A Weakly Over-Penalized Non-Symmetric Interior Penalty Method

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1 Europan Socity of Computational Mthods in Scincs and Enginring (ESCMSE Journal of Numrical Analysis, Industrial and Applid Mathmatics (JNAIAM vol.?, no.?, 2007, pp.?-? ISSN A Wakly Ovr-Pnalizd Non-Symmtric Intrior Pnalty Mthod Susann C. Brnnr 1 Dpartmnt of Mathmatics and Cntr for Computation and chnology, Louisiana Stat Univrsity, Baton Roug, LA 70803, USA Luk Owns Dpartmnt of Mathmatics, Univrsity of South Carolina, Columbia, SC 29208, USA Rcivd 15 Dcmbr, 2006; accptd in rvisd form Abstract: A wakly ovr-pnalizd nonsymmtric intrior pnalty mthod for scond ordr lliptic boundary valu problms is considrd in this papr. his mthod is consistnt, stabl for any choic of th pnalty paramtr and satisfis quasi-optimal rror stimats in both th nrgy norm and th L 2 norm. Furthrmor, thr xists a simpl block diagonal prconditionr that kps th condition numbr of th discrt problm at th ordr of O(h 2. Both thortical and numrical rsults ar prsntd. c 2007 Europan Socity of Computational Mthods in Scincs and Enginring Kywords: intrior pnalty mthods, wakly ovr-pnalizd, prconditionr, quasi-optimal rror stimats Mathmatics Subjct Classification: 65N30 1 Introduction Lt b a convx polygonal domain in R 2 and f L 2 (. In this papr w will considr a wakly ovr-pnalizd nonsymmtric intrior pnalty (WOPNIP mthod for th following modl problm: Find u H0 1( such that u v dx = fv dx v H0 1 (. (1.1 Bfor introducing our nw mthod, w first rviw som wll-known intrior pnalty mthods for (1.1 to motivat our approach. Lt h b a quasi-uniform triangulation of whr h is th msh siz. W dfin V h to b th discontinuous P 1 finit lmnt spac with rspct to th triangulation h. hat is, V h = {v L 2 ( : v = v P 1 ( h }. Also, w dfin 1 Corrsponding author. brnnr@math.lsu.du

2 2 S.C. Brnnr and L. Owns th jumps and mans in th usual way [2, 3]. Lt b an intrior dg shard by th triangls 1, 2 h. hn w dfin on, [[v]] = v 1 n 1 + v 2 n 2, (1.2 {{ v}} = 1 2( v1 + v 2, (1.3 whr v 1 = v 1, v 2 = v 2 and n 1 (rsp. n 2 is th unit normal of pointing towards th outsid of 1 (rsp. 2. On an dg along, w dfin [[v]] = (v n, (1.4 {{ v}} = ( v, (1.5 whr n is th unit normal of pointing outsid. h variational problm can b solvd by th symmtric intrior pnalty Galrkin (SIPG mthod [13, 1] and th nonsymmtric intrior pnalty (NIPG mthod [12], which ar dfind as follows. Find u ± h V h such that whr a ± h (u± h, v = a ± h (w, v = w v dx {{ w}} [[v]] ds ± h E h E h + [[w]] [[v]] ds, E h fv dx v V h, (1.6 {{ v}} [[w]] ds E h is th st of all th dgs of h, and > 0 is a pnalty paramtr. h function u + h (rsp. u h in (1.6 is th NIPG (rsp. SIPG approximat solution of (1.1. It is wll-known that both mthods ar consistnt, th NIPG mthod is stabl for any choic of, and th SIPG mthod is stabl for sufficintly larg. Furthrmor, whn th mthods ar stabl, w hav [1, 12] u u ± h a + Ch f h L2(, whr providd v 2 a + h = a + h (v, v = v 2 L + 2( E h 1 [[v]] 2 L 2(, 0 > 0. (1.7 From now on w assum that th pnalty paramtr satisfis (1.7, i.., it is boundd away from 0, and w will us C (with or without subscript to dnot a gnric positiv constant indpndnt of f, h and that can tak diffrnt valus at diffrnt occurrncs. Sinc th SIPG mthod is symmtric, it is also adjoint consistnt. Consquntly th Aubin- Nitsch duality argumnt can b applid and w hav u u h L 2( Ch 2 f L2(. (1.8 On th othr hand th NIPG mthod is not adjoint consistnt and th analog of (1.8 dos not hold for u + h. c 2007 Europan Socity of Computational Mthods in Scincs and Enginring (ESCMSE

3 A Wakly Ovr-Pnalizd Intrior Pnalty Mthod 3 o rcovr th quasi-optimal L 2 rror stimat for th NIPG approach, th following ovrpnalizd mthod was introducd in [12]. Find ũ h V h such that whr ã h (w, v = ã h (ũ h, v = fv dx v V h, (1.9 w v dx {{ w}} [[v]] ds + {{ v}} [[w]] ds E h E h + 3 [[w]] [[v]] ds. E h h mthod (1.9 is stabl for any > 0 and u ũ h L2( + h u ũ h ãh Ch 2 f L2(, whr v ãh = ã h (v, v. Of cours, th gain in th L 2 rror stimat is at th xpns of incrasing th condition numbr of th discrt systm from O(h 2 for (1.6 to O(h 4 for (1.9. Our goal is to dsign an intrior pnalty mthod such that (i it is consistnt, (ii it is stabl for any choic of th pnalty paramtr, (iii it satisfis quasi-optimal rror stimats in both th nrgy norm and th L 2 norm, and (iv w only hav to solv a systm of linar quations whos condition numbr is of ordr O(h 2. Our ida for th nw schm is basd on th following obsrvation on th ovr-pnalizd mthod (1.9. Sinc {{ w}} and {{ v}} ar constant vctors along th dgs of h, w can rwrit th bilinar form as ã h (w, v = w v dx {{ w}} Π 0 [[v]] ds + {{ v}} Π 0 [[w]] ds E h E h + 3 [[w]] [[v]] ds, E h whr Π 0 is th orthogonal projction oprator from L 2( onto P 0 (. hat is, Π 0 v = 1 v ds v L 2 (. (1.10 Accordingly, w only nd to ovr-pnaliz th intgral Π0 [[w]] Π 0 [[v]] ds. h rsulting wakly ovr-pnalizd mthod is: Find u h V h such that whr a h (w, v = a h (u h, v = fv dx v V h, (1.11 w v dx {{ w}} [[v]] ds E h + {{ v}} [[w]] ds + E h E h 3 Π 0 [[w]] Π 0 [[v]] ds. (1.12 c 2007 Europan Socity of Computational Mthods in Scincs and Enginring (ESCMSE

4 4 S.C. Brnnr and L. Owns Rmark 1.1. Huristically, th WOPNIP mthod (1.11 (1.12 works bcaus th wak ovrpnalization forcs Π 0 u h to b almost 0 and it is wll-known [8] that wakly continuous P 1 functions can b usd to solv th Poisson problm (1.1. Also, at first glanc th condition numbr of (1.11 is as bad as th condition numbr of (1.9. Howvr, thr is a simpl block diagonal prconditionr (cf. Sction 3 blow that will rduc th condition numbr of th systm back to O(h 2. Rmark 1.2. It follows from th midpoint rul that Π 0 v = v(m v P 1 (, (1.13 whr m is th midpoint of th dg. hrfor th natural nodal basis for th WOPNIP mthod is associatd with th midpoints of th dgs of h. h rst of th papr is organizd as follows. W driv quasi-optimal rror stimats for th WOPNIP mthod in Sction 2 and construct th block diagonal prconditionr in Sction 3. W thn prsnt numrical rsults in Sction 4 and nd with som concluding rmarks in Sction 5. 2 Error Analysis First w not that th solution u of (1.1 satisfis, via intgration by parts, a h (u, v = fv dx v V h, (2.1 i.., th schm (1.11 is consistnt. Furthrmor, w hav th lliptic rgularity stimat [10] u H2 ( C f L2(. (2.2 by W will carry out th rror analysis using th msh-dpndnt norms h and h dfind v 2 h = v 2 L + 2( {{ v}} 2 L + 2( E h E h v 2 h = v 2 L + 2( E h 3 Π0 [[v]] 2 L 2(, (2.3 3 Π0 [[v]] 2 L 2(, (2.4 for all v H 2 ( + V h. It is asy to s that v 2 h = a h (v, v v H 2 ( + V h, (2.5 i.., th schm (1.11 is stabl for any choic of th pnalty paramtr. W bgin our analysis with svral lmmas, whr w us th notation h, to dnot th st of th triangls of h that hav E h as an dg. Lmma 2.1. h bilinar form a h (, is boundd on H 2 ( + V h with rspct to h, i., a h (w, v C w h v h v, w H 2 ( + V h, (2.6 whr th positiv constant C dpnds only on th minimum angl of h. Proof. Lt E h and v H 2 ( + V h b arbitrary. W hav 1 ( 1 [[v]] 2 L C 2( Π0 [[v]] 2 L + 1 2( [[v]] Π0 [[v]] 2 L 2( c 2007 Europan Socity of Computational Mthods in Scincs and Enginring (ESCMSE

5 A Wakly Ovr-Pnalizd Intrior Pnalty Mthod 5 C 1 Π0 [[v]] 2 L + C 1 2( v Π 0 v 2 L 2( (2.7 ( 1 C Π0 [[v]] 2 L + 2( v 2 L 2(,, whr th constant C dpnds only on th shap of th triangls in h,. In th last stp w hav usd th trac thorm (with scaling and th Brambl-Hilbrt lmma [4, 6]. h stimat (2.6 follows from th Cauchy-Schwarz inquality, (1.7, (1.12 and (2.7: a h (w, v ( w L2( v L2( + 1/2 ( {{ w}} L2( 1/2 [[v]] L2( E h + ( 1/2 {{ v}} L2(( 1/2 [[w]] L2( + E h E h ( w 2 L + 2( {{ w}} 2 L + 1 2( [[w]] 2 L + 2( E h E h E h ( v 2 L + 2( {{ v}} 2 L + 1 2( [[v]] 2 L + 2( E h E h E h C w h v h. 3 Π0 [[w]] L2( Π 0 [[v]] L2( 1/2 3 Π0 [[w]] 2 L 2( 1/2 3 Π0 [[v]] 2 L 2( It is clar from (2.3 and (2.4 that Morovr, th two norms ar quivalnt on V h. v h v h v H 2 ( + V h. (2.8 Lmma 2.2. It holds that v h v h v V h. (2.9 Proof. Lt v V h b arbitrary. It follows from (1.2, (1.4 and an invrs stimat that {{ v}} 2 L C 2( v 2 L C 2( v 2 L, 2( E h E h, whr th constant C dpnds only on th minimum angl of h. hrfor w hav v h C v h v V h. Lmma 2.3. It holds that Proof. It follows from (1.11 and (2.1 that Using (2.5, (2.6, (2.9 and (2.11, w find u u h h C inf v V h u v h. (2.10 a h (u u h, v = 0 v V h. (2.11 u h v 2 h = a h(u h v, u h v = a h (u v, u h v C u v h u h v h. c 2007 Europan Socity of Computational Mthods in Scincs and Enginring (ESCMSE

6 6 S.C. Brnnr and L. Owns hrfor, w hav and hnc, in viw of (2.8, u h v h C u v h v V h, which implis (2.10. u u h h u v h + v u h h C u v h v V h, Lt Π h b th nodal intrpolation oprator for th conforming P 1 finit lmnt. It follows from (2.2, a standard intrpolation rror stimat [7, 6] and [[u]] = [[Π h u]] = 0 that u Π h u h = (u Π h u L2( Ch u H 2 ( Ch f L2(. (2.12 Furthrmor, by th trac thorm (with scaling and standard intrpolation stimats, w hav {{ (u Π h u}} 2 L C ( 2( u Π h u 2 H 1 ( + h2 u 2 H 2 ( E h E h C h 2 u 2 H 2 ( Ch2 u H 2 ( (2.13 E h which, togthr with (2.2, (2.3 and (2.12, implis u Π h u h Ch f L2(. (2.14 Combining (2.10 and (2.14, w hav th following rsult for th nrgy rror. horm 2.4. It holds that W can also masur th rror in th h norm. Corollary 2.5. It holds that Proof. From (2.9, (2.12, (2.14 and (2.15, w hav u u h h u Π h u h + Π h u u h h u u h h Ch f L2(. (2.15 u u h h Ch f L2(. (2.16 Ch f L2( + C Π h u u h h Ch f L2( + C ( Π h u u h + u u h h Ch f L2(. Finally, w can obtain an L 2 rror stimat by a duality argumnt. Lmma 2.6. It holds that Proof. Lt φ H0 1 ( satisfy v φdx = hn φ H 2 (, u u h L2( Ch ( u u h h + u u h h. (2.17 (u u h v dx v H 1 0(. (2.18 φ = u u h in, (2.19 c 2007 Europan Socity of Computational Mthods in Scincs and Enginring (ESCMSE

7 A Wakly Ovr-Pnalizd Intrior Pnalty Mthod 7 and by lliptic rgularity φ H 2 ( C u u h L2(. (2.20 Applying (2.13 and (2.14 to φ, w hav φ Π h φ h Ch u u h L2(, (2.21 E h {{ (φ Π h φ}} 2 L 2( Ch2 u u h 2 L 2(. (2.22 It follows from (2.18, (2.19 and intgration by parts that (u u h φdx = (u u h u dx u h φdx = (u u h 2 dx [[u h ]] {{ φ}} ds, E h and hnc, in viw of (1.12, (2.1 and th fact that [[φ]] = 0 = [[u]], u u h 2 L 2( = = (u u h φdx + [[u h ]] {{ φ}} ds E h (u u h φdx [[u u h ]] {{ φ}} ds E h = a h (u u h, φ 2 {{ φ}} [[u u h ]] ds (2.23 E h = a h (u u h, φ Π h φ 2 {{ (φ Π h φ}} [[u u h ]] ds E h + 2 {{ (Π h φ}} [[u u h ]] ds. E h W now bound ach of th thr trms on th right-hand sid of (2.23 sparatly. h first trm can b boundd using (2.6 and (2.21: a h (u u h, φ Π h φ C u u h h φ Π h φ h Ch u u h h u u h L2(. (2.24 h scond trm can b boundd using th Cauchy-Schwarz inquality, (1.7, (1.12, (2.20 and (2.22: E h {{ (φ Π h φ}} [[u u h ]] ds E h ( 1/2 {{ (φ Πh φ}} L2( ( 1/2 [[u uh ]] L2( (2.25 ( 1/2 ( {{ (φ Π hφ}} 2 1/2 L 2( [[u u h]] 2 L 2( E h E h Ch u u h L2( u u h h. c 2007 Europan Socity of Computational Mthods in Scincs and Enginring (ESCMSE

8 8 S.C. Brnnr and L. Owns Finally, using th Cauchy-Schwarz inquality, (1.7, (1.12, (2.21 and a standard invrs stimat, w obtain {{ (Π h φ}} [[u u h ]] ds E h = E h (Π h φ Π 0 [[u u h ]] ds E h 1/2 3/2 (Π h φ L2( 1/2 3/2 Π 0 [[u u h]] L2( ( 1/2 ( 1/2 1 3 (Π h φ 2 L 2( 3 Π 0 [[u u h]] 2 L 2( (2.26 E h E h C (h 1/2 2 (Π h φ 2 L 2( u uh h C (h [ 1/2 2 (φ Π h φ 2 L + 2( φ 2 L 2(] u uh h Ch u u h L2( u u h h. h dsird rsult follows from (2.23 (2.26. Combining horm 2.4, Corollary 2.5 and Lmma 2.6, w hav th following rsult for th L 2 rror. horm 2.7. It holds that u u h L2( Ch 2 f L2(. ( h Prconditionr Lt A h : V h V h b dfind by A h w, v = a h (w, v v, w V h, (3.1 whr, is th canonical bilinar form on V h V h. In trms of A h, th discrt problm (1.11 can b writtn as A h u h = φ h, (3.2 whr φ h V h is dfind by φ h, v = (f, v L2( v V h. h prconditionr for A h is th oprator B h : V h V h dfind by B h w, v = wv dx + Π 0 [[w]] Π 0 [[v]] ds E h v, w V h. (3.3 Rmark 3.1. It follows from a standard quadratur rul for quadratic functions that wv dx = w(mv(m w, v P 1 (, (3.4 3 m M c 2007 Europan Socity of Computational Mthods in Scincs and Enginring (ESCMSE

9 A Wakly Ovr-Pnalizd Intrior Pnalty Mthod 9 whr M is th st of th thr midpoints of. Furthrmor from (1.2, (1.4 and (1.13 w s that, for an intrior dg, 1 Π 0 [[w]] Π 0 [[v]] ds (3.5 = w 1 (m v 1 (m + w 2 (m v 2 (m w 1 (m v 2 (m w 2 (m v 1 (m, whr w j = w j and v j = v j for j = 1, 2, and 1 and 2 ar th two triangls in h sharing as a common dg, and for a boundary dg, w hav 1 Π 0 [[w]] Π 0 [[v]] ds = w(m v(m. (3.6 Lt B h b th matrix rprsnting B h, i.., v B h w = B h w, v, (3.7 whr v (rsp. w is th coordinat vctor for v (rsp. w in V h associatd with th midpoints of th dgs of h. In viw of Rmark 3.1, th matrix B h is block diagonal with 2 2 blocks (corrsponding to th midpoints of intrior dgs and 1 1 blocks (corrsponding to th midpoints of boundary dgs. hrfor it is trivial to comput B 1 h. Lt th oprators S h, N h : V h V h rprsnt th symmtric and antisymmtric part of th bilinar form a h (, : S h w, v = w v dx + 3 Π 0 [[w]] Π0 [[v]] ds v, w V h, (3.8 h E h N h w, v = ( {{ w}}[[v]] {{ v}}[[w]] ds v, w Vh. (3.9 E h It is clar from (3.1, (3.8 and (3.9 that B 1 h A h = B 1 h S h B 1 h N h. (3.10 hrfor w can stimat th condition numbr of th prconditiond systm B 1 h A h by xamining th oprators B 1 h S h and B 1 h N h. Lmma 3.2. All of th ignvalus of B 1 h S h ar ral and th minimum and maximum ignvalus of B 1 h S h satisfy c λ min (B 1 h S h λ max (B 1 h S h Ch 2, (3.11 whr c and C ar positiv constants that dpnd only on th minimum angl of h. In particular, w hav κ(b 1 h S h = λ max(b 1 h S h λ min (B 1 h S h Ch 2. (3.12 Proof. Sinc th oprator B 1 h S h is symmtric with rspct to th innr product B h, on V h, all th ignvalus of B 1 h S h ar ral, and it follows from th Raligh quotint formula [9] that λ max (B 1 h S S h v, v h = max v V h \{0} B h v, v, (3.13 λ min (B 1 h S S h v, v h = min v V h \{0} B h v, v. (3.14 c 2007 Europan Socity of Computational Mthods in Scincs and Enginring (ESCMSE

10 10 S.C. Brnnr and L. Owns Lt v V h b arbitrary. By (3.3, (3.8, a standard invrs stimat and th fact that h, w hav S h v, v = v 2 L + 2( 3 Π0 [[v]] 2 L 2( E h Ch 2 v 2 L + 2( Ch 2 Π0 [[v]] 2 L 2( Ch 2 B h v, v, E h which togthr with (3.13 implis that λ max (B 1 h S h Ch 2. In th othr dirction w hav th following stimat from [5]: ( v 2 L C 2( v 2 L + 2( E h 1 [[Π0 v]] 2 L 2(. (3.15 Combining (1.7, (3.3, (3.8 and (3.15, w find B h v, v = v 2 L + 2( Π0 [[v]] 2 L 2( E h ( C v 2 L + 1 2( [[Π0 v]] 2 L 2( + C 3 Π0 [[v]] 2 L 2( E h E h C S h v, v, which togthr with (3.14 implis λ min (B 1 h S h c. Lmma 3.3. h ignvalus of B 1 h N h ar purly imaginary and B 1 h N hw, v Ch 1 B h w, w 1/2 B h v, v 1/2 w, v V h, (3.16 whr th positiv constant C dpnds only on th minimum angl of h. In particular, th spctral radius of B 1 h N h satisfis ρ(b 1 h N h Ch 1. (3.17 Proof. Sinc th oprator B 1 h N h is antisymmtric with rspct to th innr product B h, on V h, all th ignvalus of B 1 h N h ar purly imaginary. Using (1.7, (3.3, th Cauchy-Schwarz inquality and standard invrs stimats, w find {{ w}}[[v]] ds = 1/2 1/2 {{ w}} 1/2 Π 0 [[v]] ds E h E h ( 1 {{ w}} 2 L 2( E h 1/2 ( 1 Π 0 [[v]] 2 L 2( E h ( C 1/2 Bh v, v 1/2 ( C w 2 L 2( h 2 w 2 L 2( 1/2 Bh v, v 1/2 Ch 1 B h w, w 1/2 B h v, v 1/2, and similarly {{ v}}[[w]] ds Ch 1 B h v, v 1/2 B h w, w 1/2. E h 1/2 c 2007 Europan Socity of Computational Mthods in Scincs and Enginring (ESCMSE

11 A Wakly Ovr-Pnalizd Intrior Pnalty Mthod 11 W conclud from Lmma 3.2 (rsp. Lmma 3.3 that B 1 h S h (rsp. B 1 h N h bhavs lik a scond (rsp. first ordr diffrntial oprator. In viw of (3.10, th condition numbr of B 1 h A h is also of ordr O(h 2. 4 Numrical Rsults Lt b th unit squar (0, 1 (0, 1 and th xact solution of (1.1 b givn by u(x, y = xy(1 x(1 y. W solvd (1.1 using th WOPNIP mthod (with = 0.1, 1, 10 and 100 on uniform grids 1,..., 7, whr th lngth of a horizontal/vrtical dg in k is h k = 2 k, and computd th rlativ rrors k (u u k 2 L2( u L2( in th picwis H 1 smi-norm and th rlativ rrors u u k L2( u L2( in th L 2 norm. h rsults ar plottd against k in Figur 1 and Figur 2. h rror bounds (2.15 and (2.27 ar clarly visibl. Furthrmor th rlativ rrors for = 1, 10 and 100 ar vntually indistinguishabl =0.1 =1 =10 = Error k Figur 1: Rlativ rrors in th picwis H 1 smi-norm for 1 k 7 and =0.1, 1, 10, and 100 W also computd th condition numbr κ(b 1 k S k and th spctral radius ρ(b 1 k N k for 1 k 6, and th numbrs h 2 k κ(b 1 k S k and h k ρ(b 1 k N k ar tabulatd in abl 1 and abl 2, which clarly dmonstrat th stimats (3.12 and (3.17. c 2007 Europan Socity of Computational Mthods in Scincs and Enginring (ESCMSE

12 12 S.C. Brnnr and L. Owns =0.1 =1 =10 =100 Error k Figur 2: Rlativ rrors in th L 2 norm for 1 k 7 and =0.1, 1, 10, and 100 k = 0.1 = 1 = 10 = abl 1: h 2 k κ(s k for 1 k 6 and =0.1, 1, 10, Concluding Rmarks h rsults in this papr can b xtndd to gnral scond ordr lliptic boundary valu problms. hy can also b xtndd to nonconforming mshs with hanging nods and highr ordr lmnts. Multigrid algorithms for th WOPNIP mthod can b dvlopd using a smoothr built upon th block diagonal prconditionr in Sction 3, and th quasi-optimal L 2 rror stimat (2.27 is crucial for th convrgnc analysis of th multigrid algorithms. hs and othr issus concrning th wakly ovr-pnalizd intrior pnalty mthods will b addrssd in [11]. Acknowldgmnt h work in this papr was partially supportd by th National Scinc Foundation undr Grant No. DMS h first author would also lik to thank th Alxandr von Humboldt Foundation for support through hr Humboldt Rsarch Award. Part of th rsarch in this papr was carrid out whil th first author was visiting th Humboldt Univrsität zu Brlin and sh would lik to thank th mmbrs of th Institut für Mathmatik for thir hospitality. c 2007 Europan Socity of Computational Mthods in Scincs and Enginring (ESCMSE

13 A Wakly Ovr-Pnalizd Intrior Pnalty Mthod 13 k = 0.1 = 1 = 10 = abl 2: h k ρ(n k for 1 k 6 and =0.1, 1, 10, and 100 Rfrncs [1] D.N. Arnold. An intrior pnalty finit lmnt mthod with discontinuous lmnts. SIAM J. Numr. Anal., 19: , [2] D.N. Arnold, F. Brzzi, B. Cockburn, and D. Marini. Discontinuous Galrkin mthods for lliptic problms. In B. Cockburn, G.E. Karniadakis, and C.-W. Shu, ditors, Discontinuous Galrkin Mthods, pags Springr-Vrlag, Brlin-Hidlbrg, [3] D.N. Arnold, F. Brzzi, B. Cockburn, and L.D. Marini. Unifid analysis of discontinuous Galrkin mthods for lliptic problms. SIAM J. Numr. Anal., 39: , 2001/02. [4] J.H. Brambl and S.R. Hilbrt. Estimation of linar functionals on Sobolv spacs with applications to Fourir transforms and splin intrpolation. SIAM J. Numr. Anal., 7: , [5] S.C. Brnnr. Poincaré-Fridrichs inqualitis for picwis H 1 functions. SIAM J. Numr. Anal., 41: , [6] S.C. Brnnr and L.R. Scott. h Mathmatical hory of Finit Elmnt Mthods (Scond Edition. Springr-Vrlag, Nw York-Brlin-Hidlbrg, [7] P.G. Ciarlt. h Finit Elmnt Mthod for Elliptic Problms. North-Holland, Amstrdam, [8] M. Crouzix and P.-A. Raviart. Conforming and nonconforming finit lmnt mthods for solving th stationary Stoks quations I. RAIRO Anal. Numér., 7:33 75, [9] G.H. Golub and C.F. Van Loan. Matrix Computations (third dition. h Johns Hopkins Univrsity Prss, Baltimor, [10] P. Grisvard. Elliptic Problms in Non Smooth Domains. Pitman, Boston, [11] L. Owns. Multigrid Mthods for Wakly Ovr-Pnalizd Intrior Pnalty Mthods. PhD thsis, Univrsity of South Carolina, (in prparation. [12] B. Rivièr, M.F. Whlr, and V. Girault. A priori rror stimats for finit lmnt mthods basd on discontinuous approximation spacs for lliptic problms. SIAM J. Numr. Anal., 39: , [13] M.F. Whlr. An lliptic collocation-finit-lmnt mthod with intrior pnaltis. SIAM J. Numr. Anal., 15: , c 2007 Europan Socity of Computational Mthods in Scincs and Enginring (ESCMSE