Ph) and z E W(fp,Jare arbitrary functions, and 77is a scaling parameter that will render Ilwllx ~ IIAllx, we get

BH(A,W) = 77 ( L (Iulr,f'!, - (u, yh,n.,) + L (ZiU + PiY)O"EP" xiErD

+ Xi~Ul (z;[u] + P;[y])) ,

(21)

Diffusion Problems

and picking Z E W(rph) such that

109

Xi E rD,

Xi E riot,

Xi ErN,

(22)

..

where h is that defined in (7). Selecting Y E U(Ph) such that

{

hiPi, Xi E rD naoe,

Y(Xi) = ~he(n: ne)pi' Xi E riot n aoe,

0, Xi ErN n aoe,

with a linear interpolation on Oe between the boundary values, we get

)

22 Pe+ 1 - Pe 1 2 2Iyll,n. ~ (2 he ~ "2he (Pe+1 + Pe) ,

and using again the definition of h given in (7), we get

L IYltoe ~ L (hiP;) ~ IIpll?v·n.E'Ph xiErDur,nt

Noting that the terms (n· ne) assume the values ±1, we get [y] = (hp)i on rint, and

L (ZiU + PiY) + L (zdu] + pdY]) (Xi)X,ErD X,Erlnt

= L (hi1u2 + hip;) + L (hil[U]2 + hip;) (Xi),X,ErD xiErint

and also the following conditions:

IlzlI?v = L (hi1u2) (Xi) + L (hil[u]2) (Xi) ~ lIuliEr -Iuli,x,ErD x,Eriut

IlyllEr ~ Iyli + L (hiP;) ds ~ 2I1pll?v,xiE'YDurint

we can estimate." as follows:

Ilwll~ = .,,2 (liu - yliEr + lip + zll?v)~ 2.,,2(Ilullb + IlyliEr + IIpll?v + Ilzll?v) ~ 6.,,211>'11~.

(23)

(24)

Therefore, with." = 1/../6 we obtain Ilwllx ~ 1I>'llx'Using (21), (24), and (u,yh,o. ~ 1/2{luli,l1. + IYI?,I1J, we get

IBH(>.,W)j ~ ~ (~ L (Iultn. -lylT.I1J + L (hi1u2 + hip;) (Xi)n.E'Ph xiErD

+ L (hil[u]2 + hip;) (Xi)) ~ 2~1I>'1I~ ~ 2~11>'llxllwllx'xiErlnt

as asserted. IIn this section, we have proven that the variational formulation (16) in the space X is stable,

therefore, there exists a unique solution to (16).

110 1. BABUSKA et al.

3. FINITE-DIMENSIONAL APPROXIMATIONThe discontinuous Galerkin approximation (2) is now restricted to a subspace Vp(Ph) of the

Hilbert space V(Ph) analyzed in the preceding sections. Using Vp(Ph) instead of V(Ph), thegeneral form of the error estimate can be written as follows [20,21]:

Ilu - uhllv S (1 + M) inf Ilu - wllv,'Yh wEVp(25)

where M is the continuity constant, which is unity from (9), and 'Yh is the inf-sup parameterdefined as

inf sup IIB(u,v)11 > 'Yh.uEVp vEVp Iluliv Ilvllv -

What must be established to apply (25) is how the inf-sup parameter 'Yh depends upon thediscretization parameters hand p. We analyze this behavior in some detail in the followingsections. We establish that for hp approximations, 'Yh is independent of hand p for two-pointboundary-value problems.

Let us consider the finite-dimensional subspace Vp(Ph) C V(Ph) defined as

N(Ph)

Vp(Ph) = IT pp,(ne),e=1

(26)

where for every element ne E Ph, the finite-dimensional space of real-valued shape functionsPpr. (ne) is the space of polynomials of degree S Pe.

The discontinuous Galerkin approximation in Vp(Ph) is:

I find Uh E Vp(Ph) such that B(Uh,Vh) = L(Vh), VVh E Vp(P) I,

where B(.,.) and L(.) are defined in (3) and (4), respectively.

(27)

3.1. Discrete Stability Analysis

In this section, we prove the stability of formulation (27) defined on the finite-dimensionalsubspace Vp(Ph)'

THEOREM 3.1. In the space Vp(Ph), ifpe 2: 3, 1 S e S N(Ph), the bilinear form (3) of thediscontinuous Galerkin approximation satisfies

. f IB(u,v)1 (1 - (1/2)>'0)III sup > ,

uEVp vEVp lIullvllvllv - (1+~)

where 1.23 < >'0 < 1.24 independently of the discretization parameters Pe and he.

REMARK. Theorem 3.1 is proven below for polynomial basis functions of degree;::: 3. Numericalexperiments indicate, however, that the method is stable and convergent for polynomial basisfunctions of degree;::: 2.

PROOF. For convenience in the proof, we shall work with the norm 1I.lIv which is equivalent tothe norm 11.llv given in (6), except for the use of iii1 = 1/2(h;1 + hj1) rather than hi1 in thefirst term of the summation with Xi E Cnt:

(28)


Let v· = u+w, where wE P3(ne), 'v'ne E Ph(n) is uniquely defined by the following boundaryvalues:

Xi E ane n rD,

Xi E ane n rN, (29)

Xi EanenrD,Xi E ane n rN,

Xi E aDe n rint,

(30)

where ne is the unit outward vector to De at aDe1 and he is the length of De. The terms (n· ne)assume the values ±1. Note that given the above boundary conditions, w E P3(De) exists and isuniquely defined.

Using the definitions (29) and (30), we get the following identities which are valid 'v'Xi E f;nt,.. i.e., for every boundary point Xi between two adjacent elements with indices e and J, we have

and

(~:) = ii;1[u], h h~-1 1 (h-1 h-1)were i = 2 e + f .

Substituting the above definitions and identities in the bilinear form, and using the definitionhi = he/2 at Xi E aDe n rD, we obtain

Next, we observe that

where the value of Iwll is estimated using the following eigenvalue problem.Find >'0 such that

r (dw)2 dx -::;>'0 L (he (dw)2 + 2-(W)2) ,In. dx xiEan, 2 dn he

(31)

where the subindex e denotes a generic element in the partition. Note that (31) is in nondimen-sional form, therefore >'0 is independent of he = Xe+1 - Xe, Le., it is a constant 'v'ne E Ph(D).


Using the values of w on the boundary of each element as specified in (29) and (30), we havethe following estimate:o~J(:)'dx ,,,. L~~r"(2h;IU'+ ~.(~:n

+..:~~r"(Hh;' +hJI}(UI'+~(I"+hf)(~:)')} (32)

~ >'0{ L (hilU2 + hi (~~r)(Xi)xiErJ)

+ Xi~U. (hil [UJ2 + hi \ ~~ r) (Xi) } ,

and now

B(u,v") ~ ~Iuli+ (1- ~o)L~f)(hi1u2+hi (~~r)(Xi)

+ Xi~nt (hi! [UJ2 + hi \ ~~) 2) (Xi) } ,

B (u, v") ~ (1 - ~o) lIull~.From the definition of w, which is based on four boundary conditions for each element, and

using the estimate (32), we bound Ilwllv as follows:

IIwll~ = Iwli + L (hi lu2 + hi (~~ r) (Xi) + L (hillUJ2 + hi \ :~) 2) (Xi)X,ErD xiEriu•

<; (i,. + 1) C~D(h, 'u' + h,(~:n(Xi) +J~",(h, I(ul' +h, ( ~:) ') (Xi)) ,

I1wll~ ~ (>'0 + 1) lIull~,and

Finally,

Ilv"lIv ~ Iluliv + IIwllv ~ (1 + J>.o + 1) lIullv·

B(u, v") (1 - (1/2)>'0) II IIsup " ~ _.rr-:::Ll\ u v'

v·EVp

IIv Ilv (l+v>'o+l)v·~O

and given that hi 1 ~ hi! on fint implies that 11.11 v ~ 11.11 v, we get the stability conditionB (u, v") (1 - (1/2)Ao) I I

sup 11.,"11 ~ {, , . ~ u Lnr

v·EVp

v·~O

(33)

as asserted. IREMARK 3.1. It has been found numerically that the exact solution >'0to the eigenvalue prob-lem (31) can be bounded as 1.23 < Ao < 1.24. Using this bound, we can write

B (u,v")sup II "II ~ 0.15I1ullv. I

v.EVp v Vv·~o


...

3.2. Numerical Evaluation of the Inf-Sup Condition

This section presents the results of several numerical experiments designed to evaluate theinf-sup condition for the model problem analyzed in the preceding sections. First, the inf-supcondition is evaluated as described in [7], for different number of elements N(Ph) and for uniformPe = p, 1 :::;e :::;N(Ph)' Note that from the inf-sup condition standpoint, the cases in which someelements in the partition have 2 <= Pe < P is just a particular case of Pe = p, 1 :::;e :::;N(Ph).Results are collected in Table 1. It is clear that for this class of model problems the inf-supconstant is asymptotically independent of hand p.

Table 1. Inf-sup constant for Laplace's equation with Dirichlet boundary conditions.Uniform mesh.

Number of Elementsp

2 4 8 16 32 64

2 .3333333 .3333333 .3333333 .3333333 .3333333 .3333333

3 .5031162 .5031162 .5031162 .5031162 .5031162 .5031162

4 .5031162 .5031162 .5031162 .5031162 .5031162 .5031162

5 .5039530 ,5039530 ,5039530 ,5039530 .5039530 .5039530

6 .5039530 .5039530 ,5039530 .5039530 .5039530 .5039530

7 .5025293 .5025293 .5025293 .5025293 .5025293 .5025293

8 .5025293 .5025293 .5025293 .5025293 .5025293 .5025293

Table 2. Inf-sup constant for Laplace's equation with Dirichlet boundary conditions.Random distribution of mesh points (1/4).

Number of Elementsp

2 4 8 16 32 64

2 .3333485 .3365059 3586421 .3460370 .3500695 .3506928

3 .5031302 .5032297 .5034426 .5034566 .5034003 .5033587

4 .5037151 .5032127 .5036960 .5037744 .5035089 .5036255

5 .5039498 .5037831 .5038352 .5037374 .5038389 .5037868

6 .5039400 .5038976 .5038288 .5038620 .5037359 .5037929

7 .5024107 .5024598 .5025075 .5024672 .5023690 .5023519

8 .5025245 .5024703 .5023743 .5024000 .5023425 .5023330

Table 3. Inf-sup constant for Laplace's equation with Dirichlet boundary conditions.Geometric distribution of mesh points (hmin/hmax = 10-7).

Number of Elementsp

2 4 8 16 32 64

2 .3440400 .3572851 .3650082 .3658733 .3658805 .3658805

3 .5032378 .5033161 .5033382 .5033402 .5033402 .5033402

4 .5034964 .5037628 .5038515 .5038596 .5038596 .5038596

5 .5038662 .5038452, 5038392 .5038387 .5038387 .5038387

6 .5038321 .5037621 .5037426 .5037409 .5037409 .5037409

7 .5024515 .5024277 .5024210 .5024204 .5024204 .5024204

8 .5024281 .5023737 .5023585 .5023572 .5023571 .5023571

Table 2 shows the effect of a random distribution of nodal points, where the worst aspect rat.iobetween adjacent elements is 1/4, and Table 3 shows the effect of a geometric distribution ofmesh points such that hmin/hmax = 10-7.

The asymptotic values obtained are higher than the analytic value because the latter is only alower bound of the exact inf-sup constant.


4. A PRIORI ERROR ESTIMATION

(34)

The approximation properties of Vp('Ph) will be estimated using standard local approximationestimates (see [22]). Let u E HS(D.e); there exist a constant C depending on s but independentof U, he, and Pe, and a polynomial up of degree Pel such that for any 0 ~ r ~ s the followingestimate holds:

hµ-rIlu - u II 11 < C.....!:..-Ilull 11 s > 0p r, e - 8-r 8, c' _ 1

Pe

where 11.llr,l1edenotes the usual Sobolev norm, and µ = min (Pe + 1, s).An a priori error estimate is given by the following theorem.

THEOREM 4.1. Let the solution u to (27) E H"(Ph(f!)), with s > 3/2. Given that the approxi-mation estimate (34) holds for the spaces Vp(1'h), the error of the approximate solution UDa canbe bounded as follows:

(35)

where µe = min (Pe + 1, s), E: ----+ 0+, and the constant C depends on s but is independent of u,he, and Pe·

PROOF. Let us first bound Iluliv as follows:

lIull~ ~ C L (luli.l1e + h-lllull~/2+e,l1e + hllull~/2+e,l1e) ,I1,EPh

(36)

and using the approximation estimate presented in (34), there exists a local polynomial approx-imation up of u E H"(Ph(O)) in the norm 1I.llv such that

Finally, using the continuity and inf-sup parameters, and assuming that the exact solutionu E HS(Ph), we arrive at the following a priori error estimate:

where s > 3/2, µe = min (Pe + 1, s), and C depends on s but is independent of u, he, and Pe. •

REMARK 4.1. The error estimate (35) is a bound for the worst possible case, including all possibledata. For a wide range of data, however, the error estimate (35) may be pessimistic, and theactual rate of convergence can be larger than that suggested by the above bound. •

5. NUMERICAL EXPERIMENTSWe first analyze test cases of the following type:

on [0,1],

at x = 0 and x = 1.(37)

1

> 1e-05_IQ)

1e-10

'-'- ...........

.....---- -..._-

Diffusion Problems

- ... _-....-.......... ------ ....---- -----

-'--'- -.- .... -'-

p=2-

Ip~~~.-.~~

=5··..·=6 -_.~=7 ----=8 H __ ,

- .._-

-'- -'- -'-,

115

8 161/h

32 64

9 iii i I

8

7

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

~

p~§=-=4 ~:5 ···..-6 ---g~~~..-..~

Q)

1ti0:Q)uc::Q)OJ....Q)>c::oo

6

5

4

3

" -.-.:'::':- ,_. -.-.-.-.- ::':::::.::-._.:;;:;::. ::::.: :~.,-

l -----1

o8

I

161/h

32...L

64

Figure 2. V-norm of the error and convergence rate with uniform meshes: - ~ = s,S = (471")2 sin (471"x).

116

1

~I 1e-05~

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

1. BABUSKA et al.

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

------- -, --.--- -------------

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

--------

~

=2 -=3 ----.=4 .5 ..

p=6 -.-.-P:? - -p-8 ..

-"- ..

1e-1 0"

' ..~

............ - .... -.- -'- - ...........

-" - ........j,

8 161/h

32 64

9 t·... iii'

8

7

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

Q)caa:Q)Colc:Q)Ol~~c:o(.)

6

5

4

3

2

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

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

-

• .. • •• h ••••••••

.-.-.-.-.-.-.-.- .-.-.~:.~~.~...:... ...;::::.:::::::.:. ...~..-

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

.. - -- ..

o8

..L.16

1/h

,32 64

Figure 3. V-norm of the error and convergence rate with nonuniform meshes:-~ = S, S = (41T)2 sin(47l"X), 6h = ±20%.


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

1e-05

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

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Figure 4. L2-norm of the error and convergence rate with uniform meshes: -~ =S, S = (2rr)2 sin(211'x).


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Figure 5. L2-norm of the error and convergence rate with nonuniform meshes:- ~ = S, S = (611')2 sin(67rx), 6h = ±20%.

9


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Figure 7. hand p convergence rates in the H2 seminorm: a2u = S, S-p(31l')2 sin(31l'x).


First, we consider problem (37) with S:= (471')2sin(47Tx),for which the exact solution is Uexact(x) := sin(471'x). Figure 2 shows error in the norm 11.\lv and h convergence rate for uniformmeshes. Figure 3 shows error and h convergence rate for nonuniform meshes obtained by succes-sive refinements of an initial grid with a subsequent random displacement of value ±O.20h to eachinterior node. These figures show an asymptotic convergence rate of order O(hP) in agreementwith Theorem 4.1. Note: the h convergence rate is given by

The next test cases measure the error in the L2-norm. Problem (37) is solved with S :=

(271')2sin(271'x)on uniform meshes, Figure 4 shows error in the L2-norm and h convergence rate.These figures indicate an asymptotic convergence rate of order O(hp+1) for p odd and O(hP) for peven. This test did not involve p > 7 because after the first mesh refinement the error was Ilell <10-13. Next, problem (37) is solved on a nonuniform grid (±O.20h) with S := (671')2sin(671'x).Figure 5 shows error and h convergence rate, it is clear from these figures that the asymptoticconvergence rate do not deteriorate for nonuniform grids. The numerical convergence rates agreewith the upper bound oforder O(hP) obtained in [231.

The following test case deals with hand p convergence rates in the H1 seminorm. This testcase is the solution to problem (37) with S := (37T)2sin(371'x), for which the exact solution isUexact(x)= 1+ sin(37Tx).

We define the p-convergence rate (CRp) as

log(epl ep+I)CRp := log(l + lip) ,

...

Figure 6 shows hand p convergence rates in the HI seminorm, and Figure 7 in the H2

seminorm. From these figures, it is evident that the h convergence rate is optimal. The behaviorof the p-convergence rate can be estimated by considering that the error in 1.11 is ep ::::::hP I (p!),then CR.p ::::::p log ((p + 1)Ih), which is the asymptotic convergence rate shown in the figure.

We conclude that the convergence rates are optimal in all the norms considered except in theL2-norm, in which a loss of accuracy is observed for even powers of Pe.

6. SUMMARYThis paper presented the mathematical analysis of a new discontinuous Galerkin method for

the approximation of diffusion terms in the context of convection-diffusion problems. Utilizingthe method analyzed in this paper, we extend the class of problems that can be solved bydiscontinuous Galerkin techniques to problems involving dissipative or diffusive terms, such asthose appearing in the Navier-Stokes equations.

The stability study for diffusion problems shows that the method is very robust, optimal hand p convergence rates are obtained, and high-order accuracy is obtained when the regularityof the solution is high enough.

Even though this study was carried out in a one-dimensional setting, the mathematical analysisdemonstrated the distinguishing characteristics of the formulation.

REFERENCES1. J. Nitsche, Uber ein variationsprinzip zur Losung von Dirichlet problemen bei Verwendung von Teilriiumen,

die keinen Randbedingungen unterworfen sind, Abh. Math. Sem. Univ. Hamburg 36,9-15, (1971).2. P. Percell and lvI.F. Wheeler, A local residual finite element procedure for elliptic equations, SIAM J. Numer.

Anal. 15 (4), 705-714, (August 1978).3. lvI.F. Wheeler. An elliptic collocation-finite element method with interior penalties, SIAM J. Numer. Anal.

15 (4), 152-161. (1978).


4. D.N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal.19 (4), 742-760, (August 1982).

5. L.M. Delves and C.A. Hall, An implicit matching principle for global element calculations, J. Inst. Math.Appl. 23, 223-234, (1979).

6. J.A. Hendry and L.M. Delves, The global element method applied to a harmonic mixed boundary valueproblem, J. Compo Phys. 33, 33-44, (1979).

7. C.E. Baumann, An hp-adaptive discontinuous finite element method for computational fluid dynamics, Ph.D.Thesis, The University of Texas at Austin, (August 1997).

8. J.T. Oden, 1. Babuska and C.E. Baumann, A discontinuous hp finite element method for diffusion problems,TICAM Report 97-21, (1997).

9. C.N. Dawson, Godunov-mixed methods for advection-diffusion equations, SIAM J. Numer. Anal. 30, 1315-1332, (1993).

10. T. Arbogast and M.F. Wheeler, A characteristic-mixed finite element method for convection-dominatedtransport problems, SIAM J. Numer. Anal. 32,404-424, (1995).

11. F. Bassi. R. Rebay, M. Savini and S. Pedinotti, The discontinuous Galerkin method applied to CFD problems,In Second European Conference on Turbomachinery, Fluid Dynamics and Thermodynamics, ASME, (1995).

12. F. Bassi and R Rebay, A high-order accurate discontinuous finite element method for the numerical solutionof the compressible Navier-Stokes equations, J. Camp. Physics 131 (2), 267-279, (1997).

13. I. Lomtev, C.W. Quillen and G. Karniadakis, Spectral/hp methods for viscous compressible flows on un-structured 2d meshes, Technical Report, Center for Fluid Mechanics Turbulence and Computation, BrownUniversity, Providence, RI 02912, (December 1996).

14. 1. Lomtev and G.E. Karniadakis, Simulations of viscous supersonic flows on unstructured meshes, AIAA-97-0754, (1997).

15. r. Lomtev, C.B. Quillen and G.E. Karniadakis, Spectral/hp methods for viscous compressible flows on un-structured 2d meshes, J. Compo Phys. 144 (2), 325-357, (1998).

16. I. Lomtev and G.E. Karniadakis, A discontinuous Galerkin method for the Navier-Stokes equations. Int. J.Num. Meth. Fluids (submitted), (1997).

17. T.C. Warburton, I. Lomtev, RM. Kirby and G.E. Karniadakis, A discontinuous Galerkin method for theNavier-Stokes equations on hybrid grids, In Center for Fluid Mechanics 97-14, Division of Applied Mathe-matics, Brown University, (1997).

18. B. Cockburn and C.W. Shu, The local discontinuous Galerkin method for time dependent convection-diffusionsystems, SIAM J. Numer. Anal. (submitted), (1997).

19. B. Cockburn, An introduction to the discontinuous Galerkin method for convection-dominated problems,School of Mathematics, University of Minnesota, (1997).

20. A.K. Aziz and I. Babuska, The Mathematical Foundations of the Finite Element Method with Applicationsto Partial Differential Equations, Academic Press, (1972).

21. J.T. Oden and G.F. Carey, Texas Finite Elements Series Vol. IV-Mathematical Aspects, Prentice-Hall.(1983).

22. I. Babuska and M. Suri, The hp-version of the finite element method with quasiuniform meshes, MathematicalModeling and Numerical Analysis 21, 199-238, (1987).

23. I. Babuska, J.T.Oden and C.E. Baumann, A discontinuous hp finite element method for diffusion problems:1-D Analysis. TICAM Report 97-22, (1997).

adiscontinuous hp finite elementmethod ...oden/dr._oden... · per.gamon computers and mathematics...

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