I'~-ELSl:VIER Compu!. Methods App!. Mech. Engrg. t:n (1996) 25<)-286

Computer methodsin applied

mechanics andengineering

hp- Version discontinuous Galerkin methods for hyperbolicconservation laws

Kim S. Bey". J. Tinsley Oden b. *"NASA LlIllgley Resellrch Cemer. Hamptoll. VA 23681. USA

bn,e Texlls Institmc for ComplltlitiolllllllmJ Applied Mll/hemlitics. Ullil'f!rsity of Texlls (II A,,-,till. AIIStill. 7X 78712. USA

Received 21 July 19<)5

Ahstract

Thc devclopment of hp·version discontinuous Galerkin methods for hyperholic conservalion laws is presented in this work. Apriori error estimates are dcrived for a model class of linear hyperbolic conservation laws. These estimates arc obtained using ancw mesh-dependcnt norm that rel1ects thc dependcnce of the approximate solution on thc local element size and the local orderof approximation. The results generalize and extend previous results on mesh-dependent norms to hp-version discontinuousGalerkin IIlcthods. A posteriori error estimates which provide hounds on Ihe actual error ,lrC also developed in this work.Numerical experiments verify the a priori estimates and demonstrate the effectiveness of the a postcriori estimates in providingreliable estimates of the actual error in the numerical solution.

I. Introduction

Both the practitioner using computational tluid dynamics in engineering design calculations and thescientist grappling with gaps in the theoretical foundations are aware that much remains to be donebefore the subject can bc put on firm ground. This is particularly true in the theory and numericalanalysis of hyperbolic conservation laws. vital in gas dynamics and compressible fluid mechanics and afundamental component in the solution of the Navier-Stokes equations for compressible flow. Therethe numerical solution of hyperbolic systcms is confronted with a list of major difficulties and questionsthat have been under study for many years .

These include classical problems of numerically resolving shocks and discontinuities. characteristic ofsolutions of hyperbolic problems. while simultaneously producing high-order. non-oscillatory resultsnear shocks and elsewhere in the solution domain. Moreover. the basic issue of quality of numericalsolutions is fundamentally important: how accurate arc the numerical simulations and how does oneobtain the most accurate results for a fixed computational resource? These questions lie at the core ofmodern adaptive methods that aim to control the error in the computed solution and \0 optimize thecomputational process. In addition. methodologies that attempt to address thcse issucs cannot belimitcd to one-dimensional cases: they must be extendible to problems involving realistic geometries.boundary and initial conditions in arbitrary domains in two and three dimensions. Finally. there is theissue of computational efficiency. Modern numerical schemes for large-scale applications should bereadily parallelizable for implementation in emerging multi-processor architectures.

• Corresponding author.

0045-782S/96/S15.IKI © 1996 Elsevier Scicnce S.A. All rights rescrved5S DJ 0045- 7825(95 )(1l1944- 2

260 K.S. Bey. J. T. (Jdell I Cumpl/t. Mt'tllOds Appl. Mech. Ellgrg. 133 (1996) 2.'i9-186

This work addresses some of these issues for a model class of hyperbolic conservation laws for whichconcrete mathematical results. methodologies. error estimatcs and convergence criteria can bedeveloped. The basic approach dcveloped in this work employs a new family of adaptive. lip-version.finite element methods based on a special discontinuous Galerkin formulation for hyperbolic problems.The discontinuous Galerkin formulation admits high-order local approximations on domains of quitegeneral geometry. while providing a natural framework for finite element approximations and fortheoretical developments. The use of hp-vcrsions of thc finite element method makcs possibleexponentially convergent schcmes with very high accuracies in certain cases. The use of adaptivelip-schemes allows h-refinement in regions of low regularity and p-enrichment to deliver high accuracyin smooth regions. while kceping problem sizes manageable and dramatically smaller than manyconventional approaches. Thc use of discontinuous Ga1erkin methods is uncommon in applications. butthe methods rest on a reasonable mathematical basis for low-order cases and have local approximationfeatures that can bc exploited to produce very efficient schemes. especially in a paralleL multi-processorcnvironment.

Among the carliest work on finite element approximations of hypcrbolic problems is thc classicalpaper of Lesaint and Raviart [14\ which introduccd the discontinuous Galerkin method for lincar scalarhyperbolic problems. This work contained the first a priori error cstimates for II-version methods basedon elements of arbitrary. but uniform. polynomial order p. In their work. sub-optimal error estimates.with a loss in global accuracy of O(h) in the L c-norm. were obtained.

A detailed analysis of II-version discontinuous Galcrkin methods for linear scalar hyperbolicconservation laws was contributed by Johnson and his collaborators [12. 131. There. quasi-optimal apriori estimates showed a global accuracy of 0(htJ+1

/c) in mcsh-dependent norms. This work provided a

gcneral approach to the mathematical analysis of these methods that proved to be invaluable in thepresent work. Among the results established in the prescnt study are dcvclopmcnts of ncw a priorierror cstimatcs for hp-vcrsion discontinuous Galcrkin finite clement approximations of lincar. scalarhyperbolic conscrvation laws. Thus. this study extends and gcncralizcs thc rcsults of Johnson and othcrsto p- and hp-version finite clements.

Discontinuous Galerkin methods have also becn extcnded to non-linear hyperbolic conscrvationlaws. Cockburn. Shu and collaborators [8. 91 usc Runge-Kulla schemes for advancing the solution intime and a local projcction to guarantce that the total variation of the solution remains boundedthroughout the evolution proccss. The cmphasis of their work. however. trcats the discontinuousGalcrkin methods as finite volume methods, that is. focusing on the accuracy of the element meanvalues. In recent work on p-adaptive. parallel. discontinuous Galerkin methods, Biswas et al. [61 definea broader class of projections for imposing total variation boundedncss on the entirc solution in anelement by limiting higher moments of the solution. Preliminary work on parallelization stratcgies forhp-adaptive discontinuous Galerkin methods as well as alternate local projection strategies are alsoinvestigated by Bey [51.

The power of adaptivity to efficiently improve solution accuracy was recognized early on in thedevclopment of unstructured grid mcthods for hyperbolic conscrvation laws. Thcse II-adaptive methods,based on refinement/derefinement of an initial mesh [10. 151 or a complete remeshing of the domain[19), continue to be the preference for realistic How simulations. With an emphasis on resolving certainfeatures of the solution. many refinemcnt indicators have been proposed whieh are based on some apriori knowledge of the behavior of certain phcnomena. Typically. thcse indicators are loosely based oninterpolation error estimatcs applied to key variables. While this approach may provide somc relativemeasure of the local error in the solution. it does not in general provide a reliable estimate of the actualerror in the approximate solution and can be grossly in error. In this work. the element residual methodis applied to the model hyperbolic conservation law to derive a posteriori error estimates which arecomputed locally on a single element and contribute to a global estimate which is accurate enough toprovide a reliable assessment of the quality of the approximate solution.

Following this introduction. a new formulation of the discontinuous Galcrkin method is given for amodcl class of steadY-Slate, scalar. linear hypcrbolic problems in two dimensions. There. a notion oflip-dependent norms is introduccd which generalizes to lip-methods the idea of mesh-dependcnt normsused by Johnson and Pitkaranta [13]. Conceptually. one considers a partition of a domain fl C PIlc into

K.S. /J.,y. J. T. Odell I Compllt. Methods Appl. Mech. ElIgrg. 133 (1996) 259-286 261

finite clements and assigns to each element K positive numbers hl\ and Pl\ which are dcsigned to appearin coefficients of a mesh-dcpendent norm in a way to optimize subsequent estimated convergence rates.The numbers "K are identified with the element size and PK arc identified with the maximum spectralorJers of the shapc functions used in approximations over K. A priori error cstimates are derived inthcse norms.

In Section 3. the subject of a postcriori error estimates for the model problem is investigated. Ancxtension of the error rcsidual method to hyperbolic conservation laws is described. In the presentinvestigations. two typcs of estimates are produced. one which delivers an upper bound to the globalerror in a suitable mesh-dependent norm and a lower bound in anothcr rclated norm. Theorems arcproven which establish that these cstimates are indecd valid bounds on appropriate measurcs of theapproximation error.

Section 4 is devotcd to numerical experiments and testing of the theoretical results developed inearlier sections. Several model problems in two dimensions are studied. The numerical results exhibitsignificant features of the thcory and the methodologies devcloped: (I) the asymptotic rates ofconvergence predicted by our theory of a priori estimates arc fully confirmed by the computer ratcs; (2)exponential rates of convergcnce or super algebraic rates are observed. justifying finally the decision touse non-uniform lip-meshes for these types of problems: (3) the a posteriori estimation methodsproduce good estimates of the actual error. with effectivity indices near unity in many cases. and withremarkably good local crror indicators in most of the cases considercd.

2. The discontinuous Galerkin method

The discontinuous Galerkin methods are valid for hyperbolic systcms of conscrvation laws in multiplespace dimensions. The methods are based on a Galerkin formulation applied to a single clementresulting in an approximation that is discontinuous across element interfaces where continuity ofclement boundary fluxes is weakly enforced. The methods can also be viewed as finite volume schemeswhere the flux on an element interface is replaced by a numerical flux which is a function of the solutionin cach of the interfacc elements. The degrees of freedom for the higher-order polynomial approxi-mation are obtained by solving the conservation law. however. not by reconstruction of element meanvalues.

For clarity of presentation and for the purposes of analysis. we limit the discussion to a scalar linearhyperbolic conscrvation law. Wc begin with a detailed dcscription of the method for a linear modelproblem and prove some important properties. Next. we dcscribe a finite clement approximation andderive a priori error estimates for lip-version discontinuous Galerkin methods. The analysis of themcthod relies upon thc linearity of the model problem. and requires some modification for thenon-linear case.

2. 1. A linear model problem

We consider a linear scalar hyperbolic conservation law on a convex polygonal domain n. LetP = (Pl' /3z)T denote a constant unit velocity vector. The domain boundary an with an outward unitnormal vector II consists of two parts: an inflow boundary L = {x E an I fJ . n(x) <O} and an outflowboundary r, = an\L. Let II denote the quantity that is to be conscrved in fl and considcr the followinghyperbolic boundary-value problem:

P .VII + au = f in!l C '!Ii ~

/3 . 1/11 = fJ . Ilg 011 r_(1)

(2)

where f E L ~(!l). gEL ~(L). a = a(x) is a bounded measurable function on n such that 0 <au ~ a(x).While this is the simplest of hyperbolic conscrvation laws. solutions to (1) may contain discontinuiticsalong characteristic lines x(s) defined by ilx/ as = (3. Solutions to (I) belong 10 the space of functions

262 K.S. Bey. J. T. Odell I Compl/l. Methods Appl. Mech. L~·ngrg. J33 (l99(,) 259-286

2.2. Notation

Throughout this work. notations and convcntions standard in the literature on the mathematics andapplication of finite elements are used. Particularly. H'(il) denotes the usual Sobolev space of functionswith distributional derivatives of order s in L ~(n), equipped with the norm.

{

( } 1I~HulL-.f) = Jj L ID",,12dx

f) I"'''''Other notations and norms are defined in this section and where they first appear in the text.Throughout. C is used to denote a generic positive constant. not necessarily the same at eachoccurrence.

The starting point for the discontinuous Galerkin method is (1) defined on a partition of n. Let fiP"denote a partition of il into N = N(r?h) subdomains K with boundaries ilK such that:

(i) N(r?Jr)<'X(ii) f1=U{K: KEfiPJr}

(iii) For any pair of elements K. L E PPJr such thai K ':/: L. K n L = 0(iv) K arc Lipschitzian domains with piecewise smooth boundaries. The outward unit normal to aK

is denoted by "K,

(v) aK_ = {xEilKIP 'IlK<O} and aK .. =ilK\aK_ and no boundary aK coincides with a stream-line. 11 K • P ,= 0

(vi) T'~ = U Z~I aK n T_ coincides with J~ for every" > 0(vii) Tn = ilK n ilL is an entire of both K and L

(viii) the elements K E r;}h are affine maps of a master clement K = [- I, II x [-1. II. K = FK( K) asillustrated in Fig. 1.

(ix) '?Jh E:!f where :!f is a family of quasi-uniform refinements. Let "K = diam(K) and PK denote thesupremum of all spheres contained in K: then for all '!PI, E:?f. there exist positive constants aand T. independent of " = maxKe"'h "K' such that

(3)

r- _

----

lL. , ----".;;.r~-

~ 0 11.1),·1.1'_-t-_2.~1".1 Ll

: "I

(·1.·11 (1.·11

Fig. I. The affine map of the master element K 10 a typical clement 1\ E i'I' •.

K.S. Rey. J. T. (h/ell I Campll/. Methods Appl. f1feclz. Ellgrg. 133 (}1)96) 259-286 26~

(4)

REMARK. The discontinuous Galerkin methods admit more general partitions than those describedabove. In particular. the properties (iv). (vii) and (ix) are standard assumptions used in the analysis offinite element approximations. These properties as well as property (v) simplify the proofs in thefollowing sections. 0

The space of admissible solutions V(n) is extended to the partition using the broken space

V(rJPh) = n V(K)KE:i'~

V(K) = {VEL~(K)lvI3EL~(K)}

which admits discontinuities across element interfaces. The following notations arc used concerningfunctions v. IV E V(gtlh):

inl K I ( ) KV =vKx. xEiJ

eXl K I ( ) E "K n . Iv =vLx. X u rI,

v'" =Iim v(x:t Eµ)f-O

(v,I\')'1= !'1vll'lfJ·"'1ldS. yCilK

«v»'1=Y(u.u)y

(V.lV)K=! vlI'dxK

We define the following norms for functions v E VU1lh):

Illvlllll.K ,~f {lIv/l1I ~ + IIv II~ + «v» ~K} Ii ~

= {IIV/3I1;" + Ilvll~ + «v+»~" + «V-»~K.} II~

IIIv 1111./1."~f {lIvllll;" + Iv II;" + «v" - v - »;c \1'_ + «v» ~Kna!J} I' 2

IIlvlllll '~I t~/,IIIvlll;,.K } 1/2

Illvlll- ~f{.2: [lIvll~'+ (I +s 1I~)«v+»~K_]}I'~ (5)KE~h PK

IIlvlll+ '~f{.~[lIvll~ + (1 + S 1I;)«V-»~K.J}1f2KE~h PK

lIIvllltJ.1l ~f{.~ [OKllvlllli+ IIvlli+ (1 +8 1I;)«(v+ -V-»~C'L + «v»~"n.HI)J}· U~KE3'/, PK

where 8 is a parameter and fJ is used to indicate the value of the parameter fJK:

hKIf 0 = iihp then II" = ii 2

PK

II"If (J = lip then OK =2PI;

If () = II then OK = II"If () = 0 then OK = 0If fJ = I then U" = I

264 K.S. Be)'. J. T. Odell J Compllt. Met/lOcJs Appl. Mech. J:'llgrg. 133 (Jl)96) 259-286

(6)

The paramcter PK appearing in the definition of OK in the mesh-dcpcndent norm (5) will laterrepresent the spectral order of the polynomial approximation in K. The casc in which the coefficientOK = IIK/P~ in (5) plays an important role in the stability and error of the method, as we show later.

2.3. Weak formulation

Weak solutions to the linear hyperbolic model problem arc sought. since (1) admits discontinuoussolutions. Discontinuities arise through discontinuous inflow data which propagates along streamlines.Property (v) of the partition, rcquiring that no element boundaries coincide with the streamlines.implies that solutions II E V(fl) to (1) are con! inuous across elcment interfaces. Since the broken spaceVU9',,) admits discontinuities along element interfaces. we havc the following problem corresponding to(I) on the partition fiPh:

Find II E V(gIl"l such that for every K.E gil" 'jIIfJ + all = f in K

int K a eXl K a \,.J E ilK '" £}II ,..,. 11 K = U ,.., • 11 K v X _ IIJint K a a \,.J K n ['II ,., . "1.' = g,.., . "1\ v x E il _ _

for which weak solutions are sought satisfying

Fin.d II E V(;J>,,) such that for cvery K E '5'.".)(lIfJ + all, IV)K = (f. W)K V IV E V(K)

( inl K) ( ex, K) \,.J E V(K)II . v ;.K.'Ml = II ,v ~K_\M! V V

(lI,nt K. v) ,'LnL = (g. v) ;.,cnr. V v E V(K)

(7)

(8)

(9)

\"here we have taken the absolute value of P . 11 K for convenience. Next. we introduce a globalparamcter 8 which has a value (~feithcr 0 or I. Recall that for any v E V(K) we have that vf3 E L 2(K) sothat we can set no= v + 8(IIK/p;'JvP in (7) and then add the boundary intcgral equations multiplied byany constant. It will be convcnient to choose this constant to be (1 + D(h K / p~» and to write thc methodin the following abstract form. Let

dd ( II K ) ( II K ) + _. +BK(II.u)= IIp+all.v+2vfJ + 1+52 (II -II.V )"K.\I'PK K PK

(11K) .. •+ 1+{j2""(II'U)dKJlI'

PK

lId ( 11 K) ( 11 K )L,;(v)= /.u+82uµ + 1+8---:;- (g,V)ilK.nr.PK K PK

where. by definition, u + = Uinl K and 11- = u"" K on ilK .. Summing over all thc elements in the partitionyields the variational boundary value problcm for weak solutions to (1) on the partition:

Find liE V(:?P,J such that

B(u, v) = L(v). for cvery u E V(gIl,,)

where

dd '"8(11. v) = L.J Bdu. v)Ke~"

dd '"L(u);::: L.J Ldu)Ke:f'"

(10)

(11)

(12)

REMARKS(i) The case D ;::: 0 in (8) is referred to as the 'standard' discontinuous Galerkin method which can be

K.S. Bey. J.T. Oden I Complll. Methods Appl. Mech. Eng,.};. 133 (/<)96) 259-286 265

(13)

(14)

viewed as a standard Galerkin method for a single clement with weakly imposed boundaryconditions for elements lying on the inflow boundary and weakly imposed continuity for elementson the interior of the domain.

(ii) The case 0 = 1 in (8) is the lip extension of the so·called 'streamlinc upwind' discontinuousGalerkin mcthod [12]. The modification of the test function is important when approximatingsolutions with sharp gradients as the additional term in the test function adds diffusion in thestreamline direction without modifying the conservation law, i.e. without destroying accuracy inregions where the solution is smooth. 0

LEM MAl. Let the bilinear form B(- .. ) be defined by (11) and (8). Then, there exists positive constantsa, MI' and M2' independe1l1 of hK and PK' s/lch that

B(v. v)~alllvlll~hP.11

{ (hK) , } 112{ > h K , .< > } J /2

B(\\', v).s; M, L 1+ 0 ~ IIIwllli./I.K Illvlll::.+ 5 L ~ 1Ilv/lllK + «v »~KIKE~h PK KE"'hPK

B(W.V).s;M1{ L Illwlll:+8 L h~[lIW/III~+«W-»~K.]}lI2{ 2: (1+8h~)lllvlll~.#.K}"2KE~h KE;?I" P K KE!fh P K

(15)

{ (h) } 1I1{ ( h)} 1/2B(w. v).s; M2 L I + 8 ~ Illwlll~.K L 1 + 8 : IlIvlll~.KKE~(, P K KE~h P K

for every w. v E V(f1lh).

(16)

PROOF(i) From the definition of B(-. .),

. . ~ {( hK) hK, 2B(v.v)~mm(l.mma(x» L.. 1+52 (v.v/l)K+5~lIv/lIIK+lIvlIKzEll KE,f'h P K P K

( hK) + 2 - + 2}+ 1+8 P~ «(v »~K_\r_ - (v ,v LK_\I'_ + «v».lK_nr.)

Eq. (13) follows by substituting the results of applying Green's formula to the term (v. Vp)K' thatIS.

(17)

266 K.S. Bey. J. T. Oden I Comp/ll. Methods Appl. Mech. EIl~rg. 133 (1996) 159-186

x {.~ [25 "~ IIvplli +2I1vll~+ (I +S II~)«v·»~K_ J}"~f\E.~/, PK p"

Eq. (14) follows by selecting M, = v'2llalkff'(iii) Eq. (15) is obtained by applying Grcens formula to thc term (W,Vf:l)K and (wµ.vh in (II). and

applying the Cauchy-Schwarz incquality.(iv) Eq. (16) is obtained by adding (14) and (15), applying the Cauchy-Schwarz inequality to thc

result. and selecting M~ = ~lIall%./l' 0

COROLLARY J. Let S = I ill (8). Tllell

B(v. v) ~ alllvllli.t •.µ

and thcre exists a constant I'll sltcll tllat if II" Ip~ ~ r(J'VK E ;;Ph'

{, II" ,}In

B(w. v) ~ M;IIIII'IIII.f:l lllvlll~ + 2: ---:;-IIvpllK"€off'" p"

{". }li2

B(w, v) ~ M; Illwll/: + L ~ IIwf,l1l ~ IIIvlIIl./l"e,,,,, p"

B(w. v) ~ M~(1 + 1'0)11111'111 11 IIIV III [/PROOF. Set S = 1 in (13)-( 16) and choose M; = Ml~ max( I. yr;;). 0

COROLLARY 2. Let 8 = 0 ill (8). Tllell

B(v. v) ~ alllvlll~./lB(w. v).s; Mllllwllll.pll/vili.B( w. v) ~ M ,IIIIVIII+ Illvllll.'1B(w, v}.s; M21111V1lI1IIlIvllln 0

(18)

(19)

(20)

(21 )

(22)

(23)

(24 )

(25)

REMARK. Note thalmodiflcd test function for the streamline upwind discontinuous Galerkin methodresults in improved slability of the bilinear form when compared 10 the standard Galerkin method (see(18) and (22». Thc coercivity of the bilinear form for the streamline upwind discontinuous Galerkinmethod contains IhlII" lerms which do not appear in the coercivity condition for the standard Galcrkinmethod. The significance of this additional stability is less important as IIK/p~ approaches zero. 0

2.4. lip Finite elemellt approximation

We seek approximatc solutions to (10) in the finite dimensional subspace Vp(:?Pt,) C VU¥'t,) defined asfollows:

(26)

where QPK(K) is the space of tensor products of polynomials of degree PK defined on the mastcrelcment. We use the notation Vf\EQl'K(K) to mean uKEQl'K(K) and vf\=uf\oFf\' The basis forQPK( k) is formcd by tensor products of Legendre polynomials. Wc have the following inverse estimatesfor polynomials on a single element:

LEMMA 2. Let K E '3l2 be all affine map of a master element k = r -1.1] x r -1. I]: that is K = FK(k).Let y delloFe allY edge of aK which is all affille map of a master edge y = 1- I. I]. Let Ii,f\ be (/ polynomial

K.S. IJey. J. T. Oden I Complll. Methods Appl. Mech. Engrg. JJ3 (IY9fJ) 259-286 267

of degree p K defined on the master elemenr. Let w K = ~VK 0 FK denote tile image of Ii'K under thetransformation FK. Tllen. {3 .VwK satisfies tile following:

where the constalllS C are independent of "K, PK and WK'

PROOF. For polynomials of degree PK on the master elemellt. we have that (see [11])

I~\'KL.K~ IIIi'KI\...K ~ Cp;III\'KII,.;-IlvKI'''1 ~ 11~1·1\1I•. y ~ Cp~1111'l\lly

(27)

(28)

(29)

(30)

where the constants C >0 depends on 5, but not on P K or IV K'

For affine mappings FK• a standard scaling argument (sec (7]) yields that for an integer s;;=. O. thereexists constants C> 0 such that

IIVKI..,K ~ CJl~- 'lw KL.K

IIVKI •. y ~ CJl~-'/21I1'KI.1'

(31)

(32)

(33)

(34)

where C depend on s. CT and 'T (see (3)). but not on "K, PK or "'K'

The first cstimate (27) follows hy combining (31). (29) and (33). The second estimate (28) followsfrom (32). (30) and (34). 0

We have the folIowing result concerning polynomial approximations of functions with sufficientregularity:

LEMMA 3 [3]. Let K E '?J,", y denote any edge of aK. and u E H'(K). Then there exists a constantC = C(s .•. CT) independem of u. PK and hK. and a sequence zf. E QPK(K). PK = 1.2 ..... such that forevery O~r.s;,PK'

I "-1/2'KlIu - zf.llo.1' .s;, c~ Ilull".K .

PK

where v = min(PK + 1. s).

s;;='O

1s;;=''2

(35)

(36)

We use this rcsult in deriving the a priori error estimate for hp-version discontinuous Galerkin methods.Note that the first estimate is valid for solutiolls to the model problem. u E V(n) ::> HO(n) = L 1(!l). Wemllst make additional regularity assumptions on the solution to the model problem for the secondestimate to hold.

The approximate solution to (10) is obtained by replacing the exact solution u E V(9J>h) by ll~ EVp(fJ>/.> and the test function v E V(fJ>,,) by vf. E V,(fJ>h)'

268 K.S. Bey. J. T. Odell I COli/put. Afe/hods Appl. Mecll. Ellgrg. I.U (1996) 251)-286

Find u~ E Vp(9Ph) such that

B(/(~, v~) = L(v~), V v~ E Vp(g>,,) (37)

The improved stability of the strcamlinc upwind discontinuous Galcrkin method, S = I in (8). isrecovcred by the standard discontinuous Galcrkin method. S = 0 in (X). on the finite dimensional spaceVp(@lll)'

LEMMA 4. Let S = 0 in (R). Then, for every v~ E \-~(gp,J there exists a w~;E VI'(9PII) slIch that

B(v~, w~) ~ a'lllv~III~".p (38)

and

(39)

(40)

where the positive conSlants a' and C are independent of h K' PK alld v f,.

PROOF. Define the restriction of I\'~ E V/9Ph) to an element K E?Ph as

"I 1'\ hK Il ..., PIIV h K = V h K + 'Y 2 ,...\'v II KPK

where 'Y E (0. 1] is defincd later in the proof. Dropping the h. p and K scripts for ease in notation, wehave

where ao = minrEn a(x). Noting that

(41)

and that from Lemma 2

we have

2 hK 2 I + 2 (I ) ~2/JK(v,w)~(a()-c''Y)lIvIIK+'Y21IvIlIIK+2«V ))'llur_ + 2-C2'Y «V))"cnr_PK

i + -I I ilK f + - + I I- v V fJ'IIK d5+'Y2 (v -v )Vp fJ'''K dsilK_\/'_ P/\ iiK_\I'_

K.S. Bey. J. T. Odell I Compw. Methods Appl. Alec/I. ElIgrg. 133 (1996) 259-286

Using the Schwarz inequality and the previous inequalities. one can show that

I h K f + - + I 3c z +: - :y~ (v -v )v"IP'IlKld\' ~-2y«(v »ilK \1' + «v »iJK-\I')P K .JK _ \1'_" - • -

Now summing over all the elements K E~" and realizing that

results in

Choosing y == mine t. a./2c I' 1/6c 2) yields the first inequality.The sccond incquality easily follows from the definition of w~ and Lemma 2. 0

2.5. A priori error estimate

269

The discontinuous Galcrkin method (37) with l) == 0 was first analyzed by Lesaint and Raviart [141 fora given fixed value of PK' i.e. for the case in which PK == P for every element K E ~". The error in asolution II" to (37) approximating an exact solution liE H'(fl) to (10) was shown to be

1111- lI"lIn ~ Chs-llIlIll,.fJ

This estimate is not optimal in the sense of interpolation error estimates and was improved by Johnsonand Pitkaranta [131. Using a mesh-dependent norm. they showed that

11111- 11,,111".13 ~ O,s-1I2111111dlWhile this estimate is not og!!mal in the sense of interpolation error estimatcs for lie IIn == 1111 - 11" II {/. itis optimal with rcspect to Vh Kllepll K and «e + - e -» ilK _\1'.' We shall derive cstimates similar to Johnsonand Pitkaranta [131 taking into account that p K is not constant.

THEOREM 1. Let 11 E HS(a) be a solutioll to (10) and let 'I~ be a solution to (37). Then. there exists apositive constant C. independent of hK• PK and u. such that the error. e == II - II~. satisfies the followingestimate

(42)

where VK == min(PK + 1. s).

PROOF. Let n~II E Vp(~") be an approximation of II that satisfies the estimates in Lemma 3 and write

e == II - 1I~ == U - n~lI + n~u - II~

which implies that

Illelll"p./l ~ 11111 - n~lIlll"p.f3 + Illu~ - n~lIlII"p./ld,;f 11111111"p./J + IlIwlll"p.f3

(43)

(44)

270 K.S. Bey. J. T. Oden I COtnl'lII .• Hethods Appl. Mech. EIIW/.:. J33 (f996) 259-286

where. to simplify the notation. we set 1) = LI - J/~/l and I\' = L1~: - [J~;1I. Realizing that

combined with Lemma 3 yields bounds for the first term in (44):

(45)

where IJ-K = min(PK + L s). Bounds for the second term in (44) follow from the orthogonality conditionwhich is obtained by subtracting (37) from (10):

B(e. u~) = B(1). u~) - 8(1\'. un = O. v u~ E ~'(W'h) (46)

We choose vf. = v.l in (46) where the particular choice for v6 depends 011 the parameter 0 in (H). For(j = O. we choose v6 to be the function which satisfies Lcmma 4. For 8 = 1. \\le choose u6 = wandcombinc (46) with Corollary 1. The result for either case is

C11111'111;'p./l ~ B(w. Vii) = B(1). VO

)

Intcgrating the terms (1)/l' v6) K and (1). v~) K by parts in the definition of Be .. ) in (11) yields

o "" [ 0 PK ~ IiB(1).u )~lIall".!1 K~"" 111)IIKlluIIK+ ~ 1I1)IIK'hIlU/lIiK

11K Ii 11K 0+(2111)/lIlKllu/lIIK+82'111)/lIlKliu 11K

PK PK

(11K) _ 0+ 0-+ I+l)p~ «1))aK_\I'.«U -U »aK.\1.

(11K) - 0- J+ 1+15 P~ «1) »aK.nl'. «u ))i.K.nr.

{ "" [( p~) , I1K( 11K) )~lllIlIx.I1(I+o).L.. I+h. 111)111.-+152' 1+82 Ih/lll1.-J;E'~h K PK PK

(11K) _ 2 ]}1I2 II+ 1 + 15P~ «1) » aK. Illv 1I1"1'.1l

(47)

(48)

Recall that for our choice of VII we have Illvollllll'f:l = 11111'111111"# when 8 = 1 and Illullllllll'./3 ~ C111II'IIL,p./lwhen 15= O. Eqs. (48). (47) and the estimates in Lcmma 3 imply that

(49)

Combining (49). (45) and (44) completes the proof. 0

REMARKS(i) For I1Klp~ ~ L the estimate becomes

K.S. lJt'y. J. T. (Jilt'll / CompUl. Methods Appl. Mt'c". EIIKrg. /33 (1<J96) 159-28(1 271

(ii)

(iii)

For p" = constant. the a priori error estimatePitkaranta [131.Lct II=max"Ef' "" andp=min"Ef' I" then

. I, ." I\.

hµ-1I2

IlielllilP.~~~lIulldl· 0P

reduces to the one derived by Johnson and

2.6. Implementation issues

In the preccding scctions. the discontinuous Galcrkin methods werc represented as global methodsfor the purpose of analysis. The approximate problem is actually a local one since the only couplingbetween elements occurs weakly through the Iluxes on the elemcnt inflow boundary. Assuming that 1I~;-is known on ilK _. then the approximate solution in element K satisfies

(50)

whcrc

(51)

(52)

In order to solve (37) in this fashion. one must define an ordering of clements that starts at the domainintlow boundary and sweeps through the partition in such a way that u~:-is known on aK_ prior tosolve (50). Such an ordering always cxists (sec P4]) and is fairly straightforward to construct. This is theoptimal solution technique for solving the linear model problem where element inflow boundaries canbe identilled a priori. However. an alternate approach is needed for solving non-lincar hyperbolicconservation laws where the fluxes dcpend on the solution. and thus. clcment inflow boundaries cannotbe identitled a priori.

With the aim of solving more general problems in mind. the linear modcl problcm is solvcd in a waythat is easily extendible to thc non-linear case. that takes full advantage of the discontinuousapproximation. and is amenable to parallel computations: that is. by solving the time-dcpendentconservation law for the steady state solution. Sincc time aceuracy is not important in obtaining thestcady solution. we use the classical forward or backward Euler time marching with a truncation errorof O(.ltC

). Let U"TI = II~(-. t,,+I) where 1"+1= (n + 1).l1 and .ll is the time step increment. Assumingthe solution at time I" is known. then the forward Euler version of the scheme is given by

L (U"+I. v)" = L (u". v)" + .It[L(v) - 8(11". v)]"E~h "E~h

and the backward Euler vcrsion is given by

L (U,,+I.V),,+.lt L B,,(II,,+I.V)+ L (u".v),,+.lt L [,,(v)"EiPh "E3't, "E'~h "E·l'h

(53)

(54)

To prescrve thc local character of the method. the inllow boundary tcrms appcaring in thc detlnition of£,,(v) (sec (52» are evaluated at time level (", The schemes (53) and (54) are conditionally stable.although gencral stability analyses are not available. The work of Cockburn and Shu 191 suggests a timcstcp restriction for (53) given by

272 K.S. Bey. 1. T. Odell I COli/Put. Methods Appl. Alec/I. Ellgrg. /33 (1996) 259-286

[I IKI]ut ~ I!lax (21)' + I) 111.11 ....1hE"'h h

(55)

for Ph = 1.2. Numerical experimentation shows that the time step restriction for (54) is atlcast an orderof magnitude larger than the linearly stable time step for (53).

The initial data. 1If,(·. 0). needed to complete the initial-boundary-value problem is taken to be auniform field with a value associated with the intlow boundary conditions.

3. A posteriori error estimation

The a priori estimates derived in the previous section are useful for predicting how the error innumerical solutions behaves with h-refinement or p-enrichment. Unfortunately. their usefulness inassessing the aecllTacy of a given numerical solution is limited since the estimate involves unknownconstants and the exact solution we are approximating. Nevertheless. a priori error estimates such as(42) and interpolation error estimates such as (45) have been used extensively as error indicators todrive adaptive methods for hyperbolic problem [10.15. 191. Typically. the unknown constant is set tounity and some post-processing of thc approximate solution is used in place of the exact solution. Whilethe elcment contributions to these global estimates may provide some relative measurc of the localerror. this approach in general fails to provide a reliable estimate of the actual error in a particularnumerical solution and can be grossly in error.

In this section. we dcrive error estimatcs which are computed locally on a single element andcontribute to a global crror estimate which is accurate enough to provide a reliable assessment of thequality of the approximate solution.

3.1. Elemellt residual method

The estimates derived here. based on the element residual method. are similar to those proposed byBank and Weiscr (4) for elliptic problems and aden et al. 117] for solid and tluid mechanics problems.The element residual method was extended to lip-approximations for elliptic problems by aden et al.[16]. A global estimatc of the error is obtained by summing c1emcnt indicators which are the solutionsto a local problem with the c1cment residual as data. In [16. 171. the local problem is of the same formas the global problem.

For continuous finite element approximations. the element residual involvcs fluxes on the boundaryof an clement. Since the tluxes are multi-valued. an averaged flux is used. Recently. Ainsworth andOden [1. 21 have shown that it is possible to use a self-equilibrating average flux that results in an errorestimate which is equivalent to the actual error and can be asymptotically exact for certain ellipticproblcms. For a discontinuous approximation. the jump in the element boundary flux arises naturally inthe residual. eliminating the need for flux balancing.

The main difficulty with our formulation for hyperbolic conservation laws is that the norms associatedwith the continuity and cocrcivity of the bilinear form are different. The use of different norms makes itimpossible to construct a single local problem which results in an upper and lowcr bound of the error inthe same (or an equivalent) norm.

In the following sections we show that it is possible. however. to construct one local problem with asolution that provides a lower bound on the actual error and another local prohlem with a solution thatprovides an upper bound on the actual error. We also show that a local problem based on the originalproblcm results in a local lower bOllnd. Moreovcr. if the approximation of thc solution to this localproblem is limited to a certain class. then the estimate is equivalent to another commonly usedapproach: estimating the error as the difference between a newly constructed (and hopefully moreaccurate) solution and the approximate solution on hand.

K.S. Be)'. J. T. Odell I COlllp/ll. All'/hods AppJ. Mech. Ellgr~. J33 (1996) 251)-286

3.2. A global lower bound all 1he error

273

A local problem is constructed which results in a lower bound on the error in a sense to be definedprccisely later. Let Ii,; E QPI\(K) denote the approximate solution in an c1cment K and ip,; E V(K) bethe solution to the following local problem.

where

L dcf h,; _A ,;(ip,;, v,;) = 2 (13 . VipK' 13 . VV';)K + a(ipK' v,;),; + (ip,;. VK) il';

p,;

and ii > 0 is a constant. Then

induces a norm on V(~(,) which will be referred to as the A L -norm:

(56)

(57)

(58)

(59)

Thc solution to the local problem (56) provides a lower bound on the error in the following sense.

LEM M A 5. Let ip E V(£¥l(,)be the SOllllioll to the following problem:

AL(ip, v) = B(e. v) V v E V(£¥l,,) (60)

There exist positive cOllstants k I and ro such that if h,;lp~ ~ r() V K E ~(, then

IlipIlAL~k,IlJeIIILtl (61)

PROOF. Ilipll~L = AL(ip, ip) = B(e, ip)

{ ( hK') , }112{ --. [itA' 2 '~MI Kti'h 1 + 8 P~I 111e1111/3.K ,;t~h 8 pi 11%11,; + Iliplli;

(h.) J}112+ I+Op~ «lpt»~K_ from (14)

~MI(I + or() max( 1. ~)IIIeIIILllllipll\LThe desired inequality (61) follows by choosing kJ =M,(I +8ro)max(1.1IVa). 0

3.3. A global upper bound of tlte error

For simplicity, the estimates arc derived for the case when 8 = 1 in (8). A local problem isconstructed 10 result in an upper bound on the error. Let 1/1,; be the solution to the following localproblem,

wherc

U dcf h,; _AK(l/I,;· UK) = 2 (p .Vl/I,;, fJ . Vt',;),; + a(l/I,;. ud,;

p,;

and ii > 0 is a constant, Then, the A U -norm is defined as

(62)

(63)

274 K.S. Bey. J. T. DIlen I COlllplll. Methods Appl. Mech. Engrg. /33 (1996) 2.l9-2S6

(64)

The solution to the local problem (62) provides an upper bound on the error in the following sense:

LEMAIA 6. Let 1/1 E V(BbIl) be the solwion to the following problem:

A u(l/1,u) = R(e, u) VuE V(3I',,)

where 8 = 1 in 1he definition of B(', .) iI/ (I I). Then there exists a positive constlllll k ~ sllch that

11l/IIIAlI ~ kzlllelllllP.1l

PROOF. Using (18) of Corollary 1,

- ualllelllhP./l ~ B(e. e) = A (l/I. e)

~ 1Il/1I1Allliell.\lT= 1Il/111"IL1{ 2: h~ llepll~+ allell~}\!~

J;.E.'" P J;.

~lllax(1. Vli)IIl/III.\lI{.2: h; lIe/lIl~+ 11e11~}1/:}( E.'I'h P A

~ max( 1. va HIl/IIL.lulllelllllt ..1.l

Choosing k2 = a/(max( I.va)) completes the proof. 0

3.4. A locallolVer bow/(I on the error

(65)

(66)

Recall the bilinear form (50) which characterizes the space marching forlll of the discontinuousGalerkin method:

- d"l ( h ,. ') ( h ,. )I' t'.:. p I' I' ~ I' ~ 1'+ " •.B J;.(II II' U Ir) - 1I11{1 + (//111 ' U II + 8 2 U Irfj + I + D 2 (It II • U" ).lKPJ;. A PK

Now consider the following local problem:

Find 'PK E V(K) such that

iiK('PJ;.· UK) = BK(eJ;.' UK)' V UK E V(K)

then <PK provides a local lower bound on the error in the following sense.

LEM M A 7. Let <PI\, E V(K) be the so{lItiol/ to «()l)). Theil there exi.\'t.\'1I COl/SWill k 3> 0 .\'lIch that

(67)

(68)

(69)

(70)

K.S. Hey. J. T. Odell I Comptll. Methods Appl. Alec". LII}i'}i. J33 (1996) 25'1-286 275

(72)

PROOF. Setting UK = IfK in (69) and using (68) yields, -C1lcpKllii".;:; BK(If". If,,) = B"(e,,. cp,,) (71)

Sctting M" = max( L lIall",.K) and applying Young's inequality. lib.;:; 1/4e aC + eh~. e >O. to <.:ach termin B "(e,,. cp,,) yields

A1,,(. h,,) , 1

B"(e,,. IfK)';:;TE 1 + 8 ~ lilellll.µ." + 2M"elllf"llii"p"

Selecting e < C/2M" in (72) and eombining with (71) completes the proof. 0

3.5. Approximatioll of the local problems

An approximate solution to the local problem measurcd in the corresponding norm serves as a localerror indicator for an element. Since the discontinuous Ga\crkin solution satisfies the orthogonalitycondition.

(73)

wc must approximatc the error indicator with a polynomial of dcgrce PK + UK wherc UK~' I in ordcr forthc discrcte local problem to havc a non-trivial solution. If a complctc polynomial of degrec p" + (T"

(on the master element) is used to approximate the solution to the local problem. then the discrete localproblem requires the solution of a system of order (p" + UK + 1( This system can be fairly largecompared to the system of order (p" + 1)2 equations used to obtain the approximate solution for whichwe are estimatin~ the error.

Since (p" + It tcrms on the right-hand side of the discrete local problem (corresponding to (73)) arezero. we can make a simplification by approximating the solution to the local problem in the spaceQI'Ii+""(K)\QPIi(K). In other words. the solution to the local problem can he approximated withincomplete polynomials of degree p" + (TK by neglecting the terms associated with polynomials ofdegree p". The simplification results in a system of uK«(T" + 2p" + 2) equations for each element.

The size of the local problem can be further reduced by approximating its solution using only the'bubble' functions in the enriched space denoted by Q~"+""(K)\Q~Ii(K). These are thc polynomials inQP"+"K(K)\QP"(K) which are zero on the boundary of an element. This additional simplification resultsin a system of uK«(T" + 2p" - 2) equations which is smallcr than system of equations used to obtain theapproximate solution.

3.6. Remarks concernillg all alternate approach

Suppose that an approximation V to the exact solution II can be constructed which is more accuratethan the approximate solution on hand. II~:. Then a simple estimate of the error. e = II - 1If, = II - V +V -1If,. is () = IIV -1If,1I whcre 11·11 is any suitable norm. Using the triangle inequality. we have

lIell -1111 - V II.;:; 0 .;:;Ilell + 1111 - V IIor equivalently

1111 - VII 8 1111 - VII1- llell ';:;ij;jf';:;l + Ilell

If 11/1 - VII ~ llell = 1111- 1If,11. then 8 = IIV -1If,11 is a good error estimatc with an effectivity index.01 lie II. ncar unity. The main difficulty with this approach is to efficiently construct such a U. Oncobvious strategy for constructing a more accuratc approximation. V. is to re-solvc the approximateboundary-value problem on an enriched space.

For continuous finite element approximations. this Icads to a global system of cquations that is muchlarger. and therefore much more costly to solve. than the original problem. For a discontinuous

276 K.S. Ill',\'. J. T. Odell I Complll. Methods Appl. Mech. i:;llgrg. J33 (1996) 259-2X6

approximation. re-solving the problem on an enriched space of complete polynomials is still more costlythan the original problem, but is no morc costly than solving the local problems in Section 3.2-3.4 onthe complete polynomial space.

The computalional cost of this approach can be further reduced by 'freezing' the lower-order solutionand re-solving the problem on an incomplcte polynomials space of bubble functions. In other words. IctU/\ = u~l/\ + 11'/\ where 1\1/\ E Q~K+UK(K)\Q:;K(K) satisfies

(74)

In this case. the error estimate is 8 = IIV - l/~II = 1111'11.Note that this is equivalent to solving the localproblcm (69) on the space of bubble functions. We remark that Peraire and Morgan [20] simplypost-process the approximate solution to obtain the degrees-or-freedom (higher-order derivatives)corresponding to thc bubble functions.

4. Numcrical cxamples

The discontinuous Galcrkin method is used to solve several examples to verify the a priori errorcstimates derived in Section 2 and to invcstigate the performanee of the a posteriori error estimatcs ofSection 3. The performance of the error estimate is measured by the effectively indcx which is the ratioof the estimate error to the actual crror in the same norm. A reliable estimate is one for which theeffectively index is near unity.

Results are presented for the final mesh obtained using an lip-adaptive strategy developed by Oden etal. [18] for a large class of elliptic problems and extended to discontinuous solutions of hyperbolicproblems hy Bey [51. The lip-adaptive strategy is based on a rcliable a postcriori error estimate fordetcrmining the error in the approximate solution and an a priori error estimate for determining how tomodify the mesh to improve the solution accuracy to a specified level. The goal of the lip-strategy is todeliver a solution with a specil1ed error in only three steps:

(i) Construct an initial partition PJll eontaining N(PJo) clements. The elements in PJlI can he ofuniform p/\ = PII and essentially uniform in 11/\= "0' Solve the problem of interest on V (PJ() and

Poestimate the error.

(ii) Construct a partition PJ1 by subdividing each element in 'lPo into the number of elementsrequired to equi-distribute the error and reduce it to a spccified level. Solve thc problem onV,'Il(PJ,) and estimate the error.

(iii) Enrieh the approximation space by increasing p/\ for every K E PJ, in such a way to equi-distribute the error in smooth rcgions and reduce it to the specified level. Solve the problcm onthe enriched space V

P1(PJ,) and estimate the error.

Complete details of the lip-adaptive strategy and other numerical results can be found in [5].

4.1. Example 1

The linear model problem (I) is solved with the following data:(i) fl=(-I.l)X(-l.l)(ii) f3 = (0.8. 0.6)1'

(iii) a(x) = 1.0(iv) g=l.O

The source term f is chosen so that the exact solution to (1) is the C"(fl) function.

I/(x. y) = 1+ sin(; (I +.\')(1 + y);) (75)

The a priori error estimate (42) is verified by solving the problem for a sequcnce of uniformII-refinements and p-enrichments of a mesh of square elements and quasi-uniform II-refinements andp-enrichments of the mesh of quadrilateral elements shown in Fig. 2.

(

K.S. Rcy. 1.T. Odell I COII/p"t. Methods Appl. Mech. Engrg. 133 (1<)96) 259-286 277

litII

• p=1r~~·· p=2

1.~5~"@"""~.'./.'::::: :;:I . .

25 [d. 1 .•....•.

.. ·-3.5~ .. · .... squareelemcnls

•.••. '-- quad elements

10'

10'

10'

Fig. 2. Quadrilatcral clement mesh used for quasi·uniform refinements.

Fig. 3. Example I-Rate of convergence of error with "·rcfinements.

First. consider the case when P/\ is fixcd and "" is varied. According to (42). /IlelllhP./l ~C,,~+"211111Ir.n' This is vcrified in Fig. 3 where lilellll,p.µ is shown as a function of "K' On the log-logscale. the slope of the lines corresponding to a fixed value of p/( is indced p" + -! for both thc uniformand quasi-uniform meshes. -

Ncxt. consider thc case when 11/\ is fixcd and PK is varied. In this case. the estimate (42) reduces tolilelll"p.µ ~ Cp~-T+I)III1I1,.II· Since u E (."'''(fl), exponential rates of convergence are expected. This isconfirmcd in Fig. 4 wherc the curvcs corresponding to 1llelllhp.Jl as a function of PK have a slope on thelog-log scale which increases as PK increases. These results are combined in Fig. 5 where llieIIL,tJ.Jl isshown as a function of the total numher of unknowns in the solution. The' solid lines representIt-rcfinements for a fixed p and the dashed Jines represent p-enrichment for a fixed II. Clearly, forsmooth solutions. higher-order accuracy is achieved for the same numbcr of unknowns by usinghigher-order elcmcnts.

Next. the performance of the a posteriori error estimates in Section 3 is investigated. Recall that thecomplcte polynomial space QPK+<7K(K) or the incomplete space QPK+rrK(K)\QPK(K). fT/\;;;' 1 can be usedto approximate the solution to the local problem. The effect of approximating the local problem on theperformance of the error estimate for thc lower bound (56) is shown in Table 1. The effcctivity indiccslisted in Table I are greatcr than one for all values of u/( when the complete polynomial space is usedand less than one for small values of UK when the incomplete polynomial spacc is used. Note that theeffectivity index is closest to onc for the complete polynomial space with tT/( = I. The effect ofapproximating the local problcm on the pcrformance of the error estimate for the uppcr bound (62) is

10"1

IIlelll"oJl10'

10'

p

10·)

lIIelll"",10·

10'

10number or unlmowns

p--4.....10'

Fig. 4. Examplc t-Rate of convergcnce of crror with p.cnrichmenls.

Fig. 5. Example I-Rate of convergencc of error with number of unknowns.

278 K.S. Bey. J.T. Oden I Comput. MeThods AppJ. Mech. Engrg. 133 (/996) 259-286

Table!Example l-Effcet of the approximation of the local problem for the lower bound on the effecti\'ity index

Mesh PK frK .p" E QP" "'''(/\) .pK E QP."",,(/\ )\QP"(/\)

1/1- 1/1-

8x8 I I 1.(1938 0.76288x8 I 2 1.1272 0.8'J2!8x8 1 3 1.I3-l7 0.97888x8 I -l 1.1372 1.0175

t6 x 16 t 1 1.1765 0.793316x 16 ] 2 1.222<) 0.949216x 16 I 3 1.2340 1.(465lox t6 I 4 1.2378 1.0898!lx8 2 I 1.1224 0.95558x8 2 2 1.200<) 1.0711

shown in Table 2. The effectivity indices for thc upper bound estimate arc significantly larger than onewhen the complete polynomial space is used and close to one when the incomplete polynomial spacewith UK = I is used to approximate the solution to the local problem.

Next. we verify that the error estimate exhibits the same rates of eonvergence as the actual error withJr-refinement or p-enrichment. Based on the results from above. the error is estimated by solving thelower bound local problem in QPK+1(K) and by solving the uppcr bound local problem inQ'/K+1(K)\QPK(K). The estimate error for a sequence of refinements of a meshes with fixed p is shownas a function of the mesh size in Fig. 6. The slope of the lines is p + ~ as in the ease of the actual crror(see Fig. 3). The estimated error for a sequence of p-enrichments of a uniform mesh is shown in Fig. 7where the same behavior as the actual error is observed (see Fig. 4).

While the theory developed thus far applies to global error estimates. local effectivity indices near

Table 2Example I-Effect of the approximation of the local problem for thc upper bound on the effectivity index

Mesh

8xg8X8

HxS!lxg

t6 x 16t6 x 16t6 x 1616 x 168x88x8

22

IT,.;

I2J-lI234t2

h

,fr" E QPc""(K)

1/u

4.19114.36035.12975.32186.444t6.67297.'X)488.11572.63533.7052

,frKE QP""'K(K)\QPK(K)

1/e

1.1)5361.35512.l1931l2.68031.12381.491;02.82983.6llOl;1.19571.4012

I)'

I)'

~.,83~"~16x 16mesh

I

2P

Fig. 6. Example I-Rale of convergcnce of the estimated error with uniform h·rclinemenls.

Fig. 7. Example I-Ratc of convergence of the estimated error with uniform p-enrichmenls.

K.S. Bey. J. T. Odl'll I Complll. MelJlOds App/. Mech. Ellgrg. 1.13 (19%) 259-286 279

unity are desired in order to use the estimate to drive an effective adaptive strategy. The local (element)effectivity index for the error estimate based on the upper bound local problem (62) using theincomplete polynomial space QI'Ii' t(K)\QI'Ii(K) is shown for a uniform I{ x 8 element p = J mesh inFig. 8. for a uniform 8 x S clement p = 2 mesh in Fig. 9. and for a uniform 16 x 16 clement p = I meshin Fig. 10. For all of these cases. the local effectivity index is close to one. except in a few isolatedelements. indicating that the error cstimate is reliable enough to drive an lip-adaptive strategy. Thesolution to the upper bound local problem on the incomplcte polynomial space with a,.,.. = I is presentedthroughout the remainder of this section.

The results of applying the lip-adaptive strategy lSI to this problem are listed in Table 3. The error isnormalized by the sum of the upper bound norm of the approximate solution and the estimated error sothat it has some physical relevance. Note that the specified kvel of error is achieved at each step and

1.251.221.191.171.141.111.081.05

1.071.020.970.920.880.830.780.73

i ,_=-L __I

.----'--- I ~ I

tTl.:--':-

Fig. 8. Example I-Local effectivity index for error estimale based on the upper bound local problem (8 x 8 mesh. I' = I).

Fig. 9. Example I-Locat effectivity index for error estimate hased on the upper bound local problem (8 x 8 mesh. p = 2).

1.111.040.970.900.840.77

0.700.64

I1~'

1.'.' '------'---' '. . --r- '--j '--r--l.-'.1m. . J. i .!~. I. •. I. f-------r-1. 1 •-I j t (-+--: . ~:

. ;' -j _.- -mi' '-'1---:-;-, . ffi' i i - .1 • 1....:_

Fig. 10. Example I-Local effectivity index for error cstimate based on the upper bound local problem (16 x 16 mesh. p = 1).

Tahle 3Example I-Error history for an adaptive hp solution

Adaptive Target Achievedstep crror error

Effectivityindex

tnitial 4 x .t mesh p = Ilr·refinemcnt,,-enrichmcllt

1.5%0.1%

3.79%1.21o/rU.11%

1.251.()6

0.94

280 K.S. Bey. J.T. Odell I Complll. Methods Appl. Mecll. Ellgrg. J33 (1996) 259-2Rfi

that the cffectivity indcx is close to unity at each stage. The final lip-mesh, the exact and estimatedcrror, and local effectivity indices are shown in Fig. 11. The results in Fig. 11 show some degradation inthe performance of the local error indicators for lip-meshes when compared with the uniform meshes.however, the global effectivity index is close to one.

4.2. l:xanlple 2

The linear model problem (1) is solved with the following data:(i) n = (-1,1) x (-I, 1)

T(ii) fJ = (1.0, 0.0)(iii) a(x) = 1.0

{

3e-5(1".I'~) if / < 0(iv) g(y) = )

-5 1... ~ •- 3e ( -,) othcrwlse

The source term f is chosen so that the cxact solution to (1) is the discontinuous function,

{3e -5(.r~ ..}'~) if y < 0

/leX, y) = ~ '- s ( ... - .- 3e . (. -,) otherwise(76)

The discontinuity is aligncd with clemcnt intcrfaces at y = 0 to illustratc the advantage of using adiscontinuous method to capture shocks, particularly if thc adaptive schemc includes somc shock fitting

hp-adapted mesh

Estimated error

PK4.5

3.5

2.51.50.5

9.6E-4

6.8E-43.9E-41.1E-4

Exact error

Local effectivity index

9.6E-4

6.8E-43.9E-41.1E-4

llK1.591.200.810.42

Fig. 11. Example I-Results from an hp-adaptive solution.

K.S. Bey. J. T. Odell I Complll. lIfethods Appl. Mech. Ellgrg. 133 (1996) 259-286

- exact

number of unknowns

Fig. 12. Example 2-Rate of convergence of the error with respect to the total number of unknowns.

281

to align the grid with the discontinuity. The problem is solved using a variety of uniform meshes withII-refinements. p-cnrichments and the lip-adaptive strategy with no special treatment at the shock.

The rate of convergence of the estimated and exact error is compared in Fig. 12 for uniformII-refinements. uniform p-enrichments and two lip-adaptive solutions. The exact error (denoted by asolid line in the figure) and the estimate error (denotcd by a dashed line) are in c10sc agreement.indicating the reliability of the estimate. Notc that with the discontinuity aligned with elementinterfaces. the error behaves as if the solution is smooth. that is. algebraic rates of convergence areachicved with respect to mesh refinements, and exponential ratcs of convergence arc achicved withrespect to p-enrichments. In this case. the most significant error reduction with fewest degrees offreedom will result by specifying a target error for thc II-step which is closer to the initial error than tothe final target error. This is verified, by two curves corresponding to two hp-adaptive solutions in Fig.12. The error history for the hp-adaptive solution starting with an 8 x 8 mesh of p = 1 elements is listedin Table 4. Results on the final mesh are shown in Fig. 13. Poor local error estimates are observed inthe two parallel vertical regions indicatcd by the darker shades in Fig. 13. Morcover. the local error issignificantly undcrestimated in these regions. This under-estimation is also present aftcr thc h-refincment step of the adaptive strategy. This under-estimation is possibly due to a failure of theprocedure to adequately handle the very high changes in gradients in these regions. The globaleffcctivity indices. however. arc quite satisfactory with effectivity indices very near unity as listed inTable 4.

4.3. Example 3

The following data is used in (1):(i) fl=(-l.l)x(-I,I)

(ii) p=(Yj-, Yj-)T(iii) a(x) = 1.0

. {5e-H+Y21 + 3e-(1+(y-WI(tv) g(x. y) = 5[( 1)2+1]-1-8e- X-'i :\

x =-1

Y =-1

Table 4Example 2-Error history for an adaptive lip solution starting with a uniform 8 x 8 mesh. p = 1

Adaptive Targct Achieved Effectivitystcp crror crror index

Initial 8 x 8 mesh p = 1 - 15.4% 0.998II-refinement 7.5% 3.3% 0.<)96p-enrichment 0.5% 0.55% 0.901

282 K.S. lley. J. T. 0111'11I Complil. Methods Appl. Mech. ElIgrg. 133 (1996) 259-286

hp-adapted mesh

Estimated error

PK3.52.5

1.5

0.5

3.1E-3

2.2E-31.3E-34.4E-4

Exact error

Local effecti vi ty index

3.1E-32.2E-31.3E-34.4E-4

llK1.44

1.030.630.23

Fig. t3. Example 2-Results from an hp.adaptive solution.

The source term f in (1) is chosen so that the exact solution is a function which is discontinuous alongthe domain diagonal:

(77)

The global effectivity index for the estimate obtained by solving the upper bound local problem in thespace QI'K+1(K)\QI'K(K) for several uniform lip meshes is listed in Table 5 which shows that the globalerror is slightly under-estimated.

Table 5Example 3--Error estimale obtained by approximating the upper bound local problem in QP""(K)\QP"(K)

Mesh p" 1IJ;IL,u Ilell"u 1)u4x4 1 3.01325 3.81716 0.7<)4x4 2 1.49536 1.97645 0.768x8 I 1.62507 2.10238 0.778x8 2 0.82209 1.35611 0.61

16 x 16 I 0.88158 1.37317 0.6416 x 16 2 0.63044 1.00438 0.6332 x 32 1 0.85472 1.01102 0.85

K.S. Bey. J. T. Odell / ComplII. Methods Appl. Mecll. ElIgrg. 133 (I<J<J6)259-286

Table 6Example 3-Error history for ~n adaptive hp solution starting from a uniform 8 x 8 mesh. p = 1

Adaptive Target Achie\'ed Effcctivitystcp error error index

Initial 8 x 8 mesh p = 1 mesh - 1.18% 0.77Ir-refincment 5% 4.8% 0.63p·cnrichment 3% 3.8% 0.55

283

The error history for an hp-adaptive solution, where only p-enrichment of clcments in rcgions whercthe solution is smooth is used as the third step of the adaptive process, is listed in Table 6. Results atcach step in thc adaptive process, shown in Figs. 14-16, reveal that the under-estimation of the globalerror is primarily duc to the under-estimation of the local crror at thc discontinuity. Notc that while theadaptive strategy is able to reduce the error to the targct value in the h-refinement step, the achievederror after the p step largely represents the remaining error in the discontinuity.

The error achieved by the adaptive strategy is compared to uniform refinements of p = 1 and p = 2meshes in Fig. 17. Thc ratc of convcrgence of the error in the adaptivc solution shows that the rate atwhich the error is reduced in the p-step of the adaptive procedure is much higher than is possible usingIz-refinements alone. Howcver, the overall reduction in the error is relatively small since thepredominant error in the solution is due to thc discontinuity.

I

flI

Exact errorIleIIAu(K)6.3E-15.5E-l4.7E-l3.9E-13.l'E-l2.3E-l1.6E-17.6E-2

Estimated error

IlwllAU (K)

6.3E-l5.5 E-l4.7E-l3.9E-l3.1E-l2.3E-l1.6E-l7.6E-2

Local effectivity index

11K1.080.990.890.800.700.610.510.42

Fig. 14. Examplc 3-Results from the initial 8 x 8 mesh. p = 1.

284 K.S. Iley. J. T. Dell'll I CUlIlpUl. Methods Appl. Mech. E/lgrg. JJ3 (1996) 251)-286

h-adapted mesh, p=l Exact errorlIellAu(K)

1.6E-11.4E-11.2E-19.7E-27.6E-25.5E-23.4E-21.3E-2

Estimated error1I\JfIIAU (K)

1.6E-l1.4E-l1.2E-l9.7E-27.6E-25.5E-23.4E-21.3E-2

Local effectivity indexllK1.671.481.291.110.920.730.540.35

Fig. 15. Example 3-Rcsults from the h·adaptcd mesh.

5. Concluding remarks

The development of lip-version discontinuous Galcrkin methods for hyperbolic conscrvation laws ispresented in this work. A priori crror estimates are derived for a model class of linear hyperbolicconservation laws. These estimates are obtained using a new mesh-depcndent norm that reflects thedependence of the approximate solution on the local element size and the local order of approximation.The results generalizc and extcnd prcvious results on mesh-dependcnt norms to lip-version discontinu-ous Galcrkin mcthods.

A postcriori crror cstimates for thc model problem are also developed. An extension of the elementresidual mcthod to hyperbolic conservation laws result in estimatcs which arc provcn to provide boundson appropriate mcasures of the approximation error.

Numerical experiments verify the a priori cstimates and demonstrate the cffcctivcness of the aposteriori cstimatcs is providing reliable estimates of the actual error in the numcrical solution. Thenumerical examplcs also illustrate the ability of an lip-adaptive strategy to provide super-linearconvergencc rates with rcspect to the number of unknowns in the problem, even for discontinuoussolutions.

Among the major conclusions drawn from this work are the following:(I) The machinery of elliptic approximation theory can be extended to lip-finite element approxi-

mations of hypcrbolic equations using thc notion of discontinuous Galerkin mcthods; this is madcpossible by the introduction of special bilinear and linear forms which depcnd upon meshparameters, thc mesh size 11K of a element Kin thc mesh and thc spectral order fJK of the shapefunctions characterizing local approximations over the cell.

K.S. Bey. J. T. Odell I Complll. Methods App/. Mech. Ellgrg. /33 (1996) 259-286 285

Exact errorIleIlAu(K)

1.6E-11.4E-11.2E-19.7E-27.6E-25.5E-23.4E-21.3E-2

: -1-1 -II:. -

- ~... -.=--. - ---;;

- -- -- .- --

- -·· =--= . -

-. -===~i --

'.-.lZIII

PK4.53.52.51.50.5

hp-adapted mesh-------- ....••

Estimated error11'VIIAu(K)

1.6E- 1l.4E- 11.2E- 19.7E- 27.6E- 25.5E- 23.4E- 21.3 E- 2

Local effectivity indexl. uO

== -==.. .. =----- ==-- =~-= --

--- - .-= =· _._----. = ._--

--

il-I-I: :'.- == - - -c dc--- .. =-

11K

3.102.722.351.981.601.230.850.48

Fig. 16. Example 3-Results from thc p-adapted mesh.

10' 10'number of unknowns

Fig. 17. Example 3-Rate of convcrgenec of the error with respcct to the total numbcr of unknowns.

(2) The use of thc new mesh-dcpendent norms makes it possible to derive, for the first time, a priorierror cstimates for non-uniform lip-approximations of linear hyperbolic problems; these csti-matcs are quasi-optimal, and the estimatcd rates of convergence arc fully confirmcd by numerousnumerical expcriments.

(3) Exponential and/or super-lincar rates of convergence are obtained, even for solutions with verylow regularity; this justifies the usc of lip-methods and dcmonstratcs their superiority overconventional mcthods for a model class of problems.

(4) Rigorous a postcriori error estimates are derivcd using a new version of thc elemcnt residualmethod; thcsc cstimatcs provide computablc measures of local (element-wise) error with

286 K.S. Bey, J. T. Odell I Complll. Methods Appl. Mech. Engrg. 133 (19%) 259-286

remarkable accuracy in smooth regions and provide a reasonable basis for assessing solutionquality and for adaptivity.

Acknowledgment

The second author (JTO) gratefully acknowledges support of his work by the Army Researeh Office.

References

[1] M. Ainsworth and J.T. Oden, A unified approach to a posteriori error estimation using element residualmcthods. Numer.Math. 65 (1993) 23-50.

(2J M. Ainsworth and J.T. Oden, A procedure for a posteriori crror estimation for hop finite element methods, Compu\.Methods Appl. Mech. Engrg. 101 (IW2) 73-96.

[31 1. Babuska and M. Sun. The hp-version of the finite element method with quasiuniform meshes. Math. Modeling Numer.Anal. 21 (1987) 1<)9-238.

[4J R.E. Bank and A. Weiser. Some a posteriori error estimatcs for clliptic partial differential equations. Math. Compu\. 44(1985) 283-301.

(51 K.S. Bey. An Itp-adaptive discontinuous Galerkin method for hyperbolic conservation laws. Ph.D. Disscrtation. TheUniversity of Texa~ at Austin. Austin. TX. May 1994.

[6] R. Biswas, K. Devine and J. Flahcrty, Parallel. adaptive tinite element methods for conservation laws. Department ofComputer Sciencc Technical Report No. 93-5, Rensselacr Polytechnic Institute. Troy. New York. February, 19<)3.

(7] P.G. Ciarlet. The Finite Element Method for Elliptic Problems (North-Holland. Amsterdam, 1978).[8] B Cockburn, S. Hou and C.w. Shu. The Runge-KUlla local projection discontinuous Galerkin method for conservation

laws IV: The multidimensional case. Math. Comput. 54 (1990) 545-581.[9] B. Cockburn and C.w. Shu. TVB Runge-Kutta loeal projcction discontinuous Galcrkin finitc clement methods for

conscrvation laws II: GcneraJ framework, Math. Comput. 52 (1989) 411-435.(10] L. Demkowicz, P. Devloo and J.T. Oden. On an h-typc mesh rcfincmcnt strategy bascd on minimization of interpolation

errors. Comput. Methods App!. Mech. Engrg. 53 (1985) 67-89.[11] M.R. Dorr, The approximation theory for thc p-version of the finite element method, SIAM J. Numer. Anal. 21 (1<)84)

1180-1207.[121 C. Johnson, U. Navert and J. Pitkaranta, Finite element mcthods for linear hypcrbolic problcms. Comput. Methods Appl.

Mech. Engrg. 45 (1984) 285-312.[13] C. Johnson and J. Pitkaranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math.

Compu!. 46 (1986) 1-26.(141 P. Lesaint and P.A. Raviarl. On a tinitc element method for solving the neutron transport equation. in: C. de Boor, ed ..

Mathematical Aspects of Finite Elements in Partial Differential Equations (Academic Press. New York. t974) 89-123.115/ R. Lohner. K. Morgan and O.C. Zicnkiewicz. Adaptive grid refinement for the compressible Euler equations. in: I.

Babuska. O.c. Zienkicwicz. J. Gago and E.R. dc A. Olivicra. cds., Accuracy Estimates and Adaptive Refinements in FiniteElement Computations (John Wiley and Sons. Ltd .. Chichester, 1<)86)281-2<)7.

[16] J.T. Oden. L. Demkowil'Z. W. Rachowicz and T.A. Westerman. Toward a universal h.p adaptive finitc clement strategy.Part 2: A posteriori error cstimation. Comput. Methods Appl. Mech. Engrg. 77 (1989) 113-t80.

[17] J.T. Oden. L. Demkowicz, T. Strouboulis and P. Devloo. Adaptive methods for problems in solid and l1uid mechanics. in: I.Babuska, O.C. Zienkiewicz. J. Gago and E.R. de A. Olivicra. eds .. Accuracy Estimatcs and Adaptive Rctincmcnts in FiniteElement Computations (John Wiley and Sons. Ltd .. Chichester. 1986) 249-280.

(18] J.T. Odcn. A. Patra and Y.S. Feng. An hp adaptive stratcgy, in: A.K. Noor. ed., Adaptive. Multilevel, and HicrarchicalComputationaJ Strategies AMD-Vo!. 157 (1992) 23-46.

(19] J. Peraire, M. Vahdati. K. Morgan and O.c. Zienkiewicz, Adaptive remeshing for compressible flow computations. J.Comput. Phys. 72 (1987) 449-466.

(20] J. Pcraire. K. Morgan and J. Peiro. Unstructured finite element mesh generation and adaptive procedures for eomputationalfluid dynamics, AGARD Conference Proceedings No. 464 (1990) 18.1-18.12.