~

IEL<iEVIER Comput. Methods Appl. Mcch. Engrg. 175 (1999) 111-141

Computer methodsin applied

mechanics andengineering

www.elsevier.com/locale/ Clna

•

A discontinuous hp finite elelnent method for convection-diffusionproblems

Carlos Erik Baumann), 1. Tinsley Oden*Te,I'lI.1'Institute for COlllpllttUional and Applied Mathematics. The University ot' Texas at AIlS/in. AilS/ill. TX 787/2. USA

Received 10 January 1998: rcvised 2 June 1998

Abstract

This paper presents a new method which exhibits thc best features of both tinite volume and tinite clement techniqucs. Special attention isgiven to thc issues of conservation. l1exible accuracy. and stability. The method is elemenlwisc conservative. the order of polynomialapproximation can hc adjusted elemcnt by elemcn\. and the stability is 1I0t bascd on the introduction of artificial diffusion. bIll on the usc ofa very particular tinitc element formulation with discontinuous basis functions.

The mcthud supports h-. 1'-. and hp-approximatiuns and can be applied to 'illY type uf mcshcs. including nun-matching grids. A priorierror estimates and numerical expcriments on representative model problems indicate that the method is robust and capable uf deliveringhigh accuracy. © 1999 Elsevicr Science S.A. All rights reserved.

1. Introduction

We present the treatment of convection-diffusion problems using a new type of Galerkin method (DGM). Inthis method both the approximate solution and the associated fluxes can experience discontinuities acrossinterelement boundaries. Among features of the method and aspects of the study presented here are thefollowing:

• the method does not need auxiliary variables such as those used in hybrid or mixed methods:• the method is robust and exhibits elementwise conservative approximations;• the formulation is particularly convenient for time-dependent problems, because the glohal mass matrix is

block diagonal, with ullcoupled hlocks:• a priori error estimates are derived so that the parameters affecting the rate of convergence ;.lI1dlimitations

of the method are established:• the method is suited for adaptive control of error, can deliver high-order accuracy where the solution is

smooth: and• the cost of solution and implementation is acceptable.The solution of second-order partial differential equations with discontinuous basis functions dates back the

the early I970s. Nitsche 136] introduced the concept of replacing the Lagrange multipliers used in hybridformulations with averag-;:d normal fluxes at the boundaries, and added stabilization terms to produce optimalconvergence rates. Similar approaches can be traced back to the work of Percell and Wheeler 139]. and Arnold[3]. A different approach was the p-formulation of Delves and Hall [23], they developed the so-called GlobalElement Method (GEM): applications of the latter were presented by Hendry and Delves in 125J. The GEM

* Curresponding author. Dire~tor uf TICA1\·1. Coc"rell Family Regents Chair in Engineering #2.I Research Engineer, CO:v!CO. Austin. TX.

0045· 7825/99/ $ - sec front matter © 1999 Elsevier Science S .A. All rights rescrved.PII: S004:'i-7X25(9~)003:'i9·4

3J2 C.E. BOl/llloll/l. J.T. Odell / COlllpl/t. Met/1ud,I' Appl. Meell. ElIgrg. /75 (/999) JIl-J4!

consists essentially 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 by the average fluxacross interelement boundaries. without the addition of penalty temlS. A major disadvantage of the GEM is thatthe matrix associated with space discretizations of diffusion operators is indefinite, therefore the method can notsolve time dependent diffusion problems: and being indefinite. the linear systems associated with steady stateproblems needs special iterative schemes. The interior penalty formulations presented in [391 and [3 J utilize thebilinear form of the GEM augmented with a penalty term which includes the jumps of the solution acrosselements. The disadvantages of the last approach include the dependence of stabi Iity and convergence rates onthe pcnalty parameter, thc loss of lhc conservation propcrty at clement level. and a bad conditioning of thematrices. The DGM for diffusion operators developed in this study is a modification of the GEM. which is freefrom the deficiencies mentionetl earlier. More dctails on these formulations, and the relativc merits of each oneare presented in [10.381.

Among the applications of the Galerkin method to nonlinear lirst-order systems of equations. Cockburn andShu 117-20J developed the TYB Runge-Kulla projection applied to general conservation laws: Allmaras [I]solved the Euler equations using piecewise constant and piecewise linear representations of the field variables:Lowrie [35] developed space-time discontinuous Galerkin methods for nonlinear acoustic waves: Bey and aden[13] presented solutions to the Euler equations: Hu and Shu [26] presented solution to the Hamilton-Jacobiequations: and Atkins and Shu 141 presented a quadrature-free implementation for the Euler equations. Otherapplications of discontinuous Galerkin methods to first-order systems can be found in [14.28].

The solution of second-order systems of equations using discontinuous Galerkin approximations has beendone mostly with mixed formulations, introducing auxiliary variables to cast the governing equations as afirst-order system of equations. This methodology was used by Bassi and Rebay [9.81 for the solution of theNavier-Stokes equations. also Lomtev. Quillen and Karniadakis in 131-34J and Wurburton et al. in [41] solvedthe Navier-Stokes equations discretizing the Euler fluxes with the DG method and using a mixed formulationfor the viscous fluxes. A similar approach was followed by Cockburn and Shu in the development of the LocalDiscontinuous Galerkin method 121[, see also a short-course notes by Cockburn [161. Other methodologieswhich also used mixed approximations for the diffusion terms were developed by Dawson 122[. and Arbogastand Wheeler [21. The disadvantage of using mixed methods is that for a problem in lR:,t, for each variable subjectto a second-order differential operator, d more variables and equations have to be introduced to obtain afirst-order system of equations.

In this paper we present a new method for the solution of convection-diffusion problems which is based onthe classical discontinuous Galerkin approximation for convection terms 115.17,27,29,30] (summarized inAppendix A) and on the technique developed in [7.10,38] for diffusion terms. This new methodology supports11-. p-. and I1p-version approximations and can produce highly accurate solutions. We explore the stability of themethod and we present a priori error estimates.

Following this introduction. Section 2 presents the model problem. domain partitions and spaces of basisfunctions. Section 3 introduces the variational formulation. a stability study in special mesh-dependent normsand an a priori error estimate. Section 4 presents numerical experiments which confirm the theoretical resultsfrom Section 3 and applications which include the solution of the Euler and Navier-Stokes equations. Finally,conclusions are listed in Section 5 and Appendix A contains a review of the classical t1iscontinuous Galerkill ..method for hyperbolic problems with a summary of a priori error estimates.

2. Governing equation and boundary conditions

We consider the following model problem: Let fl be an open bounded Lipschitz domain in [Rd, such as thepolygonal domain in IR:" depicted in Fig. I and let us consider a model second-order convection-diffusionproblem characterized by the following scalar partial differential equation

-v· (AVu) + V· (fJu) + (TIl = S in fl C IR:"

with boundary conditions

(7)

C.E. BaUlI/allll. J.T. Odell / COli/put. Methods Apl'l. Meck Ellgr/? 175 (/999) 311-341

/

Fig. I. Domain and boundarics-Notation.

II =f on To

(AVII)' 11 = g on To

3D

(2)

where fJ E (L x(D)" is the mass nux vector which may have contact discontinuities. (T E L x(il), (T> 0 a.e. in!l, S E L 2(11), and A E (L x(f})dXd is a diffusivity matrix characterized as follows:

TA(x) =A (x).(3)

a.e. in !l.The boundary an consists of two disjoint parts. I;) 011 which Dirichlet conditions are imposed, and 1~ on

which Neumann conditions are imposed: 1;) n /~ = 0. 1;) U ~ = an. and meas To> n. We make the followingassumption regarding the inflow part of the boundary r_:

/~) ~ T_ = {x E afl I fJ . fleX) <()a.e.} .

2./. Families of regular partitiolls

Since the discontinuous approximations are defined on partitions of the domain fl, we now establish notationsand conventions for families of regular partitions.

Let 9P = {~,(f})L >11 be a family of regular partitions of n c IRd into N='=N( 91;) convex subdomains n" suchthat for 9f, E flP.

i1 = U ii,.. and f},. n n,= 0 for e oFf·/,=1

(4)

The partitions 2Pf,({}) are regular in the sense that if 11,.= diam([~,). and Pc = sup{diam(s) : sphere s C f),.}, thenfor simplices

(5)

314 C.E. 8m/lIllill/l. IT. Odell I Cumpl/t. MetllOds Appl. Mech. Eilf?lK 175 (1999) 31 J-341

for every ~Ir(fll. and for quadrilaterals/hexahedra. if I;, = min{lengths of edges of iJfl,.}, cr,. = min{interiorangles between intersecting edges/faces of an,.}. then

cos(cr,l < I , (6)

for every ~h(fl).

Each subdomain fl,. has a Lipschitzian boundary Cifl .., which consist of piecewise smooth arcs plus verticesfor d = 2. and pieccwise smooth surfaces and edges for d = 3.

Let us dehne the interelement hound{//} by

(7)

On r.lll we define 11 = II" on (a{}~ n af2,) C T"" for indices e, f such that e >f

22 Broken spaces

Let us define the so-called broken spaces on the partition ?},,(f2):

(8)

if v E Hm(f~J, the extension of v to the boundary an.., indicated by the trace operation '}hv, is such that}'ov E H",-II\afl), In> 1/2. The trace of the normal derivative YIV E H"'-3I1(afl,), m > 3/2, which will bewritten as Vv . 11 1,111 ' is interpreted as a general ized flux at the element boundary an~..

With this notatio'n. for vln EH:l12'I'(flJ and vlnEH3:2+'(fl,), we Introduce the jump operator [.J defined- -' ,

on 1~,= fl,.n fl, ~ 0 as follows:

Ivl=(Xp)l"nnr -(Yov)l,nnr' e>f," 'f ' 'f "l

and the average operator (.) for the normal flux is defined for (AVv)' II E L 2(l,~.), as

I(AVv)' 11) = -2«(AVv) '1l)liJll nJ' .. + «AVv) 'Illl"n nr ), e >f.

" " J .:1

(9)

( 10)

where A is the diffusivity matrix. Note that Il represents the outward normal from the element with higher index.

3. The discontinuous Galerkin method

The aim of this paper is to present a discontinuous Galerkin formulation for convection-diffusion problems.Given thaI the formulation to be presented is built as an extension of the classical discontinuous Galerkinmethod for hyperbolic problems [15.17,27,29], we assume that the reader is familiar with the notation andformulations presented in Appendix A.

3. J. Weak formulation

Let W( ~,,) be the Hilbert space on the partition (jJ" defincd as the completion of H:lI2+,( PJ,,) under the noml11.11\\ defined as follows (induced by (16))

Ilull~v= III/II~+ Ilull~ + III/(T11211~n'

Ilvll~= 2:. f Vv· AVv dx + Ivl~.t;., '~E~ ~ .

(II)

(12)

and

C.E. BlJIlI/wf/tI. IT. Odetl / Compur. Methods Appl. Mccl1. ElIg,.g. 175 (/999) 311-3.JI

Illtll~ = IIIIPII/21~Jl + 2.: IVu' P/lplt f"I~.Jl. + 1111P . Ilil/"I(~.rfI,.Elf'"

Ivl~.r;" = 111-avl;ur, + Iha(Avv) '''I~ti)+ Ih-alv]I~.I;", + Ih"«AVv) ·")I~.J;"in·. "

Ivl~r= L V2

ds, for r E U;" J~. 1;n.}·

315

(l3 )

( 14)

The terms h ±". with /1' = 1/2. are introduced to IlllI1lmlZe the mesh-dependence of an otherwise stronglymesh-dependent norm. In (14), the value of 11 is 11)(2/1't) on J-;'. and the average (h, + h])/(2/1'1) on that part ofr.nl shared by two generic elements n, and D,.. the constant /1'1 was defined in (3). In (13), however, 11 is h)2on I;l' and the average (11" + 111)/2 on an, n a.q.

A consistent formulation of problem (1)-(2) is the following variational statement:

Find It E W( £1>,,) such that(15)

B(u, v) = L(v) 'rJ v E W( fJ>,,)

where

B(Il, v) = Jl'~"'h {t, lVv . Avu - (vv . fJ)u + vmJj dx

+f VU-(P'",.)cIs}+f «AVV)'I/1/-v(AVu)'Il)ds;,(],.If', to

+ f «(AVv) ''')Iu]- «AVu) 'lI)lv I) ds.fillt

It=: = lim II(X ± t::fJ), for x E T, I .E~O J I

see Fig. A.I in Appendix A, and

L(v)= 2.: f vSdx+ f (AVu)'llfd~+ f vgcls- f. vRP·Il)cIs.~E~ ~ ~ ~ ~

(16)

( 17)

REMARK. The space H,I,(fl) C W( £1>,,), and for II, \. E H ~(n) the bilinear and linear forms B(u. v) and L(v)reduce to those of the continuous Galerkin formulation. which is known to be unstable for convection-diffusionproblems. The use of discontinuous basis functions in combination with ( 16)-( 17), however. produces a methodwith superior stability properties.

3.2. COllservarion property

In this section. we prove that the formulation presented is glohally and locally conservative. Using the unityweight function 'rJ{~ E£Y>,,({})in (15). we obtain

L, (fJ 'Il)u d~+ too f(fJ' 1/) cis - LN g d~- t, (AVu)' 1/ cis = Jl'~"h L. (S - (m) dx ,

which is the expression of conservation at global level.If \¥e now use a weight function which is unity on a generic element n,., and zero at any other element, from

(15) we obtain

316 C.E. 8(/1111/(/1111. J.T. Odell I COli/put .• Helhods Appl. Meek E/lgrg. 175 (1999) 3/1-34/

1 (fJ'/)I/U~+f f(fJ'/)U~-f gU\·-f (AVII)'1Idsaf),.n}', ,'Jlen/', ;,u,.n/',. '-'Jl,.nJO

+ f . «P '11,.)11- - «AVII) '11» cis = f (8 - (TIl) d.r.dJl("n I inl fl,_

which is the expression of conservation at element level.

3.3. Existence and stability of'solllthm.\·

The existence of solutions to (15) l:an be established using the dassil:aJ Generalized Lax-Milgram theorem

15.371·

THEOREM 3.1. Let 'J{ alld CfJhe Hilbert spaces, and B : 'J{ X ~f}~ ~ a bilinear jimctional with the followingtitreI.' properties:

(i) There exist a constant M > 0 SI/clt that

B(II. u) -s;; MIIIIII/rilu II ,". V 1/ E fft. v E CfJ.

where 11.11l{' and 11.11'f: deflote tlte norms on :J(' and CfJ. respectively.(ii) There exist a constant 'Y > 0 SI/ch that

inf Slip IB(lI, u)1 ~ 'Y 'IIE:){ vEi!.'

11·"'1 Jr= 1 11"11'/;"'- t

(iii) And

Slip IB(u. u)1 > 0 .r,ER"

v=l'O. vE'fj.

Let L be a continl/ol/s linear fl/nctional 011 CfJ.Theil, there exist a I/nique sollltion to tire following problem:Find 1/ E :J{ slIch that

B(II. u) = L(v) V II E Cfj .

Moreover,

I111I11l(-s;; y IILII'f!'

An approximation of ( (8) consists of constructing families of closed (generally finite dimensional) subspaces.:Jt;, C itt. w" c CfJ. and seeking solutions IIh E Jt;, to the following problems:Find III, E ~, sllch that

(19) •Let us assume that the conditions of Theorem 3.1 hold for problem (18), and that (19) is an approximation to

(18). Then. B(.,.) and L(.) are continuous on Jt;, X CfJ"and ~,. respectively. It follows that (19) is solvable ifthere exist I'J, > 0 such that

and

inf Slip IB(I/. v)1 "'" I'J, •uE'Jfh r:E'/}'r

r"u·Jr;,= I 11"11'",,"'-1

(20)

Slip IB(II, u)1 > 0 ,IIE)("h

v=l'O, uECfJ. (21 )

A straightforward calculation reveals that the error II -II" in the approximation (19) of (18) satisfies theestimate

C.E. Bal/lllilllll, J.T. Odell / COlllplll. Method.\' Appl. Mech. Ellgrg. /75 (/999) 3/1-34/

1\11- uhll J( ~ (1 + M) inf.. 1111- 11'11 K .')'J, ",E](,.

3.4. Natllral norm

317

(22)

In this section we prove continuity of the bilinear form (16) with respect to the induced W-norm (II) whichdefines W(?P,,) through the process of completion.

THEOREM 3.2. The bilillear form (/6) call be bOllnded as follm}'s:

B(II. v) ~ 1IIIIIwllvllwwhere II.II,\! is the lIorm de.fined in (11).

PROOF. Let us first note that

2: J v(P'Il,,>II-d~= f Ivl(p'n)lI-d~+ f V(P·Il)lIds.arE (I'h ;jJ},.\/"_ lint 1'.,

then. we rewrite the bilincar form (16) as follows:

B(I/, v) = 2: f (Vv· AVu - (Vv . P)II + v(Ju) dxn,.E :1\ fl,.

+ L v(p· II)U ds + t «ftVV)' IlU - v(AVu) 'Il) ds+ n

+f «(AVv) . 1I)lu I - «AVu) . 1l)1 v 1 + Il - (p .Il)l v]) ds .1;111

(23)

•

introducing scaling factors (h"h -"') where necessary and using Holder's inequality, the above bilinear form canbe bounded as follows:

B(II. v) ~ Ilullwllvllll' 0

3.5. Altel'llillive definitioll rd'lIorms

Given that the natura] norm 11.1111' is a mesh-dependent variant of the H' norm, to perform a study ofconvergence and stability of solutions in the L 2 norm we need to use different norms for the trial and testspaces. In this section we show how the bilinear form can be bounded utilizing one norm for the trial space,which is related to the L 2 norm, and another norm for the test space which is related to the H 2 seminoml. Theset of norms to be presented in this section will he used to obtain an a priori error estimate of the en-or in thc L 2

norm .

THEOREM 3.3. The bilinear form (/6) can be bounded as .follows:

B(u, v) ~ 1IIIIIw,llvllw"where

Ilull~i = 111l11~j+ IIIIII~J+ 111I/T(j-IIII~.fI' j = I, 2 .

111I11~!1= 11I1~;jo>l + Ih(JIII~.T1)urN+ Ihoa ~ I (AVu) '"I~'/I)+lhall/ll~r +lhJa~I«AVII)'Il)I~r +lh"(II)I~r '

• lilt • 1111 • Ifll

(24)

(25)

(26)

(27)

3111

and

C.E. Bau/Ilalln. J.T. Oden / CO/llfJut. Methods Apl'l. ,Uecll. Ell!!,.!!. 175 (1999) 3/1-341

11I1~.Y'I,=L J 112dx, Ivl;f>''',= L J (V'(AVv)/dx,

~E~ ~ ~E~ ~

/ulz,.r = Lu2 d~. for r E {J;). I~. (,J.

IIIIII~, = Ih"II-I~.r~ + Ih"llIiI~.r;", + Ilhu{JII~!J'

Ilvll~" = Ih-"II3..1u+l~r_ + Ih-"I,B"lu+I~.ti", + Ilh-tIPlull~.Ii·

(28)

(29)

with 0'= 1/2,8=3/2. ,B,,=(p·n). [11]=(11- -11+) on ~nl' and u{3=IPi-'(Vu·f:J) whenlpl>O. otherwise1I{3 = O. In 11.111' and 11.11{3 the scaling parameter h is h,.l2 on afl,. n an. and the average (h,. + IIf)/2 on

J J

an,. n MH

PROOF. first. let us define a new set of norms for the diffusion operators. Considering only the diffusion termsin (16), we can usc integration by parts on the (jrst summation of the RHS to obtain

+ J «II)I(AVu) '111 + «AVv) '11)111]) d~- L J IIV' (AVu)dx,'illl JJeE ';;Ph fl,.

and the diffusive part of the bilinear form can be rewritten as follows:

8(11. u) = - .L J IIV· (AVv)dxfI,. E :1'" f 1,

+ J (2(AVv) '1111 - u(AVII)' n) d~+ f (AVu)' 1111 d~Ii, r"

+ r (2«(AVv) ·n)[IIJ - «AVII) ·n)lu 1+ [(AVu) '111(11» ds .linl

This bilinear form is now bounded using different norms for the test and trial functions. Looking for a norm onthe trial space equivalent to the L 2 -norm. we write

I I' I I) ·1" I' I I; -1 A"'" I' I" I'R(u. u) ~ II ~':"" U ;.:>\ + (h U ~./i)u/j.; + hal ( '1111)'n ~./i, + h I II 1~.t;n'

+lhl;(1<t't«AVu)'n)l~r +1h"(II)I~r )112, 1111 ' Ill!

X (Ih -I; (1<, ul~ti) + Ih-"2(AVu) . 1I1~.ri,urN + Ih -8(1<1 [u 11~.t;"1+ Ih-"(2(AVv)'n)I~" + Ih-"I(AVv)'nll~r )'n

, 11\1 • IIlI

and using the definitions of II.IIFIand 11·11,·"we obtain

8(11. u) ~ 1IIIIIv,llull,·,. (30)

Here again. we should note that the terms h "" and 11=8 were introduced to minimize the mesh-dependence of anotherwise strongly mesh-dependent norm. For a uniform mesh, (1< = 1/2 and n = 3/2 render a mesh-independentnorm. from definition (26). we see that 11.111' is closely related to the L 2 -norm,,

Let us now define a new set of norms for the convection terms. We first rewrite the bilinear form (convectionterms only) as follows:

C.E. BatlmwlII, J.T. Odell / COlllplll. Method.\' Appl. Meeh. ElIgr);. /75 (1999) 311-34/

B(u.v)= 2: {-J (Vv.{J)lItlx+f U({J'"")U-ds}~E~ ~ J~r

+ L {- f u({J '11")11 ds + J vV· ({JII) tlx + f V( {J' n,JII- ds}n,.EiY'" i,fl,. n,. ;,fl,.It'..

= - J'.. V({J '11)11 ds + fl~i' {t,. uV· ({JII) dx + f.n,\t'.. V({J '11,.)(11- -U) dI'}t· I.

= r vIP'lllud~+ J VV'({JII)tlx- r u'I{J·nl(II--II+)ds._ fl . I inl

319

Introducing scaling factors (11"11-") and (1/'11-8) where appropriate and using Hiilder's inequality. we obtain

B(u, v) ~VIIt"11 +I~r_ + II1"lujl~li,,, + Ilhllj3ll~).fl

X '/Ih -"I.Blllv +I~.r + Ih' "1.B"Iv' I~.t;", + Ilh-11{Jlvll~.1!where !3,,=({J·II).lul=(u- -li~) on li'II" and 1113= l/ll-I(VII' fJ) whcnlpl>O. otherwise 1113=0.

Using definitions (2R) and (29), we obtain

Finally. we have all the elements necessary to define a new set of norms for the convection-diffusion problem.Using the definitions of 11.1113' j = I, 2, given in (28) and (29). and the definitions of II.II\', j = 1,2, given in (26)and (27), we can bound the bilinear form (23) as follows: .'

B(u. v) ~ 1llIlIv,llvlll', + 1llIllj3lllvllj3, + 1llIlloflllv(Tllofl~ IlullllJvllw, .

By definition, 11.1111'1 is closely related to the L 2-norm, whereas II.ILI'! resembles the H2- norm with two scaling

parameters which are the local diffusivity and transport velocity D.

3.6. Poly"omial approximatiolls Oil partitiol/s

Let il be a regular master clement in IRd: e.g. il = (-1. Iyt. And let {Fn } be a family of invertible maps from[} onto fl, (see Fig. 2). "

For every element fl, E [!P". the finite-dimensional space of real-v.alued shape functions PC H"'( {l) is takento be the space Pp,,< fl) of polynomials of degree ~ p,. defined on fl. Then we define

Using the spaces PI,(fl,.). we can denne the finite dimensional space

(31 )

NI:J\I

Wp( 9I't) = [1 PI' (f1,) ,1'=1 t"

~>( [!P,,) C W( PJ,,), (32)

N( 91'1,) being the number of elements in partition g".The approximation properties of Wp( 91',,} will be estimated using standard local approximation estimates (see

/61). Let u E H"([~); there exist a constant C depending on sand 011 the conditions (5) and (6). but independentof u, II. = diam(fl,.). and p,.. and a polynomialu,> of degree p,.. such that for any 0 ~ r ~ s the following estimatehold:

Iµ'- ,

Ilu - IIplL.n, ~ C ~':-, Ilult..fl,.' S ~ 0,

where II.II..!I,. denotes the usual Sobolev norm. and µ = min(p" + I, s).

(33)

320

'I

-Aa ~

C.E. Ballmol/I/ . .I.T. Odel/ / Compllt. Methods 111'1'1.Mech. EI/grg. 175 (1999) JI f -341

Fig. 2. I\,'lappings !1~ n, and disconlinllolls approximation.

3.7. Discontinllolls Galerkin approximation

To construct approximations to the exact solution of problem (1 )-(2) in a finite dimensional space. we usethe following variational formulation in the space WI'( .':flj,):

Find l/ lJG E ~V;.(PJ,) such that

B(l/lJ(;> Vh) = L(v,) V vh E W,,( PJ,,)(34)

where B(. , .) and L(.) are defined in (16) and (17), respectively.The properties of the discontinuous Galerkin method presented in connection with (15) also hold for the finite

dimensional approximation (34): namely. solutions are locally or elementwise conservative. the associated massmatrices are block diagonal. and the spacc of discontinuous functions provides the basis to obtain solutions withpOlentially better approximation properties in the L 2_ norm.

Stability is one of the most important characteristics of a method for the solution of convection-diffusionproblems. The following section addresses this issue and provides a priori error estimation for solutions of (34).

3.8. A priori error estimation

Convection-diffusion Iransport problems usually exhibit drastic changes in naturc from diffusion dominatedto convection dominated zones. The error of diffusion dominated problems is better measured in the H I normbecause the associated physics depends on the solution gradient. such as heat transfer, viscous stresses. etc.:whereas the error in convection dominated transport is better measured in the L 2-norm, because the underlyingphysics depends almost exclusively on the solution values rather than on its gradient.

The range of local Pe values 10. I) represents a class of problems in which diffusion is dominant, and forwhich the W-norm converges to the V-norm as Pe -t O. The analysis of stability in the V-norm for diffusion

C.E. BlII1IIWfII1 . .f.T. Odell / COlIIl'lIt. Method" AI'I'I. Mecll. EII~/'g. J 75 (/999) 3/1-341 321

dominated problems was presented in 11{),3Xj, where optimal II-convergence rates are presented. Forcompleteness. Section 4 includes a numerical example which shows optimal convergence rate in the W-norm.

The following theorem presents an a priori error estimate for the range of high Pe number. where convectionis important. The reaction coefficient if is assumed to he zero. this is the worst case scenario because whenif> () the problem is more stable.

THEOREM 3.4. Let the soltltion to (15) be 1/ E fi"( p;,,(f})). wilh s > 3/2. and assl/me Ihat there exists K ~ 0and C" > {)slIch that

inf sup IB(II, v)1 ~ C p -.K, .HEW, f"EW.· P 1n.1

IIi/IIII', 11.. 11w, '" 1

(35)

lI'here Pm"x = max,.( p,). If the approximation eslimate (33) holds for Ihe spaces ~V,,(.0/1,,). then Ihe error ()f Iheapproximate .\'oll/Iion II[)(; can be hOl/nded as jillloll's:

(36)

where µ,. = min(p,. + I . .1'). € > {)arbitrarily .I'll/alt. p" ~ I. and the consla11l C depends on .\'and on condilions(5) lind (6). hw il is independent of 1/. h,. lind p",

PROOF. Let us first bound III/II\' and III/lip as follows:I I

and using (37)-(38). we hound 111/11\1' :I

111/11~1!1~ C L (111/11~.f1, + h~IIII~/l, + hlll/ll~t~hn..+ h3111111~Il+d),)·n,E'p"

(37)

(38)

(39)

Let lit, E "Y"UJ\) be a polynomial that approximates 1/ E H'( p;,,(fl)) in the norm 11.11\\.'1'To ohtain an upper houndof 1111 - 11)1 \\'1 ' we use (39) and bound each of the four terms in the summation using (33) so that the exponent ofh is the same in each term. Thus. we obtain:

s

s > 3/2. µ..-min(p. + I . .1'), € > o.

Finally, using the continuity and inf-sup parameters in (22). we arrive at the following a priori error estimate:

where.\' > 3/2, € > 0 arbitrarily small, µ", = min( p,. + I ..\'}, and C depends on s but is independent of u. h,., andp, D.

The error estimate (36) is a bound for the worsl possible case. including all possible data. For a wide range ofdata. however, the error estimate (36) may be pessimistic. and the actual rate of convergence can be larger thanthat suggested by the above bound.

322 C.E. HWllllallll. 1.T. Odell / Compl/t. Methods Al'pl. Mech. ElIgrf.:. 175 (/999) 31 J -341

0.1

0.01

e-o 0.001c;:

~c'":;;c0 0.0001'-'[]J[]J

le-05

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

p=2 -p=3 __u.

p=4 'p=5p=6 .--.

1e-061 10 100

Peelel1000 10000 100000

Fig. 3. BB( W,) "S. Peclct, discontilluous Galerkin.

3.9. Numerical evaluatio/1 of the in/-sup cOlldition

The numerical evaluation of the Inf-Sup parameter 11,associated with the nom1 II.II\\'I was done as described in[1O.38J. The bilinear form (16) with if = 0 was discretized in a one-dimensional selling with Dirichlet boundaryconditions using different number of elements N( [!Ph) and uniform p,. = p. I ~ e ~ N( 9h).

Figs. 3 and 4 show the Inf-Sup parameter '}'t, as a function of the Peclet number Pe = 1.8lh/a for different

0.1

0.Q1

0.001

c'":nc8 0.0001[]J[]J

le-05

1e,061

".' ...

".

-........'.: ....

10Peelet

1000 10000

p=2 -p=3 ----p=4 .....p=5p=6 - - -

100000

Fig. 4. HB(WI} "s. Peele!. conlilluous Galerkin.

C.E. RaUlIlalll/. J.T. Odell I Complil. Methods Appl. Mec/1. Ellgrg. 175 (1999) 31/-341

0.1

323

0.01

E0 0.001'i'

~C01'iiic0 0.00010(Q(Q

le,05

Pe=2 -Pe=10 ----

Pe=l00 .... ,Pe=1000

Pe=10000 ---

le,06o 2 3

p4 5 6 7

Fig. 5. BB(W,) v.s. spectral order. discontinuous Galerkin.

polynomial orders [1, Fig. 3 corresponds to the DG method and Fig. 4 lO the continuous ell Galerkin method.Figs. 5 and 6 shows the Inf-Sup parameter ')'J, as a function of [1 for several values of Pe number. These tiguresshow that the CO Galerkin method is unstable for large Pe numbers (a well-known fact) because the Inf-Supparameter ')'J, is inversely proportional to lhe Pe number. The DG method, however. is stable; the Inf-Supparameter ')'J, decreases only up to a given value when Pe increases.

0.1

0.01

Pe=2 -Pe=10 ----

Pe=100Pe=1000

Pe=10000 ---

E~ 0.001

~C01'iiic8 0.0001IIIIII

le-05

le-06o 2 3

p4 5 6 7

Fig. 6. HB(WI) vs. spectral order, continuous Galerkin.

324 C.E. BaUlI/allll. J.T. Odell / COli/put. Methods AI'I'I. Mech. ElIgrg. /75 (/999) 311-34/

4. Numerical tests

4. /. Scalar cm/l'ect;oll-d(fjil.\·;OIl problems

First. the h-convergence rate is evaluated solving the following convection-diffusion problem

{

a~1I all ,-..a ----;-+ -;- = a(47rr sin(47rx) + 411'cos(47rx)ih'- dx

II(X) = 0

for which the exact solution is II,.",,)X) = sine411'x).

The h-convergence rate (accuracy) is given by

on 10, IIat x = 0 and x = I

(40)

(41 )

The analysis of error in the solution of problem (40) with a = IO~ is presented in Fig. 7, which shows theW-norm of the en'or, and Fig. 8 shows the II-convergence rate for uniform meshes. The asymptotic convergencerate in this HI like norm is of order O(hP). which is optimal from the approximation point of view.

Fig. 9 shows the error in the solution of (40) with a = 10-~. In this case the error is measured in theWt-norm, which is an L2-like norm, and Fig. 10 shows an h convergence rate of order O(hp+t), which is againan optimal convergence rate from the approximation standpoint. This order of convergence rate is in agreementwith the eHor estimate (36).

The next test cases involve the solution to the following singularly perturbed convection-diffusion problem

{

(J~II (JII-a --:;-+ -;- = S on /0, II

(Jx" oX

11(0) = 1 u(l) = 0

where S = {O, I}. and the corresponding exact solutions are IImJr) = {I.x} - (exp(xla) - I )/(exp( I/a) - 1).The domain is discretized with 10 elements (h= 1/10), and a=II/Pe. Fig. II shows the DG solution for

p=2 -p=3 .m

p=4 .....p=5p=6 - ..p=7 ...p=8 '

1e'05

1e'10

8 161/h

32 64

Fig. 7. W-nonn of the error: -10' iJ'u/i!x' + all/iJx = S.

c.E. {/allJ/l{lI1I1. J.T. Odell / COIllIJ/lt. Methods Appl. Mecll. Ellgrg. 175 (/999) 311-.141 325

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

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

9

8

7e-(;c:

~ 6OJiiicr. 5

Cl><>c:Cl>

4 .~Cl> r--;·~,->c:0U 3

2

o8 16

1/h32 64

Fig. R. H convergence rate: -10' il'lI/ilx' + c1l1liJx= S.

s = () and Pe = 10, and Fig. 12 is a close up view of the two rightmost elements. These figures showcualitatively the weak approximation of Dirichlet boundary conditions. This type of approximation isfundamentally different from approximations in the H I norm; as it is shown in the following examples, in thepresence of under-resolved boundary layers the DG approximation does not pollute the entire domain withSPUriOllsoscillations. and the approximation in the L 2 -norm is very good.

p=2 -p=3p=4 ....p=5p=6 m._p=7p=8 ......

~ 1e-05_I

]j

.......

18,10

8 161/h

32 64

Fig. 9. W"norm or the error: -10' iJ"lt/ilx' + (tll/flx = S.

126 C.E. Hal/lI/lIl/lI. IT. Odl'1l / Caml'l/t. Methods Apl'l. Meek Engrg. 175 (/999) 3/1-341

E15<;:

~Ql

1il0:Qll)c'"OJiii"c8

9

8

7

6

5

4

3

2

;.c. ~ __

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

o8 16

1/h32 64

Fig. 10. If convergence rale: _10" iJ'lIfi)x' + ol//ilx = S.

Fig. 13 shows the solution for S = ] and Pe = 100 obtained with the DG method, and Fig. 14 shows the samesolution at the rightmost elements. Fig. 15 shows the solution to the same problem obtained with the ('o/lt;nuou.l'Galerkin method, and Fig. 16 shows this solution at the rightmost elements. Note that the oscillations of thecontinuous Galerkin method pollute the entire domain, whereas the DG method only presents small oscillationon the rightmost element. almost without pollution in the rest of the domain. This is very important for adaptive

1.4

1.2

0.8

:J

0.6

0.4

0.2

00 0.2 0.4 06

x0.8

exact -p=2 --".P=3 .p=4 .p=5 _m

p=6,m

1.2

Fig. II. Discontinuous Galerkin: Ii = 1/1 n. Pe = IO().

C.E. Bal/manll. J.T. Oden / Compl/t. IHetllOds Appl. Mech. Engrg. 175 (/999) 3/J-34/

1.4

127

1.2

0.8

:J

0.6

0.4

0.2

00.8 0.85

exact -p=2 ----p=3 .....p=4p=5 -.-.-p=6 ,.-.,

0.9x

0.95

Fig. 12. Discontinuous Oalerkin (rightmost elements): h = 1/10. Pe = 100.

strategies. because error indicators for the DG technique are very simple and accurate due to the lack ofpollution in the solution.

Now. we present the solution to a convection-diffusion problem with a turning point in the middle of thedomain. The Hemker problem is given as follows:

1.4

1.2

0.8

0.6

0.6x

0.8

exact -p=1 ---"p=2 .....p=3p=4 --.-p=5 _.m

p=6 ....

1.2

Fig. 13. Discontinuous Galcrkin: h = I flO, Pe = 100.

32RC.E. Baumali

ll. J.T. Oclm / Compil'. Ml'thods Al'l'l .• Heel!. EIlf{ff{. 175 (/9991 3/1-341

1.4

1.2

0.8

::>

0.6

0.4

0.2

00.8 0.85

exactp=lp=2p=3p=4p=5p=6

0.9x

0.95

Fig. 14. Discontinuuus Galerkin (rightmost elements): h = I flO. Pc = Ion.

{

fl2u flu ,a ----, + x -:-) = -a1T- COS(1TX) -1TX sin(1TX)nx- eX

1I( - I ) = - 2 . u( I ) = 0

on lO, 1\ (42)

0.6

1.2

exactp=1p=2p=3p::4p::5p=6

O.B

.: ;.

) ,: I,··t :.

\ ~'.: II I

\

,.L,."'.~:"":~

0.4

..

0.6x

Fig. 15. Continuous Galcrkin: h = 1/10. Pc =' 100.

0.2

..rt

o o

0.2

0.4

O.B

1.2

14

C.E. Balill/alll/. 1.7'. Odell I Comput. MetllO'/.I· Appl. Mech. ElIgrg. /75 (/999) 3/1-341 329

1.4

!~/. "~,

1.2~ J'"i:

0.8

0.6

0.4

0.2

o0.8 0.85 0.9

x

exact -P=l ----p=2p=3p=4 - -.-p=5 ---q

p=6

0.95

Fig. 16. Discontinuous Oalerkin (rightmost elements): II = 1/10. Pe= 100.

for which the exact solution is

u(x) == cos(m:) + erf(x/ ,,'2a)/elf( I /~) .

Figs. 17 and 18 show solutions to the above problem (a == 10-10 and" == 1110) obtained with continuous anddiscontinuous Galerkin, respectively.

exact -p=2 m_

p=3p=4 ...p=5

2.5

2

1.5

0.5

o

-0.5

,1

,1.5

,2

,2.5o 0.2 0.4 0.6 0.8

Fig. 17. Hemker pruhlcm: a = 10- '" and h = 1/10. continuous Oalerkin.

330 CE. BaumQIlIl . .I.T. Oden I Comput. MerlJods App/. Meek Ellgrg. 175 (1999) 3/1-34/

2.5

2

1.5

0.5

o

,0.5

,1

,1.5

·2

,2.5o 0.2 0.4 0.6 0.8

exact -p=2 ----.p=3 .....p=4p:5 m._

Fig. IH. Hemker problem: a = to-Ill and II = 1/ 10. discontinuous Galerkin.

The numerical experiments show that for cases where the best approximation to the exact solution in theL2-norm is better represented using discontinuous functions, the DG method performs better than the continuousGalerkin method. This is in general true for solutions to convection dominated problems with boundary layers.If the quantity of interest is the gradient of the solution at the boundary layers, the DG method provides anexcellent error indicator which allows to capture steep gradients using adaptive refinement.

The next test case involves a two-dimensional irrotational incompressible flow around a cylinder with centerat (0,0), radius I, and undisturbed flow velocity (1,0). For this case. the exact solution given in terms of thestream function is

rjJ = Y( I - x2 ~ ./ ) ,

and the velocity field is given by 1I = aI/II c1y. and v = -flrjJl (Ix. The value of the stream function rjJ does notchange along a streamline

Fig. II). Subdomain A. Fig. 20. Subdomain B.

C.E. Sal/II/alll/. iT Odell I COII/I'lit. Methods Appl. Mech. EIlgrf!. 175 (/999) 3//-341 331

1e,05

1e,lO

1e,13

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

~..-......-..-.....

....

2 4H/h

8 16

ill/I (/1/1U-iI +U-:-j =0.

X ry

Fig. 21. L' -l1orm of the error on subdomain A.

and 1/1 is also a solenoidal field, namely. it satisfies the condition !l1/l = O. Therefore, we can set the followingconvection diffusion problem: Find 1/1 such that

10

9

8

7

CIliii 6ccCIluc: 5CIlOJiii>c:0 4()

3I ,

:l=:1 4

.......

8H/h

16

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

Fig. 22. Convergence mle in L '-norm on subdomain A.

332 C.E. Hal/111I11111. J.T. Odefl / C(I/!/I'llt. Methods Appl. Mec". Eflgrg. 175 (1999) 3/1-34/

p=2 -p=3 un

p=4p=5 .p=6 - -p=7 m

u

p=8p=9 ......

le-05

le,lO

le-13

2 4H/h

8 16

{

(J ifl ell/I-D.r!/+II-;- +U-;-=O,

dx rly

1/1=)'(I-~). x' + y'

Fig. 21. L' -norm of the error on subdomain B.

III f}

on aD,(43)

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

, .

.... , ...

10

9

8

7

Ql

6'"a:<l>U

5c:<l>Cl

IQ;>

4c:0U

:~ ..................

~

J ,42 8

Hlh16

p=2 -p=3 '''--p=4 .p=5 .....p=6 -.--p=7 m

p=8 ."p=9 ......

Fig. 24. Convergence r'lte in L'-norm on suhdomain R.

C.E. !JOIIIII({III1 • .I.T. Or/ell / COIIII'III. Method.\' Al'fll. Mecll. ElIgrg. 175 (/999) 3/1-34/ 333

h-padapt I

Error

.800E-021

.738E-02

.677E-02

.615E-02

.554E-02

.492E-02

.431 E-02

.369E-02

.308E-02

.246E-02 I

.185E-02

.123E-02 I

.616E-03 i

.294E-06 I

Fig. 25. Pointwise error with adaptive: refinement: max fl. = I.

h-padapt

ErrorI

.120E-03 I

.111E-03

.102E-03

.923E-04

.831 E-04

.738E-04

.646E-04

.554E-04

.462E-04

.369E-04

.277E-04

.185E-04

.923E-05

.342E-09

Fig. 26. Pointwise: error with adaptive refinemcnt: max 1'. = 3.

334 C.E. Hili/lilli/III • .I.T. Ode" I COlllplll. Methods Appl. Meek E,,!!}'!!. 175 (/999) 3/1-341

where

(> >

1/ = I_X· - Y-)(x2 + /)2' and

arc given in fl. The numerical evaluation of l1-convergence rates (accuracy) is done on the subdomains shown inFigs. 19 and 20.

Fig. 21 shows the L 2-norm of the error and Fig. 22 11 convergence rate for uniform refinements of subdomainsA. Figs. 23 and 24 are the counterparts on subdomain B. For low p the convergence rate is 0(11"+1) for p oddand 0(11") for p even. For high p, however, the tendency secms to be towm;c!s a convergcnce rate of order0(11"+ I) regardless of p, but the asymptotic convergence rate is not reached for high fJ in the finest mesh due toround-off errors.

Figs. 25 and 26 show pointwise error in the solution of problem (43) using adaptive refinement with a maxorder fJ,. = I and p,. = 3, respectively. From these figures it is noticeable the reduction in pointwise error with theincrease in polynomial order.

Summarizing, the numerical experiments confirm the stability and high accuracy that the method can deliverfor the class of problems considered, including the lISC of 11 refinements and p enrichments.

Mach

.173E+01

.160E+01

.146E+01

.133E+01

.120E+01

.107E+01

.935E+OO

.803E+OO

.670E+OO

.538E+OO

.406E+OO

.274E+OO

.142E+OO

.972E-02

Fig. 27. i'\'ACAOO 12. M. = 1.20. incidence 7.(}o. mesh ~nd Mach number.

C.E. Bal/lllallll. iT 0111'11 I COlllpl/t. Methods Appl. /l1<'l'h, ElIgrg. /75 (/9W) 311-3·11

4.2. Erlensiolls 10 lIolllinear systems

335

The basic approach outlined thus far for convcction-diffusion problems can be cxtended to nonlinear systemsof equations with and without diffusion terms. For example, in [10-121 we prescnted extensions to the Eulerand Navier-SlOkes equations. Some representative results are shown in Figs. 27-30.

Thc first sct of figures corresponds to the transonic now around a N ACAOO 12 profi Ic at M 7. = 1.2 andincidcnce anglc 7.0°. The inviscid Euler equations are used to model this problem. Fig. 27 shows the finalmcshafter adaptation with thc Mach number distribution, and Fig. 28 shows the pressure distribution. These figuresshow the advantage of using discontinuous approximations even in cases where the shock waves are obliquewith respect to the original mesh, note that the oblique shock wave that starts at the trailing edge of the airfoil isvery well resolved.

The second sct of figures corresponds to a driven cavity problem at Re 7500 as dcscribed in [241. The solutionis obtained with a mesh of quadratic elements which is equivalent (in number of degrees of freedom) to a meshof 60 X 60 linear elements as that shown in Fig. 29, this figure also shows the pressure distribution on thebackground. Note that the pressure range shown is considerably narrower than the actual range, which is verywide because of the prescnce of singularities at the top corners of the cavity. The cutoff values [Pmin' Pm;!,]applied to the range of pressure allow to observe small changes within the domain, excluding the areas adjacent10 the top corners. Fig. 30 shows the strcamline pattern. Velocity profiles through the middle vertical andhorizontal planes are in close agreement with those rcported in 124].

M=1.20

Pressure

.124E+01

.116E+01

.108E+01

.100E+01

.926E+OO

.848E+OO

.770E+OO

.693E+OO

.615E+OO

.537E+OO

.460E+OO

.382E+OO

.304E+OO

.226E+OO

Fig. 2g. NACAOOI2. M1 = 1.20. ineidence 7.ct. pressure.

330 c.e:. /1(/11111(//111, .I.T. Odell I COli/filII. Metl!od,' I\l'fll. Meel!. Ellgrg. 175 (/999) 3/1-34/

Pressure

Rey7500

z

Lx

.833E+02

.718E+02

.603E+02

.488E+02

.373E+02

.258E+02

.144E+02

.287E+01-.862E+01-.201 E+02-.316E+02-.431E+02-.546E+02-.661 E+02

Fig. 29. Driven cavity at Rc = 7500. mcsh and pressurc contours.

5, Concluding remarks

The salient properties of the discontinuous Galerkin formulation introduced in this paper can be summarizedas follows:

• the method is capable of solving convection-diffusion problems with an lip-approximation methodology. Ifthe local regularity of the solution is high the p-approximation can be used and the method delivers veryhigh aCl:uracy; otherwise the h-approximation can be used and the error is rcduced by local refincment ofthe mesh;

• approximate solutions of unresolved flows (e.g. boundary layers) do not suffer from widespreadoscillations, for these cases the treatmcnt of interelement boundaries prevents the appearance and sprcadingof numerical oscillations;

• stability studies and numerical tests demonstrate quantitatively and qualitatively the superiority ofdiscontinuous solutions over continuous Galerkin solutions for convection-diffusion problems;

• discrete representations are stable in the sense that the real part of the eigenvalues associated with the spacediscretization are strictly negative, a property that all9ws the use of time marching schemcs fortime-dependent problems and also allows to solve steady state problems with explicit schemes. Manydiscontinuous approximations of convection-diffusion problems do not have this property;

• an a priori error estimate for high Pe numbers indicates that the method delivers optimal h-convergcnce rate(accuracy); and

C.E. BOIllIUIlI/I. J.T. Odell I COII/pllt. Methods IIppl. Meeh. ElIgrg. 175 (/999j 31/-341 137

z

Pressure

Rey7500

.833E+02

.718E+02

.603E+02

.488E+02

.373E+02

.258E+02

.144E+02

.287E+01-.862E+01-.201 E+02-.316E+02-.431 E+02-.546E+02-.661 E+02

Fig. 30. Driven eavity at Re = 7500. pressure contours and streamlines.

• contrasting other tcchniques that use artificial diffusion to improve the stability of continuous Galerkinapproximations (e.g. SUPG), thc present discontinuous Galerkin method does not introduce such mesh-sizedcpendent terms in thc governing equations, which allows for approximation with unlimitcd order ofaccuracy; namely, the order of accuracy grows lincarly with the order of the basis functions p.

Acknowledgment

The support of this work by the Army Research Office undcr grant DAAH04-96-I-0062 is gratefullyacknowledged.

Appendix A. The discontinuous Galerkin mcthod applicd to thc convection cquation

This appendix includes a summary of the discontinuous Galerkin method applied to linear hyperbolicproblems. It includes error estimates and deflnitions of norms thai are used for the analysis of hyperbolicproblems 129,27.15]. We have also included a weak formulation which has not been given the importance that itdeserves in the literature.

338 C.E. BOIIIIIOIIII. J.T. Odell I COlllpl/t. Ml'/IIOd.\' l\pl'l. Mel'il. ElIgrg. 175 (/999) 3/1-341

A.1. Review of a model govel'llillg equation and fhe associated .finite element formulation

This section describes the basic governing equation and Discolltinuous Galerkin formulation that have beenutilized in the literature to obtain a priori error estimates.

Let n be a hounded Lipschitz domain in Iffitt. slich as the polygonal domain in 1ffi2 depicted in Fig. A.I. Thegoverning equation is the following linear scalar hyperbolic conservation law (strong form)

V' (PII) + (TIl = S in fl C IR"

u =f on 1'-.(A. I )

where {J is a COllstmll 1111it velocity vector. (T E L '"n I (T > () a.e. in fl, S E L 2(fl). f E L 2(1'-.), 1'- = {x Ean I ({J . 11 lex)< O}. T+= (jfl\I'-, and 11 represents the outward normal to an. The assumption of {J being acO/l.I·fantullit vector was used in the literature to obtain a priori error estimates, but in general we do not makethis assumption.

A.I.I. DisCOlllilluous Galn'kill formulatiollThe partition of fl and all fhe related /TOmenelafure are presented in Section 2.1.Let V( 91>1» be the space of functions

(A.2)

The discontinuous Galerkin ti.mnulation can be stated as follows:

Find u E V( 9,,) such that

f u(V· ({Ju) + (TIl - S) dx - { U(II + - U -)(fJ '11) tl~fl,. r,n ;.\t'-

-J u(u+-f)({J'Il)ds=O;,n ,:nt'_

(A.3)

VuE V( ;jJ(,) .

yy

Fig. A.I. Physical problcm and symhnlic fCllrescntatiun of 1/'.

C.E. /JaIl/IU/IlII. J.T. Oelen I Compll/. Melhod.\· Appl. Mecl1. Engrg. 175 (1999) 311-341

where

iI[},~ = {x E iln.1 fJ . I/,(X) < O},

II~ = lim II(X± EfJ), for x E (Ill = ( U an,.)\afl .£-ioO+ fl ..E::jIh1fl\

A.2. D;s('otll;1/1I0Il.l' GlI/c:rkin OpproX;lIIat;Otl

339

Let Pt, (!.~,) he the space of of polynomials III on e1emenl n. E f!}h(fl) defined through a transformation"-t ' 'r/J = 110F fl,.' where r/J are P?'ynol1lials of degree ~pc defined on the master element fl, and the transformation

Fu is such that fl, = Fu.([})'Let us define a finite-~Iirnensional subspace of V(gill,) as follows:

With ~,U!l\), the linite element approximation can be statcd as follows:

Find II" E \/( gill,) such that

-J V/,(lIt~ - f)(fJ '/I) d.1 = 0;'ll ;:n/'-

Here. we should note that under the assumption of polygonal subdomains and constant transport velocity. theabove system can he solved element hy element, in one sweep.

A.3. A priori error estill/ates

For a regular discretization consisting of triangles and quadrilaterals. Lesaint and Raviarl [291 proved thefollowing a priori error estimate in the L 2 -norm:

(A.4)

wherc

h = max diam(fl,,).fl,.EY'"IU)

and 11.1/1

, + t reprcsents the classical Sobolev norm for If/'+ t (f2).Using mesh dependent norms, Johnson and Pitkaranta 1271 obtained a sharper estimate

1111" -1111" ~ 11111/, -1111113 ~ ChP-r11211111It,+1 .

IIlvlll13={ 2: [h"lIv13II~.JI,+lIvll~fl,+«v+ -V-»;ulr, + «V»~J),NJ)l}11lfI..E Y',.

where v# =Vv . fJ. and

(A.5)

340 C.E. BlI/IIIUIII1I. J.T. Odell I Call1pl/t. Methods ApI'/. Mech. E1Igrg. /75 (/999) 31/-34/

For the casc in which all elements are rectanglcs. Lesaint and Raviart 129J proved the following error estimate

(A.6)

and for the case in which all elements are triangles. Richter 140] proved (A.6) assuming uniformity in thetriangulation.

Bey and Oden 115] proved the following h-p a priori ClTor estimate

(A.T)

where

J}I12- 2 2

- V » ;'JI,lr_ + «v» ilf},.nan .

the previous estimate is valid if evcry elemcnt in the partition has order fJ,. ~ I.

A .4. Alterl/{I/ive discontin/lo/ls Galerkin fomrulalion

[n general, when the velocity field fJ is a function of x. and/or the partition !J>h(fl) contains elements withcurved boundaries. the finite clement formulation will not he uncoupled and the element by element techniquecan not be applied. It is convenient to utilize the following conservative formulation obtained when thedivergence term is integrated by parts:

Find II E V(?Ph) such that

J' 1-(VV'{J)/I+V«(TfI-S)]dx+J V/I-(fJ·Il,.)d~JI,. 01fl,. 11'_

which can also be solved e[cment by element in one sweep when the transport velocity is a constant vector andthe elements are polygons, otherwise it is a coupled system of equal ions and has to be solved as such.

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