Symmetric Interior penalty discontinuous Galerkin methods for elliptic problems in polygons

Transcription

1 Symmtric Intrior pnalty discontinuous Galrkin mthods for lliptic problms in polygons F. Müllr and D. Schötzau and Ch. Schwab Rsarch Rport No March 2017 Sminar für Angwandt Mathmatik Eidgnössisch Tchnisch Hochschul CH-8092 Zürich Switzrland Funding SNF: _149819/1 Funding: Natural Scincs and Enginring Rsarch Council of Canada (NSERC)

2 SYMMETRIC INTERIOR PENALTY DISCONTINUOUS GALERKIN METHODS FOR ELLIPTIC PROBLEMS IN POLYGONS FABIAN MÜLLER, DOMINIK SCHÖTZAU, AND CHRISTOPH SCHWAB Abstract. W analyz symmtric intrior pnalty discontinuous Galrkin finit lmnt mthods for linar, scond-ordr lliptic boundary-valu problms in polygonal domains Ω whr solutions xhibit singular bhavior nar cornrs. To rsolv cornr singularitis, w admit both, gradd mshs and bisction rfinmnt mshs. W prov that judiciously chosn rfinmnt paramtrs in ths msh familis imply optimal asymptotic rats of convrgnc with rspct to th total numbr of dgrs of frdom N, both for th DG nrgy norm rror and th L 2 -norm rror. Th sharpnss of our asymptotic convrgnc rat stimats is confirmd in a sris of numrical xprimnts. 1. Introduction Th rror analysis of discontinuous Galrkin finit lmnt mthods (DGFEMs) for lliptic problms is by now wll dvlopd. For various h-vrsion DG formulations, optimal nrgy norm and L 2 -norm rror stimats with rspct to th msh-width h ar providd in,.g., [2, 17, 18], and th rfrncs thrin. W rmark that L 2 -optimality typically rquirs adjoint-consistnt DG discrtizations, as introducd in [2]. Most of th DG rror analyss availabl in th litratur ar basd on sufficint smoothnss of wak solutions and on quasi-uniformity assumptions for th msh squncs. In addition, to driv L 2 -norm rror bounds, an H 2 -rgularity hypothsis for th solution of a suitabl dual problm is usually imposd. Whil ths smoothnss proprtis hold tru in smooth or convx domains, thy ar known to b fals in gnral polygonal domains, du to th apparanc of singular solution componnts nar cornrs [10]. On way to charactriz th singular bhavior of solutions is by mans of suitably wightd Sobolv spacs and corrsponding lliptic rgularity shifts of scond, and highr ordr in ths spacs. Hr, w shall focus on th wightd spacs H k,l (Ω) and shifts as introducd and analyzd in [4, 3] in th contxt of conforming hp-vrsion FEMs. For rlatd, finit ordr lliptic rgularity rsults in polygonal and polyhdral domains, w furthr rfr to [10, 14, 8]. To rsolv cornr singularitis in fixd ordr, conforming h-vrsion discrtizations, in rcnt yars svral typs of local msh rfinmnt stratgis hav bn proposd and invstigatd. In [5], conforming picwis linar FEMs on so-calld Ky words and phrass. AMS Subjct Classification: 65M20, 65M60, 65N30 Elliptic Boundary-Valu Problms, Finit Elmnt Mthods, Discontinuous Galrkin Mthods, Cornr Singularitis, Lipschitz Domains. Rsarch supportd by th Swiss National Scinc Foundation undr Grant No. SNF /1 and by th Natural Scincs and Enginring Rsarch Council of Canada (NSERC). Th authors thank Thomas P. Wihlr for hlpful rmarks. 1

3 2 FABIAN MÜLLER, DOMINIK SCHÖTZAU, AND CHRISTOPH SCHWAB gradd rgular simplicial mshs wr shown to captur singularitis at th optimal H 1 -norm convrgnc rat N 1/2 with rspct to N, th numbr of dgrs of frdom. Mor rcnt variants of approximation rat bounds on familis of gradd mshs, also for FE spacs of polynomial dgr p 1 on gradd, rgular simplicial mshs in Ω can b found in [7, 1]. A public domain msh gnrator for gradd mshs in polygonal domains Ω is availabl in th LNG FEM softwar packag [13]. An altrnativ approach to build locally rfind mshs is basd on local rfinmnt rfinmnt via nwst vrtx bisction; s,.g., [15, 16]. Indd, in [9], a bisction rfinmnt algorithm was proposd and it was provd that it crats msh squncs which rsolv singular solution componnts at optimal H 1 -norm rats N p/2 with rspct to N. Th work in [21] was th first to study (symmtric and non-symmtric) intrior pnalty (IP) discontinuous Galrkin mthods for lliptic problms with solutions in th wightd spacs of [4, 3]. By introducing nw tchnical tools to handl singular solution componnts, it stablishd th bounddnss and consistncy of th intrior pnalty forms. It furthr showd and vrifid numrically that on gradd mshs as in [5], algbraic convrgnc rats of th optimal ordr N p/2 ar obtaind for th DG norm rrors. In [22], ths consistncy and stability proprtis wr mployd to show xponntial convrgnc rats for hp-vrsion intrior pnalty mthods for problms with picwis analytic solutions. For othr work on th analysis of DG mthods for lliptic problms with low-rgularity solutions, w also rfr to [17, Sction 4.2.5], [23] and th rfncs thrin. In this papr, w build on, rfin and xtnd th rsults of [21]. Mor spcifically, w focus hr on symmtric IP mthods which ar adjoint-consistnt. Basd on th wightd spacs of [4, 3], w considr IP approximations on gradd and bisction rfinmnt mshs which ar locally rfind towards cornrs of th domain. Using th tchniqus of [21], w stablish continuity bounds for th IP forms with rspct to suitabl norms, show Galrkin orthogonality and driv optimal nrgy rror stimats in trms of N. In addition, w driv an optimal L 2 -norm rror bound. Whil our approach procds roughly along th lins of standard argumnts as in [2] and is basd on duality, w now mploy lliptic rgularity with rspct to th wightd spacs H l,k (Ω). Hnc, unlik in prvious works, w do not nd to impos any xtra (and unralistic) rgularity assumptions. Our continuity proprtis nsur th wll-dfindnss of th various intgral trms which appar in th drivation of th L 2 -norm convrgnc rat bound. To complt our rror analysis, w prsnt proofs that gradd and bisction rfinmnt mshs yild optimal approximation bounds, both for th primal and th dual solutions. To this nd, w rxamin th rror bound obtaind in [21, Proposition 2.5.5] for gradd mshs; w xtnd it to an stimat of th rror in a slightly strongr norm and for a widr rang of rgularity ordrs. Morovr, for bisction rfinmnt mshs, w stablish a compltly nw variant of th approximation rsults in [9, Thorms 5.2 and 5.3] for wightd spacs and with rspct to our consistncy norm. Finally, w vrify our thortical statmnts in a sris of numrical tsts for bisction rfinmnt mshs. Dtaild numrical xprimnts for gradd mshs ar availabl in [21]. Th papr is structurd as follows: In Sction 2, w introduc our lliptic modl problm in polygonal domains and rviw rgularity shifts in wightd spacs. In Sction 3, w rcall th symmtric IP mthod. Our main rsults ar statd and

4 SYMMETRIC IP METHODS FOR ELLIPTIC PROBLEMS IN POLYGONS 3 discussd in Sction 4 (Thorm 4.3). Sction 5 contains th dtaild proofs of our rror stimats. In Sction 6, numrical xprimnts ar prsntd, vrifying th sharpnss of our thortical stimats. Finally, in Sction 7, conclusions ar offrd and potntial xtnsions ar outlind. Throughout, w us standard notation. In particular, for a domain G R d, d = 1,2 and q [1, ], th Lbsgu spac of q-intgrabl functions is dnotd by L q (G). For k N, th classical Sobolv spacs of functions in L q (G) with q- intgrabl drivativs of ordr up to k will b dnotd by W k,q (G), and by H k (G) if q = Modl Problm W introduc polygonal domains, dfin our modl problm and rviw rgularity shifts in wightd Sobolv spacs Polygonal domain. AnopnandboundddomainΩ R 2 iscalldpolygonal if its boundary Ω can b writtn as a finit union of M N opn and straight lin sgmnts i of positiv surfac masur: Ω = M i=1 i and i ds > 0, 1 i M. (2.1) Th vrtics of th polygon Ω ar givn by c i := i i+1, 1 i M, with th undrstanding that M+1 = 1. W assum th vrtics to b numbrd clockwis. W introduc th st of all vrtics as S := {c i : 1 i M}. Th intrior opning angl of th domain at c i is masurd in positiv orintation and dnotd by ω i (0,2π]. Th cas ω i = 2π ariss in modls of fractur mchanics. Although in this cas th domain Ω is not Lipschitz, it can b writtn as finit union of Lipschitz domains, so that all statmnts on variational formulations rmain valid in this cas. To discuss th solution rgularity in th vicinity of cornrs, w associat with ach cornr local conical domains dfind by Ω i := {x Ω : x c i < R i }, 1 i M, (2.2) whr 0 < R i < 1 2 min i j c i c j. This implis that th cons Ω i ar mutually disjoint and Ω i Ω i i+1. Hnc, Ω i is containd in a infinit con with opning angl ω i and vrtx c i Elliptic boundary-valu problm. Lt now Ω b a polygonal domain. W dnot by D and N th indx sts of all boundary sgmnts i, on which Dirichlt and Numann boundary conditions will b applid, rspctivly. This lads to th partition Ω = Γ D Γ N, whr Γ D = i D i and Γ N = i N i. W furthr dnot by ν th outward unit normal vctor on th boundary Ω. Lt c C (Ω) b a smooth ral-valud diffusion cofficint such that c c(x) c, x Ω, (2.3) for constants 0 < c < c <. Assum givn in Ω a forcing trm f, on Γ D a Dirichlt datum g D and on Γ N a Numann boundary datum g N. Th smoothnss assumptions on th data will b mad prcis subsquntly in Proposition 2.3 and

5 4 FABIAN MÜLLER, DOMINIK SCHÖTZAU, AND CHRISTOPH SCHWAB Rmark 2.4. W considr th diffusion problm: (c u) = f in Ω, (2.4) u = g D on Γ D, (2.5) ν (c u) = g N on Γ N. (2.6) Th standard wak form of problm (2.4) (2.6) rads: Find u H 1 (Ω) such that u ΓD = g D on Γ D and a(u,v) := c u vdx = f vdx+ g N vds (2.7) Ω Ω Γ N for all v H 1 Γ D (Ω) := {v H 1 (Ω) : v ΓD = 0}. If D, problm (2.7) has a uniqu solution providd that g D can b stably liftd in H 1 (Ω) and that th linar functionals on th right-hand sid of (2.7) blong to H 1 Γ D (Ω), th dual spac of H 1 Γ D (Ω). In th pur Numann cas (i.., D = ), th trial and tst functions u and v ar takn in th factor spac H 1 (Ω)/R, i.., wak solutions ar quivalnc classs which diffr by constants. For xistnc, th data f and g N must satisfy th compatibility condition Ω f dx+ Ω g N ds = 0, which will always b assumd to hold in this cas Wightd Sobolv spacs. With ach vrtx c i of Ω, w assign a wight xponnt i R and introduc th wight xponnt vctor = { i } M i=1. For a scalar ξ R, w dfin + ξ by { + ξ} i := i + ξ. Similarly, inqualitis of th form < ξ ar undrstood componntwis. For R M, w dfin th wightd distanc function Φ by M Φ (x) := r i (x) i, (2.8) whr r i (x) = x c i. Rmark 2.1. To localiz th wight function Φ in (2.8), w dcompos Ω into i=1 Ω := Ω 0 ( M i=1 Ω i ), (2.9) whr Ω 0 = Ω\ M i=1 Ω i and whr Ω i is th con in (2.2). If [0,1) M, thn w hav C 1 dc r i(x) i C dc, x Ω 0, 1 i M, (2.10) C 1 dc Φ (x) r i i (x) C dcφ (x), x Ω i, 1 i M, (2.11) for a constant C dc > 0 dpnding on th radii R i in (2.2). Givn intgrs k l 0, w nxt dfin th wightd Sobolv spacs H k,l (Ω) as th compltion of C (Ω) with rspct to th norm v H k,l (Ω) givn by v 2 := H k,l (Ω) Th smi-norm v H k,l (Ω) is givn by v 2, l = 0, H k,0 (Ω) v 2 H l 1 (Ω) + v 2 H k,l (Ω), l 1. (2.12) k v 2 := Φ H k,l (Ω) +m l D m v 2 L 2 (Ω). (2.13) m=l

6 SYMMETRIC IP METHODS FOR ELLIPTIC PROBLEMS IN POLYGONS 5 Hr, w adopt th notation D m v 2 := α =m Dα v 2, with D α v dnoting th partial drivativ of v with rspct to th multi-indx α N 2 0. W shall also mak us of th wightd spacs H k,l i (Ω), thir associatd norms v H k,l (Ω) and sminorms v H i k,l (Ω), which ar dfind compltly analogously, but with rspct to th i wight r i (x) i. Furthrmor, ovr subdomains Ω Ω th wightd spacs and norms ar dfind by rplacing th domains of intgration by Ω. TracspacsofthwightdspacsH k,l (Ω)ardfindasfollows. Ltγ Ωb th union of som lin sgmnts i. For k 1, k l 0, w dfin H k 1/2,l 1/2 (γ) as th spac of all functions φ : γ R such that thr is a function Φ H k,l (Ω) with Φ γ = φ. Th associatd norm is dfind by φ k 1/2,l 1/2 H (γ) := inf{ Φ H k,l (Ω) : Φ γ = φ}; (2.14) s also [4, Sction 1.4]. Th following proprtis hold. Lmma 2.2. Lt [0,1) M. Thr holds: (i) W hav th continuous mbddings H k,2 (Ω) H2,2 (Ω) C0 (Ω), k 2. (2.15) (ii) For k l 1, lt v H k,l (Ω) and lt α N2 0 b such that α l. Thn w hav D α v H k α,l α (Ω) and D α v H k α,l α (Ω) v H k,l (Ω). (2.16) (iii) Lt f H 0,0 (Ω). Thn Ω fvdx is a linar continuous functional on H1 (Ω) and Ω fvdx C f H 0,0 (Ω) v H 1 (Ω), v H 1 (Ω), (2.17) with C > 0 dpnding on. (iv) Lt g N H 1/2,1/2 (Γ N ). Thn Ω g NvdS is a linar continuous functional on H 1 (Ω) and Γ N g N vds C gn H 1/2,1/2 (Γ N) v H 1 (Ω), v H 1 (Ω), (2.18) with C > 0 dpnding on. Proof. Th proof of th scond mbdding in (2.15) can b found in [5, pag 449]. Th first inclusion in (2.15) is trivial. Proprty (2.16) is straightforward. Finally, th bounds (2.17) and (2.18) ar provd in [4, Lmma 2.10 and Lmma 2.11] Rgularity in wightd spacs. W bas our analysis on th following lliptic rgularity shift in wightd Sobolv spacs, which is a consqunc of Rmark 3 and Lmma 3.2 in [4]. In th lowst-ordr cas, w also rfr to [5, Thorm 3.2]. Equivalnt rsults (with diffrntly dfind wightd spacs) ar stablishd in [8, Sctions 4 and 7.1]. Proposition 2.3. Thr xist paramtrs i > 0 (dpnding on th domain Ω, th sts D, N and th cofficint c) such that for wight xponnts [0,1) M with 1 i < i < 1, 1 i M, (2.19)

7 6 FABIAN MÜLLER, DOMINIK SCHÖTZAU, AND CHRISTOPH SCHWAB th following lliptic rgularity shifts hold: for k 1, f H k 1,0 (Ω), g D H k+1/2,3/2 (Γ D ) and g N H k 1/2,1/2 (Ω), th wak solution u H 1 (Ω) of (2.7) xists (s (2.17) and (2.18)) and blongs to H k+1,2 (Ω). Morovr, w hav th stability bound u H k+1,2 (Ω) C stab,k ( f H k 1,0 (Ω) + g D k+1/2,3/2 H (Γ + g D) N k 1/2,1/2 H (Γ N) with a stability constant C stab,k > 0 that is indpndnt of th data (but dpnds on th ordr k, th domain Ω, th sts D, N, and th cofficint c). Rmark2.4. Notic that 1 i in (2.19) can b ngativ. In this cas, th condition i > 1 i is considrd void as it imposs no rstriction on th rang of i [0,1). Morovr, in viw of Proposition 2.3, w will always assum th minimum rgularity f H 0,0 (Ω), g D H 3/2,3/2 (Γ D ), g N H 1/2,1/2 (Γ N ). (2.20) This nsurs that th solution of (2.7) blongs to H 2,2 (Ω). Rmark 2.5. In th cas of th Laplacian (whr c = 1), th paramtr i at cornr c i = i i+1 is wll-known and givn by ), { π i = ω i if {i,i+1} D or {i,i+1} N, othrwis; π 2ω i s,.g., [4, Rmark 3] or [8, Exampl 7.2]. (2.21) 3. Discontinuous Galrkin Discrtization W introduc th symmtric intrior pnalty finit lmnt mthod for th numrical approximation of problm (2.4) (2.6), and rviw th discrt corcivity and continuity of th IP bilinar form Mshs, dgs and trac oprators. Lt T b a partition of Ω into straightsidd triangls K. For as of prsntation, w considr rgular triangulations and commnt on xtnsions to irrgular mshs in Sction 7. Th triangulations ar supposd to b sufficintly fin so that ach lmnt K contains at most on vrtx c i. For K T, w dnot by P p (K) th polynomials on K of total dgr at most p, and by ν K th unit outward normal vctor on K. Furthrmor, w writ h K and ρ K for th diamtr and inradius of K T, rspctivly. Th msh-width of T is givn by h = h(t ) := max K T h K. W assum th triangulations to b shap-rgular: Thr xists a constant κ > 0 such that thr holds, for all K T, and uniformly in th msh squnc, κh K ρ K κ 1 h K. (3.1) Edgs ar dfind as follows. If K and K ar adjacnt lmnts of th triangulation T with K K ds > 0, w call th intrsction = K K an intrior dg. Elmntal dgs of K ar supposd to li at most on on boundary sgmnt i, and if K i ds > 0, w call th intrsction = K i a boundary dg; it blongs to ithr Γ D or Γ N. Accordingly, w distinguish btwn Dirichlt and Numann dgs. Th st of intrior dgs of a triangulation T is dnotd by E I (T ), th st of Dirichlt boundary dgs by E D (T ), and th st of Numann boundary dgs by E N (T ). Morovr, w dfin E ID (T ) := E I (T ) E D (T ) and

8 SYMMETRIC IP METHODS FOR ELLIPTIC PROBLEMS IN POLYGONS 7 E(T ) := E I (T ) E D (T ) E N (T ). For E(T ), w dnot by P p () th polynomials of dgr at most p on, and by h = th lngth of. With th shap-rgularity assumption (3.1), it can b radily vrifid that κh K h h K, (3.2) for all K with E(T ). Following [2], w introduc th standard trac oprators. Lt K +,K T b two adjacnt lmnts which shar th intrior dg = K + K E I (T ). For a sufficintly smooth scalar function v or vctor fild q, w dnot th tracs of v and q on takn from within K ± by v ± and q ±, rspctivly. W thn dfin th jumps and th avrags of v and q along by [v] := v + ν K + +v ν K, v := 1 2 (v+ +v ), (3.3) [q] := q + ν K + +q ν K, q := 1 2 (q+ +q ). (3.4) If E D (T ) is a Dirichlt boundary dg, w similarly st [v] := v ν, [q] = q ν, as wll as v := v, q := q Discrtization. For an approximation ordr p 1 and a givn triangulation T of Ω, w introduc th discontinuous finit lmnt spac V p (T ) := {v L 2 (Ω) : v K P p (K), K T }. (3.5) Thn, th symmtric intrior pnalty discrtization of (2.7) rads as follows: Find u N V p (T ) such that a DG (u N,v N ) = l DG (v N ) (3.6) for all v N V p (T ). Hr, a DG (, ) is th symmtric intrior pnalty form givn by a DG (v,w) := c v wdx r DG (v,w) K T K r DG (w,v)+ (3.7) j [v] [w]ds, with th off-diagonal form r DG (v,w) := E ID(T ) E ID(T ) c v [w]ds. (3.8) Th linar form l DG ( ) on th right-hand sid in (3.6) is dfind as l DG (w) := fwdx g D (c w) νds K T K E D(T ) + j g D wds + g N wds. Γ N E D(T ) E N(T ) In (3.7), (3.9), w dfin th intrior pnalty function j dgwis as (3.9) j := j 0 c h 1, E ID(T ), (3.10) whr j 0 > 0 is a sufficintly larg as spcifid blow and whr w rcall that h dnots th lngth of.

9 8 FABIAN MÜLLER, DOMINIK SCHÖTZAU, AND CHRISTOPH SCHWAB Rmark 3.1. In Sction 5.2, w show th bounddnss of all th trms in (3.7) (3.9) whnvr v or w ar wightd functions in H 2,2 (Ω). In particular, for lmnts K abutting at cornrs and dgs running into cornrs, th intgrals in (3.7) (3.9) ar undrstood as boundd bilinar forms ovr L 1 (K) L (K) and L 1 () L (), rspctivly. Rmark 3.2. Th numbr of dgrs of frdom of th discrtization (3.6) is dfind as N = N(p,T ) := dim(v p (T )). (3.11) W ar intrstd in achiving convrgnc N by rducing th msh-width h 0 at a fixd (typically low) polynomial dgr p 1, which is known as th h- vrsion of th finit lmnt mthod. In th following, w driv optimal algbraic convrgnc rats with rspct to N for th DG mthod (3.6) on locally rfind mshs DG norm and discrt stability. For a partition T of Ω, w introduc th brokn H 1 -spac H 1 (T ) := {v L 2 (Ω) : v K H 1 (K), K T }, (3.12) which w ndow with th DG nrgy norm v 2 DG := K T c 1/2 v 2 L 2 (K) +J(v), J(v) := E ID(T ) j 1/2 [v] 2 L 2 (). (3.13) Notic that for D, th mapping v v DG is a norm on H 1 (T ). In th pur Numann cas, howvr, th xprssion v DG is zro if and only if v is a constant, and hnc it is a norm modulo constants. Th following discrt stability proprtis ovr th brokn FE spac V p (T ) ar wll-known; w rfr for xampl to [2] or [17]. Lmma 3.3. Thr xists j > 0 and constants C cor > 0, C cont > 0, which ar indpndnt of th msh-widths, but dpnd on κ in (3.1), th bounds in (2.3) and th polynomial dgr p, such that for j 0 > j thr holds a DG (v N,v N ) C cor v N 2 DG, v N V p (T ), (3.14) a DG (v N,w N ) C cont v N DG w N DG, v N,w N V p (T ). (3.15) 4. Main Rsults W introduc two typs of msh familis with local rfinmnt towards cornrs. Thn,wstatanddiscussourmainrsults: optimalnrgynormandl 2 -normconvrgnc rat stimats for IP discrtizations of arbitrary ordr (s Thorm 4.3), on ithr typ of msh family with sufficintly strong rfinmnt in th vicinity of cornrs Gradd msh familis. W first rcall th dfinition of gradd msh familis as introducd in [5]. Dfinition 4.1. A shap-rgular family of rgular triangulations T β is calld gradd towards th vrtics in S with grading vctor β = (β 1,...,β M ), if thr xists a uniform constant C G > 0 such that for all lmnts K T β in ach triangulation, on of th following conditions hold: (i) If K T β \K(T β ), thn C G 1 hφ β (x) h K C G hφ β (x) for all x K.

10 SYMMETRIC IP METHODS FOR ELLIPTIC PROBLEMS IN POLYGONS 9 (ii) If K K(T β ), thn C 1 G sup x K Φ β (x) h K C G hsup x K Φ β (x). In[21, Proposition 2.5.5], it has bn shown that IP mthods for lliptic problms on gradd msh familis convrg optimally in th DG nrgy norm (but not in th L 2 -norm). W will stablish a rfinmnt of this rsult for our convrgncanalysis. Othr xampls of gradd msh familis and thir constructions ar wll-known by now. W rfr to [7, 1] and th softwar packag LNG FEM in [13]. Th finit lmnt spacs constructd on ths msh familis, howvr, ar not nstd, i.., an incras of accuracy in th numrical approximation rquirs construction of th ntir finr msh Bisction rfinmnt mshs. An altrnativ ar rgular, simplicial msh familis which ar producd by rcursiv bisction rfinmnt; s,.g., [15, 16] and th rfrncs thrin. Our analysis will b basd on th work [9], whr a bisction rfinmnt algorithm has bn proposd and analyzd in th contxt of conforming finit lmnt mthods. Givn an initial msh T 0, th algorithm thr taks input paramtrs h, p, L and a wight xponnt γ > 0. In a first loop, it nsurs that all lmntal msh-widths h K ar smallr than h. In a scond loop, th algorithm rfins 2L+1 tims into th cornrs using nwst vrtx bisction, whr L is to b slctd in dpndnc of h, p and γ. This rsults in a rgular msh dnotd by T h,2(l+1). W mphasiz that th bisction rfinmnt mshs constructd from th rgular, simplicial initial msh T 0 givs ris to a shap-rgular msh family, whr th condition (3.1) is satisfid with a constant κ dpnding on T 0. For conforming finit lmnt mthods, it has bn shown in [9] that th bisction rfinmnt algorithm basd on choosing suitabl paramtrs capturs solutions of lliptic problms with solutions which allow for dcompositions into rgular parts and cornr singularitis at optimal convrgnc ordrs in N. W will gnraliz this rsult to th discontinuous Galrkin framwork and to functions in H k+1,2 (Ω); s Proposition Nxt, w introduc th notion of a locally adaptd msh. Dfinition 4.2. Lt p 1 and [0,1) M b a wight xponnt vctor. W call a a family of triangulations T locally adaptd to S with rspct to and p if it is ithr (i) a gradd msh family of mshs T β with grading paramtrs β i (βi,1) whr βi := 1 1 i, (4.1) p (ii) or a family of bisction rfinmnt mshs T h,2(l+1) as in [9], obtaind by nwst vrtx bisction with paramtrs h, γ (0,γ ] and L with γ := 1 M max i=1 i > 0 and h [2 (L+1)γ/(p+1),2 Lγ/(p+1) ). (4.2) 4.3. Optimal rror stimats. Our main rsult stablishs optimal nrgy norm and L 2 -normrrorbounds on locally adaptd mshs with rspct to th numbr N of total dgrs of frdom. Thorm 4.3. Lt [0,1) M b as in (2.19). For p 1 and 1 k p, lt f H k 1,0 (Ω), g D H k+1/2,3/2 (Γ D ) and g N H k 1/2,1/2 (Γ N ). Considr th solution u H 1 (Ω) of (2.4) (2.6) which is in H k+1,2 (Ω) by Proposition 2.3. Lt T b a msh which is locally adaptd to S with rspct to and p ithr via msh grading or via rcursiv bisction rfinmnt as in Dfinition 4.2. Lt u N V p (T )

11 10 FABIAN MÜLLER, DOMINIK SCHÖTZAU, AND CHRISTOPH SCHWAB b th symmtric IP approximation obtaind in (3.6) with j 0 > j. Thn, w hav th rror bound u u N DG +N 1/2 u u N L 2 (Ω) CN k/2 u H k+1,2 (Ω). (4.3) Th constant C > 0 is indpndnt of N, but dpnds on κ in (3.1), on th paramtr j 0 in (3.10), th bounds in (2.3), th rgularity paramtr k, th polynomial dgr p, th paramtr C dc in (2.10), (2.11) th stability constant C stab,2 in Proposition 2.3, th vctor, and on th paramtrs for gradd and bisction rfinmnt mshs in Dfinition 4.2. Rmark 4.4. For k = p (i.., u H p+1,2 (Ω)), th stimat (4.3) givs optimal convrgnc rats of ordr N p/2 and of ordr N (p+1)/2 for th L 2 -norm rror, rspctivly. W rmark furthr that Thorm 4.3 is basd on th rgularity bound (3.1). Hnc, no additional rgularity assumptions on th domains ar ncssary for th L 2 -norm stimat. W furthr not that in th proof of th L 2 -norm bound, w implicitly us th adjoint-consistncy of th symmtric IP mthod in th sns of [2]. This proprty dos not hold for non-symmtric IP discrtizations of (2.4) (2.6). As a consqunc, L 2 -norm optimality as in Thorm 4.3 and th argumnts for its proof cannot in gnral b xpctd to apply for non-symmtric IP mthods or for othr DG formulations that do not afford adjoint-consistncy. W rfr to [18, Sction 2.8.2] for a dtaild discussion on L 2 -norm rror stimation for non-symmtric and socalld incomplt intrior pnalty mthods. Rmark 4.5. Th rsults in Thorm 4.3 rmain valid in th pur Numann cas du to th fact that th nodal intrpolants usd in our analysis rproduc constants. 5. Proofs In this sction, w dtail th proof of Thorm 4.3. W shall frquntly us th short-hand notation a b for inqualitis of th form a Cb, whr C > 0 solly may dpnd on κ in (3.1), th bounds in (2.3), th paramtr j 0 in (3.10), th polynomial dgr p, and th particular xponnt vctor undr considration Prliminaris. In this sction, w introduc discrt nighborhoods and stablish som ssntial mbdding proprtis and trac rsults for th wightd spac H k,l (Ω). W furthr introduc th brokn consistncy norm which is appropriat for our analysis; it is a gnralization to wightd spacs of th norm usd in [2, Sction 4.1] Auxiliary rsults. W first rcall a numbr of tchnical stimats which ar rlvant in our analysis. Th first rsult is th trac inquality: v q ( L q ( K) h 1 K v q L q (K) +hq K v q L (K)), v W 1,q (K), 1 q <, (5.1) q whrth implid constantis indpndnt ofh K, but alsodpnds on κ in (3.1) and on q. This bound follows radily by th trac inquality on a rfrnc triangl K combind with scaling argumnts mploying affin quivalnc. W will us th polynomial trac inquality q L 2 ( K) h 1/2 K q L 2 (K), q P p (K), (5.2)

12 SYMMETRIC IP METHODS FOR ELLIPTIC PROBLEMS IN POLYGONS 11 as wll as th invrs inquality s,.g., [17, Lmmas 1.46 and 1.50]. q L () h 1/2 q L 2 (), q P p () ; (5.3) Discrt nighborhoods. For a partition T of Ω, w introduc th discrt nighborhoods N i (T ) := {K T : K Ω i }, 1 i M, N 0 (T ) := T \ ( M i=1 N i(t ) ). (5.4) W always assum th mshs to b sufficintly rfind, so that N i (T ) N j (T ) = for i j. Rmark 5.1. For 1 i M, an lmnt K N i (T ) can b writtn as K = (K Ω 0 ) (K Ω i ) with K Ω i. Thn, du to (2.10), (2.11), w find that v 2 H k+1,2 i = (K) v 2 H k+1,2 (K Ω + v 2 i 0) H k+1,2 (K Ω i i) v 2 H k+1,2 for any rgularity indx k 1. (K Ω + v 2 0) H k+1,2 (K Ω v 2, (5.5) i) H k+1,2 (K) Th following proprtis ar gnralizations of th rsults in [21, Lmmas and 1.3.4]. Lmma 5.2. For [0,1) M, thr holds: (i) Lt v H 0,0 (K) for K T. Thn w hav v K L 1 (K) and { hk v v L1 (K) H 0,0 (K), K N 0(T ), h 1 i K v H 0,0 (K), K N i(t ), 1 i M. (ii) Lt v H 1,1 (K) for K T. Thn w hav v K L 1 ( K) and { v L v L 1 ( K) 2 (K) +h K v H 1,1 (K), K N 0(T ), v L 2 (K) +h 1 i K v H 1,1 (K), K N i(t ), 1 i M. (iii) Lt v H 1,1 (Ω). Thn for any dg E I(T ), w hav (5.6) (5.7) [v] = 0 in L 1 (). (5.8) Rmark 5.3. Th proprtis in Lmma 5.2 ar valid for lmnts abutting at cornrs, for which thy hav bn provd in [21]. W furthr rmark that th first cas inqualitis in (5.6) and (5.7) hold tru for all lmnts K T away from cornrs, but without uniform control in wightd norms. Proof. W prov ach itm sparatly. Proof of (5.6): Lt K N i (T ) b such that K = (K Ω i ) (K Ω 0 ), whr ithr on or both of th intrsctions ar non-mpty. Clarly, v L 1 (K) = v L 1 (K Ω 0) + v L 1 (K Ω i). For th first trm, w us th Cauchy-Schwarz inquality and proprty (2.10) to obtain v L1 (K Ω 0) ara(k Ω 0 ) 1/2 v H 0,0 (K Ω0) h K v H 0,0 (K).

13 12 FABIAN MÜLLER, DOMINIK SCHÖTZAU, AND CHRISTOPH SCHWAB To stimat th scond trm, w introduc polar coordinats (r i,ϑ i ) cntrd at th vrtx c i and apply th Cauchy-Schwarz in combination with (2.11). This yilds v L 1 (K Ω i) = r i i v dx r i i L 2 (K) v H 0,0 (K). r i i K Ω i To bound r i i L 2 (K), w st r K = dist(k,c i ) = inf y K y c i 0. With th shap-rgularity assumption (3.1) and by intgrating out ovr th angular variabl ϑ i, w conclud that ri=r K+h K r i i 2 L 2 (K) r i=r K r 2i+1 i dr i r 2 2i i ri=rk+hk r i=r K h 2 2i K, whr th last inquality follows sinc x α y α x y α for any x,y [0, ) and α [0,1). Ths bounds imply (5.6) in all cass. Proof of (5.7): Th inquality (5.7) is a consqunc of th standard trac inquality (5.1) with q = 1 and th mbdding (5.6) applid to v H 0,0 (K)2, taking into account that v L 1 (K) h K v L 2 (K) for v H 1,1 (K). Proof of (5.8): Th idntity (5.8) is clar for dgs E I (T ) away from cornrs. For an dg E I (T ) with c i, lt x : [0,1], t x(t) b an affin paramtrization of with x(0) = c i. It can b radily sn that 1 ε [v] dt = 0 for all ε > 0. By Lbsgu s dominatd convrgnc thorm (using (5.7)), it follows that [v] ds = 1 0 [v] dt = 0, which is (5.8) for this cas Cornr lmnts and consistncy norm. In th squl, a particular rol will b playd by th subst K i (T ) of lmnts of N i (T ) abutting at c i, dfind by K i (T ) := {K N i (T ) : K c i }, 1 i M. (5.9) W also hav K i (T ) K j (T ) = for i j. W furthr may assum that K K i (T ) is locatd in th con Ω i (i.., K Ω i ). W thn st K(T ) := M K i (T ). (5.10) For lmnts away from cornrs, w nxt introduc th wightd lmntal norm i=1 M K [v] 2 := h 2 K v 2 L 2 (K) + v 2 L 2 (K) +h2 K D2 v 2 L 2 (K), K T \K(T ). (5.11) For a cornr lmnt K K i (T ) and i [0,1), w dfin N K,i [v] 2 := h 2 K v 2 L 2 (K) + v 2 L 2 (K) +h2 2i K v 2 H 2,2 i (K), K K i (T ). (5.12) Lmma 5.4. Lt [0,1) M, and v = v 0 +v N with v 0 H 2,2 (Ω) and v N V p (T ). Thn: (i) W hav v C 0 (K) for K T and v 2 C 0 (K) { M K [v] 2, K T \K(T ), N K,i [v] 2, K K i (T ), 1 i M. (5.13) (ii) W hav v L 1 ( K) 2 for K T and v 2 L 1 ( K) { M K [v] 2, K T \K(T ), N K,i [v] 2, K K i (T ), 1 i M. (5.14)

14 SYMMETRIC IP METHODS FOR ELLIPTIC PROBLEMS IN POLYGONS 13 Proof. Th bound (5.13) followsfrom th continuousmbddingsh 2 ( K) C 0 ( K) and H 2,2 i ( K) C 0 ( K), rspctivly, formulatd for a rfrnc triangl K and combind with an affin scaling argumnt. To show (5.14), w not that for K T \ K(T ), th trac inquality (5.1) in L 1 ( K) combind with h 2 K v 2 L 1 (K) v 2 L 2 (K), D2 v 2 L 1 (K) h2 K D2 v 2 L 2 (K) radily yilds v 2 L 1 ( K) M K[v] 2. For K K i (T ), th inquality (5.14) follows from (5.7). For [0,1) M and a subst T T of lmnts, w now introduc th brokn consistncy norm v 2 T, := M M K [v] 2 + N K,i [v] 2. (5.15) K (T K(T))\K(T) i=1 K T K i(t ) W furthr show th following bound, which implis that DG nrgy norm (3.13) is boundd by th norm (5.15) for sufficintly smooth functions v. Lmma 5.5. Thr holds for all v H 1 (T ). v 2 DG K T ( h 2 K v 2 L 2 (K) + v 2 L 2 (K)), (5.16) Proof. With (2.3), it suffics to bound th jump trms apparing in th nrgy norm v DG. To this nd, lt = K K E I (T ) b an intrior dg. By applying th trac inquality (5.1) (with q = 2), th proprty (3.2) and th bounds in (2.3), w obtain j 1/2 [v] 2 L 2 () h 2 K v 2 L 2 (K) + v 2 L 2 (K) +h 2 K v 2 L 2 (K ) + v 2 L 2 (K ). A similar argumnt holds for Dirichlt dgs, which yilds (5.16) Bounddnss. W nxt stablish svral continuity rsults for th IP bilinar form a DG (, ) and for th right-hand sid l DG ( ) Continuity bounds for a DG (, ). To stablish continuity proprtis of a DG in (3.7), w first not that, by th Cauchy-Schwarz inquality and by (2.3), c v w dx v DG w DG, (5.17) K T E ID(T ) K j [v] [w] ds v DG w DG, (5.18) for all v,w H 1 (T ). Th bounddnss of th off-diagonal form r DG (v,w) in (3.8) is mor involvd and will b discussd nxt. Th first bound (5.19) blow is a standard rsult, s [2]. A proof is includd to rndr th rror analysis slf-containd. On th othr hand, th scond stimat (5.20) is a gnralization to wightd spacs of standard consistncy bounds as,.g., in [2, Sction 4.1] or [17, Sction 4.2]. To prov it, w us th approach in [21, Proposition 2.4.1], which is basd on using Höldr sinqualityin L 1 () L (); s alsolmma 5.2. Th third proprty (5.21) is nw and will b crucial to driv L 2 -norm rror stimats. Proposition 5.6. For r DG (, ) dfind in (3.8), thr holds:

15 14 FABIAN MÜLLER, DOMINIK SCHÖTZAU, AND CHRISTOPH SCHWAB (i) For v N V p (T ), w hav r DG (v N,w) C rdg,1j(w) v N DG, w H 1 (T ). (5.19) (ii) Lt [0,1) M. For vry v = v 0 +v N with v 0 H 2,2 (Ω) and v N V p (T ), thr holds for vry w N V p (T ) r DG (v,w N ) C rdg,2j(w N ) v T,. (5.20) (iii) Lt, [0,1) M. For vry v = v 0 + v N and w = w 0 + w N with v 0 H 2,2 (Ω), w 0 H 2,2 (Ω) and v N,w N V p (T ), w hav r DG (v,w) C rdg,3 w T, v T,. (5.21) In (5.19) (5.21), th constants C rdg,1,c rdg,2,c rdg,3 > 0 dpnd on κ in (3.1), th bounds in (2.3), on th paramtr j 0 in (3.10) and on th polynomial dgr p. Th constants C rdg,2,c rdg,3 furthr dpnd on th xponnts,. Proof. W procd in two stps: Proof of (5.19): For v N V p (T ) and for w H 1 (T ), th Cauchy-Schwarz inquality, th bounds in (2.3), th dfinition of j in (3.10), th quivalnc (3.2) and th polynomial trac inquality (5.2) yild r DG (v N,w) h 1/2 [w] L 2 ()h 1/2 v N L 2 () E ID(T ) ( ) 1/2 ( 1/2 J(w) h K v N 2 L 2 ( K) J(w) v N 2 L (K)), 2 K T which, togthr with (2.3), implis (5.19). Proof of (5.20) and (5.21): Lt K T w = w 0 +w N with w 0 H 2,2 (Ω) and w N V p (T ). (5.22) W invok Höldr s inquality as in [21, Proposition 2.4.1], th bound (2.3) and th discrt Cauchy-Schwarz inquality to obtain r DG (v,w) [w] v ds E ID(T ) E ID(T ) ( E ID(T ) ( E ID(T ) [w] C 0 () v L 1 () [w] 2 ) 1/2 ( L () v 2 L 1 () E ID(T ) [w] 2 ) 1/2 ( L () ) 1/2 K T v 2 L 1 ( K) ) 1/2. (5.23) In th last trm in (5.23) abov, w apply th bounds in (5.14) and find that ( v 2 1/2 L ( K)) v 1 T,. (5.24) K T Lt us now stablish (5.20) and tak w 0 = 0 in (5.22), (5.23). Thn, for ach dg E ID (T ), th jump [w] = [w N ] P p () is a univariat polynomial.

16 SYMMETRIC IP METHODS FOR ELLIPTIC PROBLEMS IN POLYGONS 15 Thrfor, th invrs stimat (5.3), th bounds in (2.3) and th dfinition of j in (3.10) yild ( [w N ] 2 ) 1/2 ( L () h 1/2 [w N ] 2 1/2 L ()) 2 J(wN ). (5.25) E ID(T ) E ID(T ) Th combination of (5.23), (5.24) and (5.25) implis (5.20). Finally, to prov (5.21), w now tak in (5.22), (5.23) a gnric w = w 0 +w N, whr 0 w 0 H 2,2 (Ω). Thn, th mbddings (5.13) and th dfinition (5.15) show that ( [w] 2 ) 1/2 ( L () w 2 1/2 L ( K)) w T,. (5.26) K T E ID(T) Th quations (5.23), (5.24) and (5.26) yild th bound (5.21). Proposition 5.6 and th bounds in (5.17), (5.18) imply th following rsult. Proposition 5.7. Lt, [0,1) M. Thn: (i) For v = v 0 +v N with v 0 H 2,2 (Ω) and v N V p (T ), w hav a DG (v,w N ) C adg,1 v T, w N DG, w N V p (T ). (5.27) (ii) For v = v 0 +v N and w = w 0 +w N with v 0 H 2,2 (Ω), w 0 H 2,2 (Ω) and v N,w N V p (T ), w hav a DG (v,w) C adg,2 v T, w T,, (5.28) Th constants C adg,1,c adg,2 > 0 dpnd on κ in (3.1), th bounds in (2.3), th paramtr j 0 in (3.10), th polynomial dgr p and th xponnts,. Proof. To show (5.27), w mploy th continuity bounds (5.19) and (5.20), rspctivly. W find that r DG (w N,v) C rdg,1j(v) w N DG C rdg,1 v DG w N DG, r DG (v,w N ) C rdg,2j(w N ) v T, C rdg,2 v T, w N DG. Ths bounds in conjunction with (5.17), (5.18) and (5.16) imply (5.27). For th proof of (5.28), w apply (5.21) twic and obtain r DG (v,w) C rdg,3 v T, w T,, r DG (w,v) C rdg,3 w T, v T,. Th stimat (5.28) follows from (5.17), (5.18) and (5.16) Continuitybounds for l DG ( ). WiththsamargumntsusdtoprovProposition 5.6, w furthr driv bounds for th right-hand sid l DG ( ) in (3.9). For th functionals associatd with th sourc trm f and th Numann datum g N, w stablish bounds, which ar xprssd in trms of wightd norms of th data. Proposition 5.8. Lt, [0,1) M, f H 0,0 (Ω) and g N H 1/2,1/2 (Γ N ). Lt v = v 0 +v N with v 0 H 2,2 (Ω) and v N V p (T ). Thn: (i) W hav fv dx C f f H 0,0 (Ω) v T,. (5.29) K T K

17 16 FABIAN MÜLLER, DOMINIK SCHÖTZAU, AND CHRISTOPH SCHWAB (ii) For T N := {K T : K E N (T ) for E(T )}, w hav g N v ds C gn g N 1/2,1/2 H (Γ v N) TN,. (5.30) E N(T ) Th constants C f > 0 and C gn > 0 dpnd on κ in (3.1), th polynomial dgr p and th xponnts,. Proof. W prov ach itm sparatly. Proof of (5.29): For K T, proprty (5.6) and Höldr s inquality yild fv dx f L 1 (K) v C 0 (K) f H 0,0 (K) v C 0 (K). K Th summation of ths bounds ovr all lmnts K T, th discrt Cauchy- Schwarz inquality and th bounds in (5.13) now imply (5.29). Proof of (5.30): Lt G N H 1,1 (Ω) b such that G N ΓN = g N. For a Numann dg K E N (T ), w apply (5.7) and obtain g N v ds G N L 1 ( K) v C 0 (K) G N H 1,1 (K) v C 0 (K). Hnc, by summing this bound ovr all K T N, using th discrt Cauchy-Schwarz inquality, applying (5.13) and taking th infimimum ovrall possibl G N as abov, th bound (5.30) follows. Rmark 5.9. W not that in th proof of (5.29) som powrs of h (i.., h 2 2i for vrtx c i ) wr droppd aftr application of (5.6). Th rsulting bound (5.29) will b sufficint for our purposs. For th Dirichlt datum g D, w introduc th natural boundary jump trm J D (v) 2 := j 1/2 v 2 L 2 (). (5.31) As in (5.18), w thn hav E D(T) E D(T ) j g D v ds J D (g D )J D (v) (5.32) for any v H 1 (T ). Th rmaining functional associatd with g D in (3.9) can b boundd compltly analogously to r DG (, ) in Proposition 5.6. Proposition Lt, [0,1) M and dfin T D := {K T : K E D (T ) for E(T )}. Thn: (i) W hav for all v N V p (T ) g D c v N ds C gd,1j D (g D ) ( c 1/2 v N 2 ) 1/2. L 2 (K) (5.33) K T D E D(T ) (ii) Lt g D P p () for E D (T ). For any v = v 0 + v N with v 0 H 2,2 (Ω) and v N V p (T ), w hav g D c v ds C gd,2j D (g D ) v TD,. (5.34) E D(T )

18 SYMMETRIC IP METHODS FOR ELLIPTIC PROBLEMS IN POLYGONS 17 (iii) Lt G D H 2,2 (Ω) b such that G D ΓD = g D H 3/2,3/2 (Γ D ). Lt v = v 0 +v N with v 0 H 2,2 (Ω) and v N V p (T ). Thn w hav g D c v ds C gd,3 G D TD, v TD,, (5.35) E D(T ) Th constants C gd,1,c gd,2,c gd,3 > 0 dpnd on κ in (3.1), th bounds in (2.3), th paramtr j 0 in (3.10), th polynomial dgr p and th xponnts,. Proof. W prov ach itm sparatly. Proof of (5.33): This bound is obtaind compltly analogously to (5.19). That is, with th hlp of (2.3), th dfinition of j in (3.10), th discrt Cauchy-Schwarz inquality, th quivalnc(3.2) and th polynomial trac inquality(5.2), w obtain for v N V p (T ) g D c v N ds j 1/2 g D L 2 ()h v N L 2 () E D(T ) E D(T ) J D (g D ) ( K T D h K v N 2 L 2 ( K) ) 1/2, which yilds (5.33). Proof of (5.34): Following(5.20),wusHöldr sinqualityinl () L 1 (), th discrt Cauchy-Schwarz inquality, th bounds in (2.3), th invrs inquality (5.3) (sinc g D is picwis polynomial) and th dfinition of j 0 in (3.10). This yilds E D(T) g D c v ds ( E D(T ) j 1/2 g D 2 ) 1/2 ( L 2 () v 2 1/2. L ( K)) 1 K T Invoking (5.14) shows (5.34). Proof of (5.35): This bound is similar to (5.21). By mploying Höldr s inquality and th discrt Cauchy-Schwarz inquality, w conclud that g D c v ds ( G D 2 ) 1/2 ( ) 1/2. L ( K) K T D E D(T ) K T D v 2 L 1 ( K) Th inqualitis (5.13) and (5.14) imply (5.35). Rmark Lt v = v 0 + v N, with v 0 H 2,2 (Ω) and v N V p (T ). Undr assumption (2.20) and from th bound (5.32), Proposition 5.8 and Proposition 5.10, w find th bounddnss of l DG ( ) in th sns that ldg (v) ( ClDG f H 0,0 (Ω) + g N 1/2,1/2 H (Γ +J ) N) D(g D )+ G D TD, v T,, (5.36) for a constant C ldg > 0 dpnding on κ in (3.1), th bounds in (2.3), th paramtr j 0 in (3.10), th polynomial dgr p and th xponnts, Consistncy. W vrify th Galrkin orthogonality proprty of th DG discrtization (3.6).

19 18 FABIAN MÜLLER, DOMINIK SCHÖTZAU, AND CHRISTOPH SCHWAB Intgration by parts. W will us th following variant of Grn s formula in [21, Lmma A.2.1]: For q H 1,1 (Ω)2, w hav q vdx = ( q)vdx+ q ν K vds, (5.37) K K for all K T and for v C 0 (K) with v L 2 (K) 2. In particular, if K K(T ) is a cornr lmnt, all intgrals in (5.37) ar wll-dfind in L 1 (K) L (K) rspctivly L 1 ( K) L ( K), du to Lmma 5.2 and Höldr s inquality. Th idntity (5.37) is radily provd using th dnsity of C (Ω) 2 in H 1,1 (Ω)2 ; cf. [21, Lmma A.2.1]. A consqunc of (5.37) is th following intgration-by-parts formula. Lmma Lt, [0,1) M. For v H 2,2 (Ω), w hav c v H1,1 (Ω)2, (c v) H 0,0 (Ω) and c v wdx = (c v)wdx+ c v [w]ds (5.38) K T K K T K K E(T) for any w = w 0 +w N with w 0 H 2,2 (Ω) and w N V p (T ). Hr, all th trms ar wll-dfind, which follows from Lmma 5.2, Lmma 5.4, and Proposition 5.8. In particular, as in Rmark 3.1, for cornr lmnts K K(T ), th volum intgrals on th right-hand sids ar undrstood in th sns of L 1 (K) L (K), and th intgrals ovr dgs E(T ) running into cornrs ar wll-dfind as bilinar forms ovr L 1 () L (). Proof. Sincv H 2,2 (Ω)andcissmooth, wclarlyhavq = c v H1,1 (Ω)2, and q H 0,0 (Ω), s Lmma 2.2. Morovr, th tst function w in (5.38) blongs to L (Ω) and satisfis for all K T : w K C 0 (K), cf. (2.15), and w K L 2 (K) 2. Hnc, by applying (5.37) and summing ovr all lmnts, w obtain c v wdx = (c v)wdx+ (c v) ν K wds. K T K K T K K T K All th trms abov ar wll-dfind thanks to Lmma 5.2, Lmma 5.4 and Proposition 5.8. In th last trm, w xprss intgrals ovr lmntal boundaris by intgrals ovr dgs using th idntity in [2, Equation (3.3)]. W conclud that (c v) ν K wds = c v [w]ds + [c v] w ds, K T K E(T ) E I(T ) whr again all trms ar wll-dfind ovr L 1 () L (). Th quality (5.38) now follows from (5.8) Galrkin orthogonality. Th following proprtis hold. Lmma For [0,1) M, lt u H 2,2 (Ω) b th solution of (2.4) (2.6). Thr holds: (i) Lt [0,1) M and v = v 0 +v N with v 0 H 2,2 (Ω) and v N V p (T ). Thn w hav a DG (u,v) = c u vdx r DG (u,v) = l DG (v), (5.39) K T K

20 SYMMETRIC IP METHODS FOR ELLIPTIC PROBLEMS IN POLYGONS 19 whr r DG (u,v) and l DG (v) ar boundd as in (5.21) and (5.36) rspctivly. (ii) Lt u N V p (T ) b th IP approximation in (3.6). Thn th rror u u N satisfis a DG (u u N,v N ) = 0, v N V p (T ). (5.40) Proof. Th idntity (5.39) is an immdiat consqunc of th formula (5.38) and th PDE (2.4), by taking into account th boundary conditions in (2.5) and (2.6), rspctivly, and by noting that [u] = 0 for E I (T ) and [u] = g D ν for E D (T ). Th Galrkin orthogonality (5.40) follows immdiatly from (5.39) and th dfinition of th IP mthod (3.6). Rmark W mphasiz that idntity (5.39) in conjunction with th symmtry of th IP form a DG (, ) yilds adjoint-consistncy of th symmtric intrior pnalty mthod (3.6) in th sns of [2] Error stimats. W now driv gnric and quasi-optimal rror bounds Enrgy norm rror. W bgin by proving th following nrgy norm bound. Lmma For [0,1) M, lt u H 2,2 (Ω) b th solution of (2.4) (2.6). Lt u N V p (T ) b th IP approximation in (3.6). Thn w hav th nrgy norm rror bound u u N DG C inf u v N T,, (5.41) v N V p(t ) with a constant C > 0 dpnding on κ in (3.1), th bounds in (2.3), th paramtr j 0 in (3.10), th polynomial dgr p, and th xponnt. Proof. W procd in a standard mannr. For v N V p (T ), th triangl inquality givs u u N DG u v N DG + v N u N DG. Hnc, from th corcivity in Lmma 3.3 and th Galrkin orthogonality (5.40), w find that C cor v N u N 2 DG a DG (v N u N,v N u N ) = a DG (u v N,v N u N ). By th continuity of th form a DG (, ) in (5.27), w hav a DG (u v N,v N u N ) C adg,1 u v N T, v N u N DG. Th bound (5.41) follows L 2 -norm rror. To prov an L 2 -norm rror stimat for u u N, w us rgularity in wightd spacs for th dual problm: (c z) = u u N in Ω, (5.42) z = 0 on Γ D, (5.43) (c z) ν = 0 on Γ N. (5.44) For [0,1) M as in (2.19), th stability bound in Proposition 2.3 and th continuous mbdding L 2 (Ω) H 0,0 (Ω) nsur that z H2,2 (Ω) and that z H 2,2 (Ω) C stab,2 u u N H 0,0 (Ω) C dcc stab,2 u u N L2 (Ω). (5.45)

21 20 FABIAN MÜLLER, DOMINIK SCHÖTZAU, AND CHRISTOPH SCHWAB Lmma5.16. For [0,1) M as in (2.19), lt u H 2,2 (Ω) b th solution of (2.4) (2.6). Lt z H 2,2 (Ω) b th dual solution of (5.42) (5.44). Assum th approximation proprty inf z z N T, C approx d(p,t,) z H 2,2 z N V p(t) (Ω), (5.46) Lt furthr u N V p (T ) b th IP approximation in (3.6). Thn w hav th L 2 - norm rror bound u u N L2 (Ω) Cd(p,T,) inf v N V p(t ) u v N T,, (5.47) with a constant C > 0 dpnding on κ in (3.1), th bounds in (2.3), th constants C stab,2 and C approx in (5.45) and (5.46), rspctivly, th polynomial dgr p, and th wight xponnt vctor. Proof. W invok idntity (5.39) for v = u u N with rspct to th dual problm (5.42) (5.44) and mploy th symmtry of th IP form. This rsults in u u N 2 L 2 (Ω) = a DG(u u N,z). From hr on, w procd in a usual mannr and apply th Galrkin orthogonality (5.40) to conclud that u u N 2 L 2 (Ω) = a DG(u u N,z z N ) for all z N V p (T ). Hnc, by mploying (5.28), assumption (5.46) and th stability bound (5.45), w conclud that a DG (u u N,z z N ) C adg,2 u u N T, z z N T, Ths bounds imply (5.47). C adg,2c approx d(p,t,) z H 2,2 (Ω) u u N T, C adg,2c approx C dc C stab,2 d(p,t,) u u N L 2 (Ω) u u N T, Nodal intrpolation. In viw of th rror bounds (5.41) and (5.47), it rmains to stablish optimal intrpolation stimats on gradd and bisction rfinmnt mshs with rspct to th norm T,. For an lmnt K T, lt I p K : C0 (K) P p (K) dnot th lmntal nodal intrpolant into polynomials of dgr at most p. By standard intrpolation in lmnts away from S and for 1 k p, thr holds M K [v I p K v]2 h 2k K Dk+1 v 2 L 2 (K), K T \K(T ). (5.48) In cornr lmnts, th intrpolation bounds in [19, Lmma 4.16] for th linar intrpolant in th wightd spacs H 2,2 i (K) for i [0,1) giv N K, [v I 1 K v]2 h 2 2i K v 2 H 2,2 i (K), K K i(t ). (5.49) Hr, w not that IK 1 v is wll-dfind for v H2,2 i (K) du to (2.15). Du to (5.48), (5.49), w thn dfin th global intrpolant I p : C 0 (Ω) V p (T ) by { IK 1 I p v K := (v K) if K K(T ), I p K (v (5.50) K) othrwis. W not again that I p v is wll-dfind for v H 2,2 (Ω) C0 (Ω). Nxt, w driv intrpolation stimats for I p v on locally adaptd mshs.

22 SYMMETRIC IP METHODS FOR ELLIPTIC PROBLEMS IN POLYGONS Intrpolation stimats on gradd msh familis. W first considr gradd msh familis and prov approximation bounds which ar slightly gnralizd vrsions of thos in [21, Proposition 2.5.5]. Analogous rsults (on diffrnt msh familis and for conforming FEMs) ar obtaind in [7]. Proposition Lt p 1, [0,1) M and v H k+1,2 (Ω) for 1 k p. Lt T β b a gradd msh with grading vctor β = (β 1,...,β M ) chosn as in (4.1), i.., with β i (βi,1) whr βi := 1 1 i. (5.51) p Lt I p v b th intrpolant dfind in (5.50). Thn w hav th approximation bound v I p v Tβ, C grad N k/2 v H k+1,2 (Ω). (5.52) Th constant C grad > 0 is indpndnt of N, but dpnds on κ in (3.1), on th constant C dc in (2.10), (2.11), on th vctor, th paramtrs C G, β and β in Dfinition 4.2, th rgularity paramtr k, and th polynomial dgr p. Proof. For an intrior lmnt K N 0 (T β ), du to (5.48) and (2.10), thr holds M K [v I p v] 2 h 2k K Dk+1 v 2 L 2 (K) h2k v 2. (5.53) H k+1,2 (K) Nxt, for 1 i M, lt K N i (T β ), th discrt cornr nighborhood dfind in (5.4). Lt first K N i (T β )\K i (T β ). Thn, th bounds (5.48), th first proprty in Dfinition 4.1, and th fact that β i (βi,1) imply M K [v I p v] 2 h 2k K Dk+1 v 2 L 2 (K) h2k r βik i D k+1 v 2 L 2 (K) h 2k r β i k i D k+1 v 2 L 2 (K) h2k r k (1 i)k p i D k+1 v 2 L 2 (K). Sinc k (1 i ) k p k (1 i) and du to (5.5), w obtain M K [v I p v] 2 h 2k r i+k 1 i D k+1 v 2 L 2 (K) h 2k v 2 H k+1,2 (K) h2k v 2. (5.54) H k+1,2 i (K) Scond, lt K K i (T β ). Sinc r i (x) h K for all x K, th scond assumption in Dfinition 4.1 yilds and thn h K h 1 1 β i, as wll as sup x K h K h sup r i (x) β β i hh i K, x K β r i (x) β i i h 1 β i = h p 1 1 i. (5.55) Th stimat (5.49), th scond proprty in Dfinition 4.1, th bound (5.55) and proprty (2.11) now yild N K, [v I p v] 2 h 2 2i K v 2 H 2,2 i (K) h 2 2i sup x K h 2 2i sup x K r i (x) βi(2 2i) v 2 H k+1,2 (K) r i (x) β i (2 2i) v 2 H k+1,2 (K) h 2p v 2 H k+1,2 (K) h2k v 2. H k+1,2 (K) (5.56)

23 22 FABIAN MÜLLER, DOMINIK SCHÖTZAU, AND CHRISTOPH SCHWAB Summing th bounds in (5.53), (5.54) and (5.56) ovr all lmnts yilds v I p v 2 T β, h2k v 2. (5.57) H k+1,2 (Ω) Finally, an lmntary counting argumnt (s,.g., [21, Lmma 2.5.6]) rvals that N = dim(v p (T )) p 2 h 2, which implis (5.52) Intrpolation stimats on bisction rfinmnt mshs. W shall stablish th following variant of [9, Thorm 5.3], which is basd solly on th proprtis of th algorithm in [9]. W also rcall from Sction 4.2 that bisction rfinmnt msh family thus constructd is shap-rgular with κ in (3.1) dpnding on th initial msh T 0. Proposition Lt p 1, [0,1) M and v H k+1,2 (Ω) for 1 k p. For paramtrs h, γ (0,γ ] and L as in (4.2), i.., with γ := 1 M max i=1 i > 0 and h [2 (L+1)γ/(p+1),2 Lγ/(p+1) ). (5.58) considr th triangulations T h,2(l+1) constructd by th bisction rfinmnt algorithm of [9] starting from an initial triangulation T 0 with #T 0 h 2. Lt I p v b th intrpolant in (5.50). Thn w hav th approximation bound v I p v T2(L+1), C bis N k/2 v H k+1,2 (Ω). (5.59) Th constant C bis > 0 is indpndnt of N, but dpnds on th shap rgularity paramtr κ in (3.1), th constant C dc in (2.10), (2.11), th vctor, th initial msh T 0, th paramtr γ, th sufficintly larg rfinmnt paramtr L, th rgularity paramtr k, and on th polynomial dgr p. Proof. Following [9, Thorms 5.2 and 5.3], w procd in svral stps. Intrior lmnts: W first bound th rrors ovr lmnts in th intrior nighborhoodn 0 (T h,2(l+1) ). Tothisnd,wrcallfrom[9, Lmma4.4]thatthfirstloop of th bisction rfinmnt algorithm nsurs K h 2 K h2. Hnc, with (5.48) and (2.10), w find that K N 0(T h,2(l+1) ) M K [v I p v] 2 K N 0(T h,2(l+1) ) h 2k K N 0(T h,2(l+1) ) h 2k v 2 H k+1,2 (Ω). h 2k K D k+1 v 2 L 2 (K) v 2 H k+1,2 (K) (5.60) Cornr nighborhoods: Lt 1 i M b a fixd cornr indx. For K N i (T h,2(l+1) ), w st r K := dist(k,c i ) = inf r i(y). (5.61) y K As in [9, pag 933], w thn considr th following concntric nighborhoods at c i : D l := {K N i (T h,2(l+1) ) : 2 l+1 2 < r K 2 l 2 }, l = 0,...,2L+1, and D 2L+2 := {K N i (Th,2(L+1) i ) : r K 2 (L+1) }. Hr and as in [9], w assum without loss of gnrality that N i (T h,2(l+1) ) 2L+2 l=0 D l.

24 SYMMETRIC IP METHODS FOR ELLIPTIC PROBLEMS IN POLYGONS 23 Elmnts in D 2L+2 : W first bound th consistncy rrors in th lmnts in th innrmost nighborhood D 2L+2. To do so, w nd to stimat th trms T 1 and T 2 givn by T 1 = M K [v I p v] 2, (5.62) K D 2L+2 r K>0 T 2 = K D 2L+2 r K=0 N K, [v I p v] 2. (5.63) Lt K D 2L+2 and first considr th cas r K > 0. By [9, Lmma 4.6], thr holds h K r K. Morovr, r K r i (x) for all x K. With (5.48), w thus obtain M K [v I p v] 2 h 2k K D k+1 v 2 L 2 (K) r 2k K r2 2i 2k K r i+k 1 i D k+1 v 2 L 2 (K) r2 2i K v 2. H k+1,2 (K) i Using th dfinition of D 2L+2 and γ, as wll as th condition on h from (5.58), w find that M K [v I p v] 2 2 (2 2i)(L+1) v 2 H k+1,2 (K) i 2 2γ(L+1) v 2 H k+1,2 (K) h2p+2 v 2. (5.64) H k+1,2 i (K) i Scond, lt r K = 0. W now us (5.49) and procd as bfor. This rsults in N K, [v I p v] 2 h 2 2i K v 2 H 2,2 i (K) 2 2γ(L+1) v 2 H 2,2 i (K) h2p+2 v 2 H k+1,2 i (K). (5.65) Summing th stimats in (5.64) and (5.65) ovr all rlvant lmnts and taking into account (5.5) yild T 1 +T 2 h 2p+2 v 2 H k+1,2 (K) h2p+2 v 2. (5.66) H k+1,2 i (Ω) K D 2L+2 Notic that th bound (5.66) corrsponds to [9, Equation (5.5)]. Elmnts in D l : Nxt, considr an lmnt K D l, for 0 l 2L+1. Sinc r K 2 l 2, th rsult in [9, Lmma 4.7] and th dfinition of γ (0,γ ] imply K h 2 K h 2 2 lp+1 γ p+1 h 2 2 lp+1 γ p+1 h 2 2 lp+ i p+1. Thn, insrting th appropriat powr of r i, mploying th bound abov and noticing that r K 2 l+1 2, w conclud that Sinc M K [v I p v] 2 h 2k K Dk+1 v 2 L 2 (K) h 2k Kr 2 2i 2k K r i+k 1 i D k+1 v 2 L 2 (K) h 2k 2 lk p+ i p+1 2 (l+1)(k 1+ i) v 2 H k+1,2 h 2k 2 lk p+ i p+1 2 l(k 1+ i) v 2 H k+1,2 i (K). 2 l ( k p+ i p+1 (k 1+i) ) = 2 l(p k)(1 i )+(1 i ) p+1 1, i (K)