Direct Discontinuous Galerkin Method and Its Variations for Second Order Elliptic Equations

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1 DOI /s z Dirct Discontinuous Galrkin Mthod and Its Variations for Scond Ordr Elliptic Equations Hongying Huang 1,2 Zhng Chn 3 Jin Li 4 Ju Yan 3 Rcivd: 15 Dcmbr 2015 / Rvisd: 10 Jun 2016 / Accptd: 9 August 2016 Springr Scinc+Businss Mdia Nw York 2016 Abstract In this papr, w study dirct discontinuous Galrkin mthod Liu and Yan in SIAM J Numr Anal 471): , 2009) and its variations Liu and Yan in Commun Comput Phys 83): , 2010; Viddn and Yan in J Comput Math 316): ,2013; Yan in J SciComput 542 3): , 2013) for 2nd ordr lliptic problms. A priori rror stimat undr nrgy norm is stablishd for all four mthods. Optimal rror stimat undr L 2 norm is obtaind for DDG mthod with intrfac corrction Liu and Yan in Commun Comput Phys 83): , 2010) and symmtric DDG mthod Viddn and Yan in J Comput Math 316): ,2013). A sris of numrical xampls ar carrid out to illustrat th accuracy and capability of th schms. Numrically w obtain optimal k + 1)th ordr convrgnc for DDG mthod with intrfac corrction and symmtric DDG mthod on nonuniform and unstructurd triangular mshs. An intrfac problm with discontinuous diffusion cofficints is invstigatd and optimal k + 1)th ordr accuracy is obtaind. Pak solutions with sharp transitions ar capturd wll. Highly oscillatory wav solutions of Hlmholz quation ar wll rsolvd. B Ju Yan jyan@iastat.du Hongying Huang huanghy@lsc.cc.ac.cn Zhng Chn zchn@iastat.du Jin Li lijin@lsc.cc.ac.cn 1 School of Mathmatics, Physics and Information Scinc, Zhjiang Ocan Univrsity, Zhoushan, Zhjiang, China 2 y Laboratory of Ocanographic Big Data Mining and Application of Zhjiang Provinc, Zhoushan, Zhjiang, China 3 Dpartmnt of Mathmatics, Iowa Stat Univrsity, Ams, IA 50011, USA 4 School of Scinc, Shandong Jianzhu Univrsity, Jinan, Shandong, China

2 ywords Discontinuous Galrkin mthod Scond ordr lliptic problm 1 Introduction In this articl, w considr to study dirct discontinuous Galrkin finit lmnt mthod [17] and its variations [18,24,27] for 2nd ordr lliptic problm, x) u) + cx)u = f, in R 2, 1.1) associatd with Dirichlt boundary condition u = u 0 on. To simplify th prsntation, w focus on modl problm 1.1) undr two-dimnsional stting. W hav x = x 1, x 2 ) with as a boundd and simply connctd polygonal domain. Diffusion cofficint matrix is dnotd as x) and is assumd bing uniformly positiv dfinit. Hr f is a givn function in L 2 ). WassumthdatainEq.1.1) satisfy standard rgularity assumptions. Th spcial cas of 1.1) is th Poisson s quation, u = f, 1.2) and Laplac s quation of 1.2) with f = 0. In litratur w hav normous amount of articls discuss numrical mthods solving problm 1.1). W skip th long rviw list. Singular solutions may aris from lliptic problm 1.1) on non smooth domains, with combind boundary conditions or discontinuous diffusion cofficints. Ths singularitis impos challngs and various difficultis on th dvlopmnt of accurat and fficint numrical mthods solving 1.1). In this papr, w study dirct discontinuous Galrkin finit lmnt mthod [17] and its variations [18,24,27] on th modl problm 1.1). Discontinuous Galrkin DG) mthod is a class of finit lmnt mthod that us compltly discontinuous picwis functions as th numrical approximations. Basis functions ar compltly discontinuous across lmnt dgs, thus DG mthods hav th flxibility that is not shard by standard finit lmnt mthods, such as th allowanc of arbitrary triangulations with hanging nods, complt frdom of choosing polynomial dgrs in ach lmnt p-adaptivity), and xtrmly local data structur. It is blivd that DG mthod is spcially suitabl to captur solutions with sharp transitions or discontinuitis, and solutions with complx structurs. W rfr to rviw articls [10,12,23] for th succssful dvlopmnts of DG mthods on convction diffusion problms and rfr to rcnt books [13,16,21] on DG mthods. Thr ar svral DG mthods for solving lliptic and parabolic problms. On class is th intrior pnalty IP) mthods, dats back to 1982 by Arnold in [1] also by Bakr in[3] and Whlr in [26]), th Baumann and Odn [5,19] and NIPG[22] mthods. Anothr class is closly rlatd to mixd finit lmnt mthods [8,20], th local discontinuous Galrkin mthod introducd in [11] by Cockburn and Shu originally studid by Bassi and Rbay in [4] for comprssibl Navir Stoks quations). W rfr to th unifid analysis papr [2] in 2002 for th rviw of diffrnt diffusion DG solvrs. Rcnt dvlopmnts of DG mthods on lliptic problms includ th ovr pnalizd DG mthod [6], th hybridizd DG mthod [9] and th wak Galrkin mthod [25], tc. In [17] w dvlopd a dirct discontinuous Galrkin DDG) mthod solving tim dpndnt diffusion quations. Th ky contribution of [17] is th introduction of numrical flux û x that approximats th solution drivativ u x at th discontinuous lmnt dg. Th numrical flux formula û x dsignd in [17] involvs th solution jump u, solution drivativ avrag {{u x }} and highr ordr drivativ jump valus of u across lmnt dg. Th schm

3 is dirctly basd on th wak formulation of th diffusion quation, thus gains its nam th dirct DG mthod. Du to accuracy loss with high ordr approximations, in [18] wfurthr dvlopd DDG mthod with intrfac corrction. Numrically w obtain optimal k + 1)th ordr convrgnc in [18] with a small fixd pnalty cofficint applid. As is wll known, th pnalty cofficint of symmtric intrior pnalty mthod SIPG) mthod dpnds on th approximation polynomial dgr and nds to b larg nough to stabliz th schm. W also hav th symmtric vrsion [24] and nonsymmtric vrsion [27] of th DDG mthod. Compard to NIPG mthod [22], nonsymmtric DDG mthod [27] obtains optimal ordr convrgnc with any dgr polynomial approximations. In this articl, w furthr dvlop DDG mthod [17] and its variations [18,24,27] to solv lliptic modl problm 1.1). Continuity and corcivity of th primal bilinar form ar obtaind. A priori rror stimat undr nrgy norm is stablishd for all four DDG mthods. A priori optimal rror stimat undr L 2 norm is obtaind for DDG mthod with intrfac corrction [18] and symmtric DDG mthod [24]. A sris of numrical xampls ar carrid out to illustrat th accuracy and capability of th mthods. With P k polynomial approximations w obtain optimal k + 1)th ordr convrgnc for DDG mthod with intrfac corrction [18] and symmtric DDG mthod [24] on nonuniform and unstructurd triangular mshs. Thn w focus on numrical studis of ths two DDG mthods. An intrfac problm with discontinuous diffusion cofficints is invstigatd and optimal k + 1)th ordr accuracy is obtaind vn th solution itslf is not vn C 1 ) across intrfac lins. For th intrfac problm w mak no modification on schm formulations and th zro flux jump condition is simply applid wakly through th numrical flux dfind on lmnt dgs. Pak solution with sharp transitions is capturd wll with ths two DDG mthods. Highly oscillatory wav solutions of Hlmholz quation ar wll rsolvd. Among th four DDG mthods, symmtric DDG mthod [24] is shown to b th most suitabl lliptic solvr not only bcaus th linar systm is symmtric for Laplac for xampl) such that fastr solvrs can b applid. Undr sam sttings th symmtric DDG mthod rsolvs th highly oscillatory wav bttr than th DDG mthod with intrfac corrction. Whn comparing to SIPG mthod [1], symmtric DDG mthod roughly savs 7 10 % on CPU tim with high ordr and on rfind msh simulations. Th rst of th articl is organizd as follows. In Sct. 2, w prsnt schm formulations of DDG mthod and its variations applid to modl problm 1.2) and problm 1.1) with variabl cofficint diffusion matrix. In Sct. 3 w prsnt stability and a priori rror stimat undr a standard nrgy norm and L 2 norm. Finally numrical xampls ar shown in Sct. 4. Throughout this papr, w lt H s ) and H s ) dnot th sminorm and norm of spac H s ), s 0, rspctivly. Lt H s ) dnot th spac of H s ) H s ) and L 2 ) th spac of L 2 ) L 2 ). 2 Discrtization of Dirct DG Mthod and Its Variations Lt T h b a shap-rgular partition of th domain into disjoint lmnts { } Th,for xampl triangls or quadrilatrals with = Th.Byh = diam ), w dnot th diamtr of an lmnt T h.wsth = max Th h as th msh siz of th partition. W dnot by Eh I th st of all intrnal dgs, and by E h D th st of all boundary dgs of T h. And w hav E h = Eh I E h D as th collction of all dgs. Th lngth of th dg E h is dnotd by h. W hav P k ) rprsnting th polynomials function spac of dgr at most k on lmnt. Th DG solution spac is dfind as,

4 V k h :={v L2 ) : v P k ), T h }. Suppos and ar two adjacnt lmnts and shar on common dg.thrartwo tracs of v along th dg, whr w add or subtract thos valus to obtain th avrag and th jump. W dnot by n = n 1, n 2 ) T th outward unit normal vctor pointing from into its nighbor lmnt. Now th avrag and th jump of v ovr dg ar dfind and dnotd as follows, {{v}} = 1 2 v + v ), v = v v, =. Lt s us Poisson Eq. 1.2) to illustrat Dirct DG [17] and its variations [18,24,27] schms formulations. Multiply Eq. 1.2) with arbitrary smooth tst function v, intgrat ovr lmnt T h, hav th intgration by parts and w obtain, u vdx u nvds = f vdx. 2.1) Th ida of Dirct DG mthod [17] is to dsign a formula to approximat th gradint u across th discontinuous lmnt dg and obtain a DG mthod that is basd dirctly on th wak formulation 2.1) of1.2). With no ambiguity, for th rst of this articl w us sam lttr u instad of notation u h to rprsnt DG numrical solution. Now th Dirct DG mthod of 1.2) is dfind as, w sk numrical solution u Vh k such that for all tst function v Vh k w hav, u vdx û n vds = f vdx, T h. 2.2) Th numrical flux û n which approximats th normal drivativ u n = u n involvs th solution jump u, th normal drivativ avrag {{u n }} and highr ordr normal drivativ jumps of u on th dg, u û n = β 0 +{{u n }} + β 1 h u nn + β 2 h 3 h u 4n +. In [17], w show it is hard to idntify suitabl cofficint β 2 to obtain optimal convrgnc for high ordr P k k 4) approximations. Thus w add xtra intrfac trms and hav th DDG mthod with intrfac corrction in [18] such that optimal convrgnc is obtaind for any ordr approximations. Furthrmor, w introduc sam format numrical flux for th tst function and obtain th symmtric [24] and nonsymmtric vrsion [27] of th DDG mthods. Now w summariz schm formulations of DDG variations for modl quation 1.2) as follows, u vdx û n vds + σ ṽ n u ds = f vdx, for all v V h, 2.3) with û n and ṽ n dfind on th intrior lmnt dg Eh I as, u {ûn = u n = β 0u h +{{u n }} + β 1 h u nn, v 2.4) ṽ n = v n = β 0v h +{{v n }} + β 1 h v nn. W drop highr ordr trms and only kp th jump, normal drivativ avrag {{u n }} and scond ordr normal drivativ jump u nn trms in th numrical flux formula. Notic that

5 th tst function v Vh k is takn bing zro outsid th lmnt. In a word, only on sid contributs to th calculation of ṽ n on. Thus trm ṽ n ssntially dgnrats to, v) ṽ n = β 0v + 1 ) h 2 v n + β 1 h v nn ). To apply Dirichlt typ boundary condition, i.. Eh D,whav, u v û n = β 0u + u n with u = u 0 u, and ṽ n = β 0v + v n. 2.5) h h If a Numann typ boundary condition is givn, i.. u n = n u = g is availabl on,w dirctly applis û n = g, on. In th numrical flux formula 2.4), h is takn as th lngth of dg = or th avrag h = h + h )/2 with h and h bing th diamtrs of lmnt and. Numrically w obsrv no ssntial diffrnc with ithr choic of h. Th cofficints β 0u,β 0v and β 1 ar chosn to nsur th stability and convrgnc of ths mthods. Dpnding on th sign of σ =+1 or σ = 1 in2.3), corrspondingly w hav th symmtric and nonsymmtric vrsion of DDG mthods. Now w list th thr variations of DDG mthods and discuss thir proprtis in dtails. 1. DDG mthod with intrfac corrction [18]: σ =+1in2.3) with ṽ n ={{v n }} in 2.4) 2.6) with β 1 = 0 in th numrical flux û n of 2.4), th DDG mthod with intrfac corrction [18] dgnrats to th symmtric Intrior Pnalty mthod. With β 1 = 0, optimal convrgnc is obsrvd with a small fixd pnalty cofficint applid for all P k polynomial approximations. For xampl, w choos fixd β 0u = 2forallP k k 9) polynomials in [18]. As is wll known, th pnalty cofficint β 0u in this cas) should b takn larg nough, roughly in th scal of k 2 for P k polynomials to stabliz th symmtric Intrior Pnalty mthod. 2. Symmtric DDG mthod [24]: σ =+1in2.3) with 2.4) 2.7) In [24], w apply sam format numrical flux for th tst function and obtain a symmtric DDG schm. Optimal L 2 rror stimat is provd. Analytically w show that any β 0,β 1 ) cofficints pair, with β 0 = β 0u + β 0v in 2.4), that satisfis a quadratic form inquality β 0 > 4 β 1 ) 2 k2 k 2 1) 2 3 β 1 k 2 k 2 1) 2 + k2 4 lads to an admissibl numrical flux, and guarants th optimal convrgnc of th symmtric DDG mthod. 3. Nonsymmtric DDG mthod [27]: ), σ = 1in2.3) with 2.4) 2.8) with β 1 = 0in2.4), th nonsymmtric DDG schm [27] dgnrats to th Baumann andodn[5] mthod β 0 = β 0u β 0v = 0) or th NIPG [22] mthod β 0 = β 0u β 0v > 0). With β 1 = 0, w obsrv optimal k + 1)th ordr convrgnc for any P k polynomial

6 approximations, s [27], which improvs th sub-optimal kth ordr convrgnc of Baumann Odn and NIPG mthods. Nxt w considr DDG schm formulation for th following variabl cofficint linar diffusion quation, x) u) = f. Th symmtric DDG schm formulation for abov variabl cofficint lliptic quation is to find DG solution u Vh k such that v V h k and on any lmnt T h,whav, x) u v dx x) u nv ds + x) v n u ds = f vdx. 2.9) Hr th diffusion cofficint matrix is dnotd as x) = k ij x)) with x. With normal vctor n = n 1, n 2 ) and cofficint k ij x) wll dfind on th dg, th numrical flux can b writtn out in dtail as x) u n = 2 i, j=1 k ij x)û x j n i. Similar to 2.4), w hav û x j and ṽ x j dfind on th dg as follows, u {ûx j = β 0u h n j +{{u x j }} + β 1 h u x j x 1 n 1 + u x j x 2 n 2, v ṽ x j = β 0v h n j +{{v x j }} + β 1 h v x j x 1 n 1 + v x j x 2 n 2. Rmark 2.1 For poisson Eq. 1.2), symmtric DDG mthod is th only on giving symmtric stiffnss matrix such that fast solvrs can b applid. Th rst thr DDG mthods lad to nonsymmtric linar systm. Rmark 2.2 W tak Taylor xpansion polynomials around lmnt cntr as basis functions in our numrical tsts. To simplify th comparisons among all four DDG mthods 2.10), w choos fixd cofficint β 1 = 1/40 in all xampls for P k 2 k 4) approximations vn thr xists a larg class of admissibl β 0,β 1 ) cofficint pair i.. symmtric DDG mthod). To simplify th discussion and prsntation, w focus on Poisson Eq. 1.2) associatd with zro Dirichlt boundary condition u = u 0 = 0 for th following thortical discussions in this articl. W can trivially xtnd th rsults to linar Eq. 1.1). Now summing up 2.3) ovr all lmnts T h, w hav th primal formulation of DDG mthod and its variations of 1.2)as:findu Vh k such that B h u,v)= Fv), v Vh k. 2.10) Th bilinar form B h w, v) is listd blow as, B h w, v) := w vdx + ŵn v ds + σ ṽ n w ) ds T h Eh I + ) β0 wv w n v σv n w ds, 2.11) h E D h with th right hand sid Fv) givn as, Fv) = f vdx.

7 Hr w hav β 0 = β 0u +σβ 0v. Again, it dgnrats to th original DDG mthod whn taking σ = 0in2.11). Coupld with 2.4) and taking σ =±1in2.11), w hav th symmtric and nonsymmtric vrsion of th DDG mthods. Th DDG with intrfac corrction is th cas with σ =+1in2.11) and with tst function numrical flux takn as th avrag ṽ n ={{v n }}. 3 Bounddnss, Stability and Error Estimat In this sction, w carry out a unifid rror stimat for th DDG mthod and its variations 2.10). W first list approximation proprtis of th solution spac Vh k and discuss th bounddnss and stability of th bilinar form B h, ). Thn w stablish a suitabl nrgy norm rror stimat for th four DDG mthods 2.10). Toward th nd of this sction, w obtain th optimal rror stimat undr L 2 norm for DDG with intrfac corrction and symmtric DDG mthods. First lt s dfin th nrgy norm for v Vh k: 1/2 v h := v vdx + v 2 ds + v 2 ds. 3.1) T h Eh I h Eh D h 3.1 Approximation Proprtis and Stability Blow w list th trac inquality and invrs inquality of th solution spac V k h.wrfr to finit lmnt txtbooks, i., [7]or[21] rgarding ths classical rsults. Lmma 3.1 Trac inquality) For any lmnt T h and v H s ) with s 1, thr xist positiv constants C g indpndnt of such that,, w hav, v L 2 ) C g h 1/2 ) v L 2 ) + h v L 2 ), v n L 2 ) = v n L 2 ) C g h 1/2 v L 2 ) + h 2 ) v L 2 ). Lmma 3.2 Invrs inquality) For any lmnt T h and v P k ), thr xist positiv constants C t, C i indpndnt of such that, and 0 j k, w hav, v L 2 ) C t h 1/2 v L 2 ), v n L 2 ) = v n L 2 ) C t h 1/2 v L 2 ), j v L 2 ) C i h j v L 2 ). Nxt w stablish th continuity and corcivity of th bilinar form 2.11). Thorm 3.1 Thr xist positiv constants C s,c b for DDG mthod and its variations 2.10) such that for any w, v V k h,whav, B h w, v) C b v h w h. 3.2) B h v, v) C s v 2 h, 3.3)

8 Proof Plug in th numrical flux formula 2.4) in2.11), w hav th bilinar form laid out in dtail as, B h w, v) = w vdx + ) β0 w +{{w n }} + β 1 h w nn v ds T h Eh I h + σ {{vn }} + β 1 h v nn ) w ds + ) β0 wv w n v σv n w ds. Eh I Eh D h 3.4) W first show th continuity 3.2) of th bilinar form. W considr {{w n}} v ds and β 1h w nn v ds as xampl trms and trat othr trms in th bilinar form of 3.4) in a similar fashion. Using Cauchy Schwarz inquality, w hav, {{w n }} v ds {{w n }} L 2 ) v L 2 and ) β 1 h w nn v ds β 1 h wnn L 2 ) v L 2 ). Now lt s study ths two trms in dtail. W intnsivly apply invrs inqualitis Lmma 3.2) to bound th polynomial intgral on dg by its intgrals ovr th lmnts and ssntially by its nrgy norm. Lt s assum dg is a common dg shard by lmnts 1 and 2. With th dfinition of avrag {{ }} and jump, whav, {{w n }} L 2 ) 1 2 wn ) L 1 2 ) + 1 wn ) L ), and wnn L 2 ) wnn ) 1 L 2 ) + wnn ) 2 L 2 ). Applying invrs inqualitis, w bound th lin intgral on dg by its intgrals ovr th lmnts as, {{w n }} L 2 ) C t 2 h 1/2 1 w L 2 1 ) + C t 2 h 1/2 2 w L 2 2 ). Similarly th lin intgral w nn 1 L 2 ) rstrictd from th 1 sid can b boundd as, wnn L 1 2 ) = w n n L 2 ) C t h 1/2 1 w n L 2 1 ) C t C i h 3/2 1 w n L 2 1 ) 2C t C i h 3/2 1 w L 2 1 ). Notic that h dnots th diamtr or th longst dg of lmnt, thus w hav h h 1 and h h 2 with = 1 2. Now combin th prvious argumnts and w obtain, {{w n }} v ds C t 2 C t 2 C t 2 v h 1/2 1 w L 2 1 ) + h 1/2 2 w L 2 2 )) L 2 ) h /h 1 ) 1/2 w L 2 1 ) + ) ) 1/2 h /h 2 w L 2 2 ) 1/2 w L 2 1 ) + w L 2 2 )) h v L 2 ), h 1/2 v L 2 )

9 and β 1 h w nn v ds β 1 h wnn L 2 ) v L 2 ) v 2β 1 h C t C i h 3/2 1 w L 2 1 ) + h 3/2 2 w L 2 2 )) L 2 ) ) 1/2 2β 1 C t C i w L 2 1 ) + w L 2 2 ) h v L 2 ). Sum up th stimats ovr all intrior dgs Eh I and w hav, {{w n }} v ds C t 1/2 w L ) + w L 2 2 )) h v L 2 ) E I h E I h E I h 2 2 C t Eh I and β 1 h w nn v ds 2C t C i β 1 w 2 L 2 1 ) + w 2 L 2 2 ) ) 1/2 1/2 h v L 2 ) 6 2 C t 1/2 1/2 w 2 L 2 ) h 1 v 2 L 2 ) Th Eh I 6 2 C t w h v h, 3.5) E I h E I h 1/2 w L 2 1 ) + w L 2 2 )) h v L 2 ) ) 1/2 2β 1 C t C i w 2 L 2 1 ) + w 2 1/2 L 2 2 ) h v L 2 ) 2 3β 1 C t C i 1/2 w 2 L 2 ) T h E I h h 1 v 2 L 2 ) 2 3β 1 C t C i w h v h. 3.6) For dgs falling on domain boundary, w us similar mthod to bound th trms and w hav, w n v) ds C t E D h T h = C t w h v h. w 2 L 2 ) 1/2 E D h h 1 v 2 L 2 ) Back to th bilinar form B h w, v) of 3.4), w apply stimats 3.5) 3.6) to th xampl trms and trat othr trms in th bilinar form similarly and finally w hav, 1/2 1/2

10 B h w, v) w L 2 ) v L 2 ) + β 0 h 1 w L 2 ) v L 2 ) T h E h C t w h v h β 1 C t C i w h v h + 2 C t v h w h + 2 3β 1 C t C i v h w h + 2C t w h v h C b w h v h, whr C b = 1 + β ) 3β 1 C t C i C t. W ar finishd with th continuity discussion of 3.2). To obtain th corcivity of th bilinar form 3.3), again w considr xampl trms 3.5) 3.6) and w apply Young s inquality. For any δ>0andε>0, w hav, E I h n }} v ds {{w 1 w 2 2ε L 2 ) εc2 t T h E I h h 1 v 2 L 2 ), and E I h β 1 h w nn v ds 1 w 2 2δ L 2 ) + 6δβ2 1 C2 t C2 i T h E I h h 1 v 2 L 2 ). Handl othr trms in th bilinar form similarly and w hav, B h v, v) 1 1 ε 1 ) v 2 δ L 2 ) T h + β 0 12δβ1 2 C2 t C2 i 5 ) 2 εc2 t h 1 v 2 L 2 ). E h W can choos ε, δ and β 0 such that 1 1 ε 1 δ > 0andβ 0 > 12δβ1 2C2 t C2 i εc2 t.now tak C S = min{1 1 ε 1 δ,β 0 12δβ1 2C2 t C2 i 2 5 εc2 t } and w obtain th stability of th bilinar form 3.3). 3.2 Enrgy Norm and L 2 Norm Error Estimats According to Thorm 3.1, it is asy to obtain th following thorm. Thorm 3.2 Thr xists a uniqu solution u V k h for problm 2.10). Proof Sinc 2.10) is a linar problm in finit dimnsional spac, xistnc is quivalnt to uniqunss. W assum that thr ar two numrical solutions u 1 and u 2.Thnwhavth diffrnc w = u 1 u 2 satisfying B h w, w) = 0. By th corcivity rsult 3.3), w hav w h = 0 which dirctly implis that u 1 = u 2. Thorm 3.3 Lt u x H k+1 ) C 2 ) b th xact solution of Poisson Eq. 1.2) with zro Dirichlt boundary and w hav u Vh k dnot on of th four DDG schms 2.10) solutions, w hav, B h u x u,v)= 0, v Vh k. 3.7)

11 Proof W dnot by u x th xact solution, thus w hav 2.1) holding tru for any v Vh k. Summing 2.1) ovr all lmnts T h and formally w hav, u x vdx u x nvds = f vdx. T h T h With u x C 2 ), whav u x = 0, {{ u x n}} = u x n and u x ) nn = 0 across ovr any intrlmnt dg.walsohavu x = 0 with th zro Dirichlt boundary condition. With boundary condition 2.5) applid, th dfinition of numrical flux 2.4) and th bilinar form 2.11), for any v V k h, w hav th xact solution satisfying th bilinar form as blow, B h u x,v) = u x n v + σ ṽ n u x ds T h u x vdx + Eh I + E D h β0 h u x v u x nv σv n u x ) ds = f vdx = Fv). This dirctly implis that 3.7) holds tru. Thorm 3.4 Lt u x H k+1 ) C 2 ) b th xact solution of 1.2) with zro Dirichlt boundary and u Vh k b on of th four DDG schms 2.10) solutions, thn w hav, u x u h Ch k u x H k+1 ). 3.8) Proof Lt Iu x V k h dnot th continuous intrpolation polynomial of u x ovr th lmnt dgs, thn w hav standard approximation rror as, 0 s k + 1, u x Iu x H s ) C I h k+1 s u x H k+1 ), T h. 3.9) Sinc both u x and Iu x ar continuous thus w hav zro jumps u x Iu x = 0 across intrlmnt dgs. From 3.1) and with zro Dirichlt boundary condition applid w hav, u x Iu x h = u x Iu x ) u x Iu x )dx C I h k u x H k+1 ). T h 3.10) Coupld with abov intrpolation rror 3.10), w s th stimat of u x u h can b asily obtaind onc w hav stimat on u Iu x h. For convninc, lt s dnot th rror btwn DDG numrical solution and xact solution intrpolation as χ = u Iu x.w hav u and Iu x Vh k and u x Iu x = 0 across intrlmnt dgs. With Thorm 3.1 and Thorm 3.3 w hav, C s u Iu x 2 h B hu Iu x,χ)= B h u u x + u x Iu x,χ) = B h u x Iu x,χ) = u x Iu x ) χdx T h + Eh I E D h ) {{ u x Iu x ) n}} + β 1 h u x Iu x ) nn χ ds u x Iu x ) nχ ds. 3.11)

12 To obtain th stimat on u Iu x h, w nd to furthr stimat th right hand sid trms of th abov quality 3.11). Using Cauchy Schwarz inquality, w obtain bounds on th last thr itms as, {{ u x Iu x ) n}} χ ds h 1/2 {{ u x Iu x ) n}} L 2 ) h 1/2 χ L 2 ), E I h E I h E I h β 1 h u x Iu x ) nn χ ds β 1 h u x Iu x ) nn L χ 2 ) L 2 ), and E D h u x Iu x ) nχds E D h E I h h 1/2 u x Iu x ) n L 2 ) h 1/2 χ L 2 ). Again w assum dg is a gnric intrior dg shard by lmnts 1 and 2, thus w hav, {{ u x Iu x )}} n L 2 ) 1 u x Iu x ) n L ) + 1 ux Iu x ) n L ), and ux Iu x) nn L 2 ) ux Iu x ) nn L 1 2 ) + ux Iu x ) nn L 2 2 ). Furthrmor, with trac inquality of Lmma 3.1 and intrpolation rror 3.9) whav, u x Iu x ) n L 1 2 ) C gh 1/2 h ux 1 Iu x ) L 2 1 ) + h 1 2 ) u x Iu x ) L 2 1 ) 6C g C I h k 1/2 1 u x H k+1 1 ), and ux Iu x ) nn L 1 2 ) C gh 1/2 ux 1 Iu x ) nn L 2 1 ) ) + h 1 u x Iu x ) nn L 2 1 ) 3C g C I h k 3/2 1 u x H k+1 1 ). Collct all stimats of th right hand sid trms of 3.11) and w hav, C s χ 2 h 9C g + 9C g β 1 + 1)C I h k u x H k+1 ) χ h, or u Iu x h Ch k u x H k+1 ). Applying triangl inquality and with 3.10), w dirctly obtain, u x u h u x Iu x h + Iu x u h Ch k u x H k+1 ). To carry out th rror stimat undr L 2 norm, w follow standard duality argumnt. For convninc, w considr continuous linar finit lmnt spac Ṽ h :={v H 1 ) : v P 1 ), T h,v = 0} to solv th auxiliary problm.

13 Thorm 3.5 Lt u x H k+1 ) C 2 ) solv th boundary valu problm 1.2) with zro Dirichlt boundary and w hav u dnot th DDG mthod with intrfac corrction 2.6) or symmtric DDG mthod 2.7) solution of problm 2.10), w hav, u x u L 2 ) Ch k+1 u x H k+1 ). 3.12) Proof W start with th following auxiliary problm: ψ = u x u, on, with Dirichlt boundary ψ = ) Standard rgularity rsult givs, ψ H 2 ) C u x u L 2 ). 3.14) W solv auxiliary problm 3.13) and dnot ψ h Ṽ h as th solution of conforming finit lmnt mthod, ψ h v h dx = u x u)v h dx, for all v h Ṽ h. 3.15) Rcall that w hav following rror stimat with linar polynomial approximations, ψ ψ h H 1 ) Ch ψ H 2 ). 3.16) With rgularity rsult 3.14) whavψ H 2 ), thus w hav {{ψ}} = ψ, ψ = 0, {{ ψ n}} = ψ n and ψ n = 0 across intrlmnt dgs. Multiply 3.13) with u x u) and intgrat ovr th domain, hav intgrating by parts ovr ach lmnt and formally w obtain, u x u 2 L 2 ) = ψu x u)dx = T h ψ u x u)dx ) ψ nu x u)ds = ψ u x u)dx + {{ ψ n}} u x u ds T h Eh I ψ nu x u)ds. 3.17) E D h Notic th numrical solution u = 0 ovr vn th xact solution is continuous across intrlmnt dgs. Now w hav u dnoting th DDGIC 2.6) or symmtric DDG 2.7) solution with bilinar form 2.11). And w hav u x as th xact solution of 1.2) andw hav Thorm 3.3 holding tru. With ψ h Ṽ h as th continuous linar finit lmnt solution of 3.15), w hav ψ h = 0and ψ h ) nn = 0 across intrlmnt dgs. Thus w hav, 0 = B h u x u,ψ h ) = u x u) ψ h dx + {{ ψ h n}} u x u ds T h Eh I ψ h nu x u)ds. E D h

14 Finally w subtract th right hand sid of 3.17) from abov quality, combin th rsults of 3.16)and3.14) and apply Cauchy Schwarz inquality as in Thorm 3.4,whav, u x u 2 L 2 ) = ψ ψ h ) u x u)dx T h + {{ ψ ψ h ) n}} u x u ds Eh I ψ ψ h ) nu x u)ds E D h Ch ψ H 2 ) u x u h Ch u x u L 2 ) u x u h. Applying th nrgy norm rror stimats 3.8), w complt th proof. 4 Numrical Exampls In this sction w provid a squnc of numrical xampls to illustrat th accuracy and capability of DDG mthod and its variations 2.10). For th linar systm, w us a rstartd GMRES mthod solving a nonsymmtric systm and conjugat gradint mthod solving th symmtric ons. To obtain machin lvl prcision, w st th stopping critrion as th rlativ rsidual norm lss than Notic that for most problms prsntd in this sction w hav analytical or xact solution availabl such that th right hand sid function f can b calculatd from th availabl function. Dirichlt boundary condition is givn with th xact solution s rstriction on th domain boundary. W us following notations to dnot th rrors btwn xact solution and numrical solution: h L := u x u L ), h L 2:= u x u L 2 ), h h := u x u h. Furthrmor, w hav h and h/2 rprsnting th rror at two conscutiv triangulations with msh siz h and h/2, rspctivly. Th ordr is calculatd with, ordr = 1 ) ln2) ln h, h/2 whr rprsnts th L norm, th L 2 norm or th nrgy norm 3.1). Exampl 4.1 Convx domain with structurd and unstructurd triangular mshs. W start with th accuracy chck of th four DDG mthods 2.10) onpoissoneq.1.2) on convx domain =[0, 1] [0, 1]. Right hand sid function is givn with f x 1, x 2 ) = 4 1 x 2 1 x2 2) xp x 2 1 x 2 2). Exact solution is availabl with u x = xp x 2 1 x2 2). W considr implmntations of th four DDG mthods on thr diffrnt mshs: structurd uniform msh and nonuniform msh Fig. 1) and unstructurd msh Fig. 2). Th structurd nonuniform msh is stup by dividing a uniform msh intrval into thr sub-intrvals in ach axis dirction. Mor prcisly, lt s dnot a uniform msh with x i = ih

15 Fig. 1 Uniform msh lft) and nonuniform msh right) Fig. 2 Unstructurd msh with 312 triangls for i = 0,...,M,whrh = 1/M and x M = 1. Th nonuniform msh nods ar gnratd and dnotd as follows, x 3i = x i, x 3i+1 = x 3i + γ 1 h, x 3i+2 = x 3i+1 + γ 2 h. Hr γ 1 and γ 2 ar positiv numbrs with γ 1 + γ 2 < 1. On uniform msh, DDG mthod 2.2) loss ordr with vn ordr P 2 polynomial approximations. This is similar to th DDG mthod for tim dpndnt problm [17] inwhichit shows it is hard to idntify suitabl cofficint β 1 to obtain optimal convrgnc. For DDGIC 2.6) and symmtric 2.7) and nonsymmtric 2.8) DDG mthods, optimal k + 1)th ordr convrgnc is obtaind undr both L 2 and L norms. To sav spac, w only list th rror tabl for symmtric DDG mthod, s Tabl 1. For implmntations on th dramatic nonuniform msh right on in Fig. 1), w obsrv ordr loss for DDG mthod 2.2) and nonsymmtric DDG mthod 2.8) with vn ordr polynomial approximations, s Tabl 2 for nonsymmtric DDG mthod. Notic that NIPG mthod obtains sub-optimal ordr convrgnc for all P k polynomial approximations, s [15] on nonuniform msh accuracy chck. Both DDGIC and symmtric DDG mthods obtain k +1)th optimal ordr convrgnc on nonuniform msh, s Tabl 3 for DDGIC and Tabl 4 for symmtric DDG mthod. Accuracy chck on unstructurd msh Fig. 2) is carrid out also and similar rsults ar obtaind. To sav spac, again w only list th accuracy tabl for symmtric DDG mthod, s Tabl 5.

16 Tabl 1 Symmtric DDG mthod 2.7) on uniform msh k,β 0,β 1 h h L Ordr h L 2 Ordr h h Ordr β 0 = β 1 = 1/ β 0 = β 1 = 1/ β 0 = β 1 = 1/ Tabl 2 Nonsymmtric DDG 2.8) on nonuniform msh with γ 1 = 1/7,γ 2 = 1/3 k,β 0,β 1 h h L Ordr h L 2 Ordr h h Ordr β 0 = β 1 = 1/ β 0 = β 1 = 1/ Tabl 3 DDGIC 2.6) on nonuniform msh with γ 1 = 1/7,γ 2 = 1/3 k,β 0,β 1 h h L Ordr h L 2 Ordr h h Ordr β 0 = β 1 = 1/ β 0 = β 1 = 1/ Tabl 4 Symmtric DDG 2.7) on nonuniform msh with γ 1 = 1/7,γ 2 = 1/3 k,β 0,β 1 h h L Ordr h L 2 Ordr h h Ordr β 0 = β 1 = 1/ β 0 = β 1 = 1/

17 Tabl 5 Symmtric DDG 2.7) on unstructurd msh with 312, 1248 and 4992 triangl lmnts k,β 0,β 1 h h L Ordr h L 2 Ordr h h Ordr β 0 = β 1 = 1/ β 0 = β 1 = 1/ Tabl 6 CPU tim comparison btwn symmtric DDG and SIPG mthods h k = 3 k = 4 Symmtric DDG β 0 = β 1 = 1/ SIPG β 0 = W also considr fficincy issus of th DDG mthods. Among th four DDG mthods, symmtric DDG mthod 2.7) is th most suitabl lliptic solvr. Th linar systm of symmtric DDG mthod has symmtric structur and is asy to apply fast solvrs. W calculat th mass matrix condition numbrs of DDGIC and symmtric DDG mthods, which ar on th ordr of Oh 1.97 ). Whn comparing with SIPG mthod, symmtric DDG mthod gains roughly 7 10 % on CPU tim for high ordr approximations, s Tabl 6. Exampl 4.2 Accuracy chck on L-shapd domain. In this xampl, w solv Laplac quation on th L-shapd nonconvx domain = [ 1, 1] [ 1, 1]\[0, 1] [ 1, 0]. Dirichlt boundary condition is applid. Exact solution is availabl in polar coordinats) with u x r,θ) = r 2/3 sin 2 3 θ ) + cos 2 3 θ )).Noticthat th rgularity of th solution is that u x H 5 3 ɛ for any ɛ>0. Th partial drivativs of th solution ar singular at th origin. W us uniform msh Fig. 3) to carry out convrgnc studis for th DDG mthod and its variations 2.10). For all four schms, w obtain clos to 5 3 th ordr convrgnc undr L2 norm. In Tabl 7 w list th rrors and ordrs of nonsymmtric DDG mthod 2.8). Slightly bttr convrgnc is obsrvd with DDGIC and symmtric DDG mthods, s Tabl 8 for symmtric DDG mthod. Exampl 4.3 Intrfac problm with discontinuous diffusion cofficints. W solv th following variabl cofficint lliptic problm, x) u) = f x), x =[0, 1] [0, 1], with Dirichlt boundary condition. Th diffusion cofficint matrix x) is diagonal x) = diagk) with k ={10, 10 1, 10 3, 1} that is picwis dfind in four subrgions, s Fig. 4

18 Fig. 3 Uniform msh on L-shapd domain Tabl 7 L-shapd domain with nonsymmtric DDG mthod 2.8), uniform msh k,β 0,β 1 h h L Ordr h L 2 Ordr h h Ordr β 0 = β 1 = 1/ β 0 = β 1 = 1/ Tabl 8 L-shapd domain with symmtric DDG mthod 2.7), uniform msh k,β 0,β 1 h h L Ordr h L 2 Ordr h h Ordr β 0 = β 1 = 1/ β 0 = β 1 = 1/ alsoin[14]). Corrspondingly th two intrfac lins ar x 1 = x c = 0.5andx 2 = y c = 0.5. Uniform triangular msh partitiond along intrfac lins is considrd. Exact solution is availabl with, u x = 1 π k sin x1 ) x 1 x c )x 2 y c ) 1 + x ) x2. Th solution itslf is continuous but th gradint is discontinuous across intrfacs lins. For th givn intrfac jump conditions u = 0and x) u n = 0, w mak no modification on our schm formulations to xplicitly nforc th jump conditions. With

19 1 0.8 k=1000 k=1 0.6 x c,y c ) k=10 k= Fig. 4 Picwis constant diffusion cofficints k ={10, 10 1, 10 3, 1} Tabl 9 Intrfac problm with DDGIC mthod k,β 0,β 1 h h L Ordr h L 2 Ordr h h Ordr β 0 = β 1 = 1/ β 0 = β 1 = 1/ zro flux jump across th intrfac, w s th flux x) u n itslf is continuous and wll dfind on th intrfac lins. For lmnt dg that falls on th intrfac, w incorporat th discontinuous diffusion cofficints x) into th numrical flux x) u n dfinition. For xampl, suppos th lmnt dg falls on intrfac lin x 2 = y c = 0.5 with outward normal n = 0, 1), th numrical flux dgnrats to x) u n = ku) x2 and w hav, k ku + u + k u x2 = β 0 + k u x 2 + k + u + x 2 + β 1 h k + u + x h 2 2 x 2 k u ) x 2 x 2. Hr w hav diffusion cofficint k = k + for x 2 > 0.5andk = k for x 2 < 0.5andu + and u corrspondingly dnot th valu of u on dg valuatd from its nighbor lmnt and from its own lmnt. Thus th zro flux intrfac jump condition is applid WEALY in our implmntations. W carry out P 2 and P 3 polynomial approximations and list th rrors and ordrs in Tabls 9 and 10 for DDGIC and symmtric DDG mthods. W obtain k + 1)th ordr convrgnc undr both L 2 and L norms. Solution simulations with P 2 polynomials and msh siz h = ar shown in Fig. 5. Exampl 4.4 Pak solution. In this xampl, w solv Poisson quation with a pak solution. Th domain is st as =[0, 1] [0, 1] and Dirichlt boundary condition is applid. Th xact solution is availabl with xprssion,

20 Tabl 10 Intrfac problm with Symmtric DDG mthod k,β 0,β 1 h h L Ordr h L 2 Ordr h h Ordr β 0 = β 1 = 1/ β 0 = β 1 = 1/ Fig. 5 Intrfac problm with DDGIC lft) and symmtric DDG right) mthods Fig. 6 Pak solution simulations by DDGIC lft) and symmtric DDG right) mthods u x = xp α x 1 x c ) 2 + x 2 y c ) 2)), whr x c, y c ) = 0.5, 0, 5) is th location of th pak and α = 1000 dtrmins th strngth of th pak. Approximations with DDGIC 2.6) and symmtric DDG 2.7) mthods ar carrid out and shown in Fig. 6 with uniform triangulation msh h = and P 2 polynomial approximations. Th sharp pak is rsolvd vry wll with ths two schms. Exampl 4.5 Highly oscillatory wav solution for Hlmholtz quation. In this xampl w solv Hlmholtz quation with variabl cofficints as follows, u 1 α + r) 4 u = f, with r = x 1 ) 2 + x 2 ) 2.

21 Fig. 7 Oscillatory solution by DDGIC lft) and symmtric DDG right) mthods Th squar domain is st as =[0, 1] [0, 1] and Dirichlt boundary condition is applid. W hav α = Nπ 1 whr th intgr N dtrmins th numbr of oscillatory wavs nar th ) origin. Exact solution is givn with u x = sin. 1 α+r W apply uniform triangular msh with h = and quadratic P 2 approximations in this xampl. Th numbr of oscillations is takn with N = 4inα. Th solution is supposd to b highly oscillatory nar th origin. As shown in Fig. 7, with sam msh and polynomial approximations applid, symmtric DDG mthod 2.7) rsolvs th highly oscillatory wav bttr than th DDGIC mthod 2.6). Acknowldgmnts Huang s work is supportd by Natural Scinc Foundation of Zhjiang Provinc Grant Nos. LY14A and LY12A01009, and is subsidizd by th National Natural Scinc Foundation of China undr Grant Nos , , and Yan s rsarch is supportd by th US National Scinc Foundation NSF) undr grant DMS Rfrncs 1. Arnold, D.N.: An intrior pnalty finit lmnt mthod with discontinuous lmnts. SIAM J. Numr. Anal. 194), ) 2. Arnold, D.N., Brzzi, F., Cockburn, B., Marini, L.D.: Unifid analysis of discontinuous Galrkin mthods for lliptic problms. SIAM J. Numr. Anal. 395), ). lctronic) 3. Bakr, G.A.: Finit lmnt mthods for lliptic quations using nonconforming lmnts. Math. Comput. 31, ) 4. Bassi, F., Rbay, S.: A high-ordr accurat discontinuous finit lmnt mthod for th numrical solution of th comprssibl Navir Stoks quations. J. Comput. Phys. 1312), ) 5. Baumann, C.E., Odn, J.T.: A discontinuous hp finit lmnt mthod for convction-diffusion problms. Comput. Mthods Appl. Mch. Eng ), ) 6. Brnnr, S.C., Owns, L., Sung, L.-Y.: A wakly ovr-pnalizd symmtric intrior pnalty mthod. Elctron. Trans. Numr. Anal. 30, ) 7. Brnnr, S.C., Scott, L.R.: Th Mathmatical Thory of Finit Elmnt Mthods, Volum 15 of Txts in Applid Mathmatics, 3rd dn. Springr, Nw York 2008) 8. Brzzi, F., Douglas Jr., J., Marini, L.D.: Two familis of mixd finit lmnts for scond ordr lliptic problms. Numr. Math. 472), ) 9. Cockburn, B., Gopalakrishnan, J., Lazarov, R.: Unifid hybridization of discontinuous Galrkin, mixd, and continuous Galrkin mthods for scond ordr lliptic problms. SIAM J. Numr. Anal. 472), ) 10. Cockburn, B., Johnson, C., Shu, C.-W., Tadmor, E.: Advancd numrical approximation of nonlinar hyprbolic quations, volum 1697 of Lctur Nots in Mathmatics. In: Quartroni, A. d.) Paprs from th C.I.M.E. Summr School Hld in Ctraro, Jun Springr-Vrlag, Brlin 1998). Fondazion C.I.M.E. [C.I.M.E. Foundation]

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