An interior penalty method for a two dimensional curl-curl and grad-div problem

Transcription

1 ANZIAM J. 50 (CTAC2008) pp.c947 C975, 2009 C947 An intrior pnalty mthod for a two dimnsional curl-curl and grad-div problm S. C. Brnnr 1 J. Cui 2 L.-Y. Sung 3 (Rcivd 30 Octobr 2008; rvisd 30 April 2009) Abstract W study an intrior pnalty mthod for a two dimnsional curlcurl and grad-div problm that appars in lctromagntics and in fluid-structur intractions. Th mthod uss discontinuous P 1 vctor filds on gradd mshs and satisfis optimal convrgnc rats (up to an arbitrarily small paramtr) in both th nrgy norm and th L 2 norm. Ths thortical rsults ar corroboratd by rsults of numrical xprimnts. Contnts 1 Introduction C948 2 Prliminaris C949 3 Th intrior pnalty mthod C951 givs this articl, c Austral. Mathmatical Soc Publishd July 1, issn (Print two pags pr sht of papr.)

2 1 Introduction C948 4 Error analysis C955 5 Nonconforming mshs C965 6 Numrical xprimnts C967 7 Concluding rmarks C970 Rfrncs C972 1 Introduction Lt Ω R 2 b a boundd polygonal domain. W considr an intrior pnalty mthod for th following curl-curl and grad-div problm. Problm 1 Find u H 0 (curl; Ω) H(div; Ω) such that ( u, v) + γ( u, v) + α(u, v) = (f, v) (1) for all v H 0 (curl; Ω) H(div; Ω), whr (, ) dnots th innr product of [L 2 (Ω)] 2, α R and γ > 0 ar constants, and f [L 2 (Ω)] 2. Th variational Problm 1 appars in lctromagntics [29, 30] and fluidstructur intractions [27, 9, 8, 10] (aftr intrchanging th rols of curl and div). Th main difficulty in th numrical solution of (1) is that standard H 1 -conforming finit lmnt mthods fail whn Ω is not convx [22]. Such mthods produc numrical solutions that convrg to a vctor fild that is not th solution of (1). Spcial tratmnts ar thrfor ncssary for capturing th corrct solution, ithr by augmnting standard H 1 finit lmnt vctor filds by singular vctor filds [11, 5, 28, 3, 4], or by solving a rgularizd vrsion of (1) [24, 25, 21]. Brnnr t al. [15] introducd a nonconforming finit lmnt mthod for (1). It uss th Crouzix Raviart wakly continuous P 1 vctor filds [26] on gradd

3 2 Prliminaris C949 mshs and has optimal convrgnc rats in both th nrgy norm and th L 2 norm. In this articl w study an intrior pnalty vrsion of th mthod by Brnnr t al. [15]. By rmoving th wak continuity condition of th vctor filds, th intrior pnalty mthod applis to mshs with hanging nods. This mthod blongs to a growing family of finit lmnt mthods for problms posd on H(curl; Ω) H(div; Ω) [16, 15, 17, 20, 18]. Sction 2 rcalls dfinitions of function spacs and proprtis of Problm 1, and Sction 3 dfins th numrical schm whos analysis is thn carrid out in Sction 4. Sction 5 discusss th xtnsion of th mthod to nonconforming mshs and Sction 6 prsnts numrical rsults that corroborat th thortical rsults. W nd with som concluding rmarks in Sction 7. 2 Prliminaris Th function spacs H 0 (curl; Ω) and H(div; Ω) ar dfind as { [ ] v1 H(curl; Ω) = v = [L v 2 (Ω)] 2 : v = v 2 v } 1 L 2 (Ω), 2 x 1 x 2 H 0 (curl; Ω) = {v H(curl; Ω) : n v = 0 on Ω}, with n bing th unit outr normal, and H(div; Ω) = { v = [ v1 v 2 ] [L 2 (Ω)] 2 : v = v 1 + v } 2 L 2 (Ω). x 1 x 2 Th uniqu solvability of problm (1), in th cas whr α > 0, is guarantd by th Risz rprsntation thorm for th Hilbrt spac H 0 (curl; Ω) H(div; Ω) with th innr product ((, )) dfind by ((v, w)) = ( v, w) + ( v, w) + (v, w).

4 2 Prliminaris C950 For α 0, problm (1) is uniquly solvabl if α is diffrnt from a squnc of xcptional valus [30]. Indd thr xists a squnc of nonngativ numbrs 0 λ γ,1 λ γ,2 such that thr xists a nontrivial solution w H 0 (curl; Ω) H(div; Ω) for th ignproblm ( w, v) + γ( w, v) = λ γ,j (w, v) (2) for all v H 0 (curl; Ω) H(div; Ω). For α 0, problm (1) is wll-posd as long as α λ γ,j for j 1. Th rgularity of th solution u of (1) is wll stablishd [6, 23, 15,.g.]. Blow w summariz th rsults as statd by Brnnr t al. [15]. First of all, u and u blong to H 1 (Ω), and u H 1 (Ω) + u H 1 (Ω) C f L2 (Ω). (3) Scondly, w hav u [H 2 (Ω δ )] 2 and th following stimat is valid: u H 2 (Ω δ ) C f L2 (Ω), (4) whr th domain Ω δ is obtaind from Ω by xcising δ-nighborhoods from th cornrs c 1,..., c L of Ω, that is, Ω δ = {x Ω : x c l > δ for 1 l L}. Thirdly, in th nighborhood N l,3δ/2 = {x Ω : x c l < 3δ/2} of th cornr c l, w hav u = u R + u S, (5) whr u R [H 2 ɛ (N l,3δ/2 )] 2 for any ɛ > 0, u S = j N j(π/ω l ) (0,2)\{1} ν l,j r j(π/ω l) 1 l [ ( sin j(π/ωl ) 1 ) ] θ l cos ( j(π/ω l ) 1 ), (6) θ l

5 3 Th intrior pnalty mthod C951 and ν l,j ar constants. stimats: Morovr, w hav th following cornr rgularity L u R H 2 ɛ (N l,3δ/2 ) C ɛ f L2 (Ω) ; l=1 L l=1 j N j(π/ω l ) (0,2)\{1} (7a) ν l,j C f L2 (Ω). (7b) Not that th rgularity of u and u imply that th boundary valu problm corrsponding to (1) is ( u) γ ( u) + αu = f in Ω, (8a) n u = 0 on Ω, (8b) u = 0 on Ω. (8c) 3 Th intrior pnalty mthod W nd gradd mshs to rcovr optimal convrgnc rats for a gnral polygonal domain Ω. W assum thrfor that th triangulation T h of Ω satisfis C 1 h T hφ µ (T) C 2 h T for all T T h, (9) whr h T is th diamtr of th triangl T, h = max T Th h T is th msh paramtr, and th positiv constants C 1 and C 2 ar indpndnt of h. Th wight Φ µ (T) in (9) is dfind by Φ µ (T) = L c l c T 1 µ l, (10) l=1

6 3 Th intrior pnalty mthod C952 whr c T is th cntr of T, and th grading paramtrs µ 1,..., µ L ar chosn according to th following rul: { µ l = 1 if ω l π, 2 µ l < π 2ω l if ω l > π. (11) 2 Th construction of T h satisfying th msh condition (9) is dscribd lswhr [1, 2, 14, 7,.g.]. Not that T h satisfis th minimum angl condition for any givn grading paramtrs. Rmark 2 Th choic of grading paramtr in (11), which is dictatd by th rgularity of th solution u of (1), indicats that grading is ndd around any cornr whos angl is largr than a right angl. This is diffrnt from th grading stratgy for th Laplac oprator, whr grading is ndd only around r-ntrant cornrs, and it is du to th singularity of th diffrntial oprator in (8a) bing on ordr mor svr than th singularity of th Laplac oprator. W tak V h to b th spac of (discontinuous) P 1 vctor filds, that is, V h = { v [L 2 (Ω)] 2 : v T = v T [P 1 (T)] 2 for all T T h }. Sinc th vctor filds in V h ar (in gnral) discontinuous, thir jumps across th dgs of T h play an important rol in intrior pnalty mthods. Blow ar th dfinitions of th tangntial and normal jumps of th vctor filds. W dnot by E h (rspctivly E i h ) th st of th dgs (rspctivly intrior dgs) of T h. Lt E i h b shard by th two triangls T ± T h (compar with Figur 1) and n + (rspctivly n ) b th unit normal of pointing towards th outsid of T + (rspctivly T ). W dfin, on, [n v] = n + v T+ + n v T, (12a) [n v] = n + v T+ + n v T. (12b)

7 3 Th intrior pnalty mthod C953 T n n T + + Figur 1: Triangls and normals in th dfinitions of [n v] and [n v]. For an dg E b h, w tak n to b th unit normal of pointing towards th outsid of Ω and dfin [n v] = n v. (13) W now dfin th discrt problm for th intrior pnalty mthod. Problm 3 Find u h V h such that whr a h (u h, v) = (f, v) for all v V h, (14) a h (w, v) = ( h w, h v) + γ( h w, h v) + α(w, v) + [Φ µ ()]2 [n w] [n v] ds E h + [Φ µ ()] 2 [n w][[n v] ds (15) E i h + h 2 1 (Π 0 [n w]) (Π 0 [n v]) ds E h + h 2 1 (Π 0 [n w])(π 0 [n v]) ds, E i h

8 3 Th intrior pnalty mthod C954 and whr dnots th lngth of th dg, and Π 0 is th orthogonal projction from L 2 () to P 0 () (th spac of constant functions on ). Th dg wight Φ µ () in (15) is dfind by Φ µ () = L c l m 1 µ l, (16) l=1 whr c 1,..., c L ar th cornrs of Ω and m is th midpoint of th dg. Rmark 4 Comparing (10) and (16) w hav Φ µ () Φ µ (T) if T, (17) whr th positiv constants in th quivalnc ar indpndnt of h. This rlation is important for th drivation of optimal a priori rror stimats. W us th Crouzix Raviart intrpolation oprator in th analysis of th intrior pnalty mthod. For s > 1/2 and T T h, w dfin Π T : [H s (T)] 2 [P 1 (T)] 2 by (Π T ζ) ds = ζ ds for 1 j 3, (18) j j whr 1, 2 and 3 ar th dgs of T. Th oprator Π T satisfis th following standard rror stimat [26]: ζ Π T ζ L2 (T) + h min(s,1) T ζ Π T ζ H min(s,1) (T) C T h s T ζ H s (T) (19) for all ζ [H s (T)] 2 and s (1/2, 2], whr th positiv constant C T dpnds on th minimum angl of T (and also on s whn s approachs 1/2). Sinc H 0 (curl; Ω) H(div; Ω) [H s (Ω)] 2 for som s > 1/2 [30, 15, cf..g.], w can dfin a global intrpolation oprator Π h : H 0 (curl; Ω) H(div; Ω) V h by picing togthr th local intrpolation oprators, that is, (Π h v) T = Π T v T for all T T h. (20)

9 4 Error analysis C955 W also dnot th picwis dfind curl and div oprator by h and h, that is, ( h v) T = (v T ) for all T T h, (21) ( h v) T = (v T ) for all T T h. (22) It follows from (18), (20) (22) and Grn s thorm that h (Π h v) = Π h 0 ( v) for all v H 0 (curl; Ω) H(div; Ω), (23) h (Π h v) = Π h 0 ( v) for all v H 0 (curl; Ω) H(div; Ω), (24) whr Π h 0 is th orthogonal projction from L 2(Ω) onto th spac of picwis constant functions associatd with T h. 4 Error analysis Th discrtization rror is masurd in both th L 2 dpndnt nrgy norm h dfind by norm and th msh v 2 h = h v 2 L 2 (Ω) + γ h v 2 L 2 (Ω) + v 2 L 2 (Ω) + [Φ µ ()]2 [n v] 2 L 2 () + [Φ µ ()] 2 [n v] 2 L 2 () (25) E h E i h + h 2 1 Π0 [n v] 2 L 2 () + 1 Π0 [n v] 2 L 2 (). E h E i h Not that a h (, ) is boundd by this nrgy norm, that is, for all v, w H 0 (curl; Ω) H(div; Ω) + V h. a h (w, v) ( α + 1) w h v h (26)

10 4 Error analysis C956 For α > 0, a h (, ) is also corciv with rspct to h, that is, a h (v, v) min(1, α) v 2 h (27) for all v H 0 (curl; Ω) H(div; Ω) + V h. In this cas th discrt problm is wll-posd and w hav th following abstract rror stimat, whos proof is idntical with our arlir proof [18, Lmma 3.5]. Lmma 5 Lt α b positiv, β = min(1, α), u H 0 (curl; Ω) H(div; Ω) b th solution of (1), and u h satisfy th discrt problm (14). Thn ( ) 1 + α + β u u h h inf u v h + 1 β v V h β sup a h (u u h, w). (28) w V h \{0} w h For α 0, th following Gårding (in)quality holds a h (v, v) + ( α + 1)(v, v) = v 2 h (29) for all v H 0 (curl; Ω) H(div; Ω) + V h. In this cas th discrt problm is indfinit and th following lmma provids an abstract rror stimat for th schm (14) undr th assumption that it has a solution. Its proof, which is basd on (26) and (29), is also idntical to our arlir proof [18, Lmma 3.6]. Lmma 6 Lt u H 0 (curl; Ω) H(div; Ω) satisfy (1) and u h b a solution of (14). Thn a h (u u h, w) u u h h (2 α + 3) inf u v h + sup v V h w V h \{0} w h + ( α + 1) u u h L2 (Ω). (30) From hr on w considr α and γ to b fixd and drop th dpndnc on ths constants in our stimats.

11 4 Error analysis C957 Rmark 7 Th first trm on th right-hand sid of (28) and (30) masurs th approximation proprty of V h with rspct to th nrgy norm. Th scond trm masurs th consistncy rror. Th third trm on th righthand sid of (30) addrsss th indfinitnss of th problm whn α 0. Sinc th intrpolation oprator Π h dfind in Sction 3 is also th on mployd in arlir work [16], w us in our analysis th following rsults from that articl [16, Lmma 5.1 and Lmma 5.2] which wr obtaind by using (11), (17), (19) and th rgularity stimats (3) (7). Lmma 8 Lt u H 0 (curl; Ω) H(div; Ω) b th solution of (1). W hav th following intrpolation rror stimats: u Π h u L2 (Ω) h 2 ɛ f L2 (Ω), (31) [Φ µ ()]2 [u Π h u] 2 L 2 () C ɛ h 2 ɛ f 2 L 2 (Ω), E h (32) for any ɛ > 0, whr [u Π h u] is th jump of u Π h u across th intrior dgs of T h and [u Π h u] is u Π h u on th boundary dgs of T h. Th approximation proprty of V h is stablishd by th following lmma. Lmma 9 Lt u H 0 (curl; Ω) H(div; Ω) b th solution of (1). Thn for any ɛ > 0. inf u v h u Π h u h < C ɛ h 1 ɛ f L2 (Ω) (33) v V h Proof: It follows from (18) that Π 0 [n (u Π h u)] = 0 for all E h and Π 0 [n (u Π h u)] = 0 for all E i h. Thrfor w hav u Π h u 2 h = h (u Π h u) 2 L 2 (Ω) + γ h (u Π h u) 2 L 2 (Ω) + u Π h u 2 L 2 (Ω)

12 4 Error analysis C958 + [Φ µ ()]2 [n (u Π h u)] 2 L 2 () (34) E h + [Φ µ ()] 2 [n (u Π h u)] 2 L 2 (). E i h Lmma 8 stimats th last thr trms on th right-hand sid of (34), and w bound th first two trms on th right-hand sid of (34) by (3), (19), (23) and (24) as h (u Π h u) 2 L 2 (Ω) + γ h (u Π h u) 2 L 2 (Ω) = u Π h 0 ( u) 2 L 2 (Ω) + γ u Π h 0 ( u) 2 L 2 (Ω) Ch 2 f 2 L 2 (Ω). Nxt w turn to th consistncy rror, whr w nd th following rsult provd in arlir work [16, Lmma 5.3]. Lmma 10 E h [Φ µ ()] 2 η ^η T 2 L 2 () Ch 2 η 2 H 1 (Ω) for all η H 1 (Ω), whr ^η T = T 1 T η dx is th man of η ovr T, on of th triangls in T h that has as an dg. Th following lmma provids an optimal bound for th consistncy rror. Lmma 11 Lt u H 0 (curl; Ω) H(div; Ω) b th solution of (1) and u h V h satisfy (14). Thn a h (u u h, w) sup Ch f L2 (Ω). (35) w V h \{0} w h

13 4 Error analysis C959 Proof: Lt w V h b arbitrary. Sinc th strong form of (1) is givn by (8), w find, by (12), (13), (15) and intgration by parts, a h (u, w) = ( u)( w) dx T T h T + T T h γ T E h ( u)( w) dx + α(u, w) = (f, w) + ( u)[n w] ds (36) E i h + γ ( u)[n w] ds. Subtracting (14) from (36) givs a h (u u h, w) = ( u)[n w] ds+ E h E i h γ ( u)[n w] ds. (37) W rwrit th first trm on th right-hand sid of (37) as ( u)[n w] ds = ( u ( u) T )[n w] ds E h E h + E h ( u) T (Π 0 [n w]) ds, (38) whr ( u)t is th man of u on T, on of th triangls in T h that has as an dg. It follows from (3), (25), Lmma 10 and th Cauchy Schwarz inquality that th first trm on th right-hand sid of (38) satisfis ( u ( ) u) T [n w]ds E h

14 4 Error analysis C960 ( [Φ µ ()] 2 u 1/2 ( u) 2 (39) L 2 ()) E h ( ) 1/2 E h 1 [Φ µ ()] 2 n w 2 L 2 () Ch f L2 (Ω) w h. For th scond trm on th right-hand sid of (38), w find, by using th Cauchy Schwarz inquality, (3) and (25), ( ( u) T Π 0 h [n w] ) ds E h E h ( 1/2 ( ) ( ) u) T L2 () 1/2 Π 0 [n w] L2 () ( 1/2 Ch ( u) T 2 L 2 (T )) (40) E h ( h ) 1/2 2 1 Π0 [n w] 2 L 2 () E h Ch u L2 (Ω) w h Ch f L2 (Ω) w h. Hr w hav also usd th fact that, if is an dg of a triangl T, thn q 2 L 2 () C T q 2 L 2 (T) for any constant function q, (41) whr th positiv constant C T dpnds only on th shap of T. Combining (38) (40), w hav ( u)[n w] ds Ch f L2 (Ω) w h, (42) E h

15 4 Error analysis C961 and similarly, ( u)[n w] ds Ch f L2 (Ω) w h. (43) E i h Th stimat (35) follows from (37), (42) and (43). Th following lmma givs an L 2 rror stimat undr th assumption that th discrt problm (14) has a solution. Lmma 12 Lt u H 0 (curl; Ω) H(div; Ω) b th solution of (1) and u h V h satisfy (14). Thn u u h L2 (Ω) C ɛ ( h 2 ɛ f L2 (Ω) + h 1 ɛ u u h h ) (44) for any ɛ > 0. Proof: Th proof is basd on a duality argumnt. Lt z H 0 (curl; Ω) H(div; Ω) satisfy ( v, z) + γ( v, z) + α(v, z) = (v, (u u h )) (45) for all v H 0 (curl; Ω) H(div; Ω). Th strong form of (45) is ( z) γ ( z) + αz = u u h in Ω, (46a) n z = 0 on Ω, (46b) z = 0 on Ω, (46c) and w hav th following analog of (3): z H 1 (Ω) + z H 1 (Ω) C u u h L2 (Ω). (47) Furthrmor w can writ (45) as a h (v, z) = (v, (u u h )) for all v H 0 (curl; Ω) H(div; Ω). (48)

16 4 Error analysis C962 It follows from (46), (48) and intgration by parts that th following analog of (36) holds: a h (u h, z) = ( u h )( z) dx T T h T + T T h γ T ( u h )( z) dx + α(u h, z) = (u h, (u u h )) + [n u h ]( z) ds (49) + E i h Combining (48) and (49) givs E h γ [n u h ]( z) ds. u u h 2 L 2 (Ω) = (u, u u h ) (u h, u u h ) = a h (u u h, z) + [n u h ]( z) ds (50) + E i h E h γ [n u h ]( z) ds, and w stimat th thr trms on th right-hand sid of (50) sparatly. Using (37) and th fact that Π 0 [n (Π h z)] (rspctivly Π 0 [n (Π h z)]) vanishs for all E h (rspctivly E i h ), w rwrit th first trm (following th notation in (38)) as a h (u u h, z) = a h (u u h, z Π h z) + a h (u u h, Π h z) = a h (u u h, z Π h z) + ( u ( ) u) T [n (Π h z)] ds E h

17 4 Error analysis C963 + E i h ( γ u ( ) u) T [n (Π h z)] ds, from which w obtain th following stimat using (3), and Lmmas 8 and 10: a h (u u h, z) C ɛ ( h 2 ɛ f L2 (Ω) + h 1 ɛ u u h h ) u uh L2 (Ω). (51) Brnnr t al. [15, pp ] gav dtails whr an idntical stimat is drivd. W now considr th scond trm on th right-hand sid of (50). First [n u h ]( z)ds = ( [n u h ] z ( ) z) T ds E h E h + E h ( Π 0 [n u h ] ) ( z)t ds, (52) whr ( z)t is th man of z on on of th triangls T T h that has as an dg. Th stimat blow follows from (25), Lmma 10, (47) and th Cauchy Schwarz inquality: ( [n u h ] z ( ) z) T ds (53) E h Ch u u h L2 (Ω) u u h h. On th othr hand, as in th drivation of (40), w obtain by th Cauchy Schwarz inquality, (25), (41) and (47), (Π 0 [n u h ])( z) T ds E h = E h ( Π 0 [n (u h u)] ) ( z)t ds (54)

18 4 Error analysis C964 Ch u u h h z L2 (Ω) Ch u u h h u u h L2 (Ω). Combining (52) (54), w obtain [n u h ]( z) ds Ch u u h L2 (Ω) u u h h. (55) E h Similarly, w hav th following bound on th third trm on th right-hand sid of (50): γ[n u h ]( z) ds Ch u u h L2 (Ω) u u h h. (56) E i h Th stimat (44) follows from (50), (51), (55) and (56). In th cas whr α > 0, th following thorm is an immdiat consqunc of Lmmas 5, 9, 11 and 12. Thorm 13 Lt α b positiv. Th following two discrtization rror stimats hold for th solution u h of (14): u u h h C ɛ h 1 ɛ f L2 (Ω) for any ɛ > 0 ; u u h L2 (Ω) C ɛ h 2 ɛ f L2 (Ω) for any ɛ > 0. In th cas whr α 0, w hav th following convrgnc thorm for th schm (14), whos proof is basd on Lmmas 6, 9, 11 and 12, and th approach of Schatz for indfinit problms [32]. Th argumnts ar idntical to thos of an arlir proof [16, Thorm 4.5]. Thorm 14 Assum α 0 is not on of th ignvalus λ γ,j dfind by (2). Thr xists a positiv numbr h such that th discrt problm (14) is uniquly solvabl for all h h, in which cas th following two discrtization rror stimats ar valid: u u h h C ɛ h 1 ɛ f L2 (Ω) for any ɛ > 0 ;

19 5 Nonconforming mshs C965 u u h L2 (Ω) C ɛ h 2 ɛ f L2 (Ω) for any ɛ > 0. 5 Nonconforming mshs For simplicity w dvlopd and analyzd th intrior pnalty mthod for triangulations (conforming mshs) of Ω. But of cours on of th main rasons for using an intrior pnalty mthod is that it can b applid to partitions with hanging nods (nonconforming mshs). Hr w indicat brifly how th schm and rsults xtnd with minor modifications to such mshs. Lt P h b a partition of Ω with hanging nods satisfying th following condition: whnvr a closd dg of a triangl in P h contains a hanging nod, thn it is th union of closd dgs of triangls in T h. An xampl of such a partition is dpictd in Figur 2. For such a partition, w modify th dfinition of E h as follows. Lt b an (opn) dg of a triangl in P h. Thn E h if and only if (i) it contains at last on hanging nod, (ii) it is th common dg of two triangls in P h, that is, its ndpoints ar th common vrtics of ths triangls, or (iii) it is a subst of Ω. For xampl, th dg of th largst triangl (th diagonal of th squar) in Figur 2 blongs to E h whil th thr dgs on th diagonal from th thr triangls on th othr sid do not. All togthr thr ar 12 dgs in E h for th partition in Figur 2. All th dfinitions in Sction 3 can b xtndd to P h in a straightforward fashion. For xampl, if E i h has at last on hanging nod, thn is th dg of a triangl T P h and also th union of dgs 1,..., m of th triangls T +,1,..., T +,m in P h that ar on th othr sid of (compar with Figur 3 whr m = 4). W dfin, on, [n v] = (n v T ) + m (n + v T+,j ) j, j=1

20 5 Nonconforming mshs C966 Figur 2: A triangular msh with hanging nods. T+,1 Figur 5.1: A triangular msh with hanging nods angl T P h and also n + th union of dgs 1,.. T+,3,..., T +,m in P h that ar on th othr sid of (c n+ ). W dfin, on, T+,4 T [n v ] = (n v T ) n+ m + (n + v T+,j ) Figur 3: A triangular msh with hanging nods. j, [n v ] = (n v T ) + m j=1 (n + v T+,j ) j. m (n + v T+,j ) j. Th analysis in Sction 4 rmains valid for th j=1 typ of nonconforming mshs undr considration bcaus th crucial rlations Π 0 [n (u Π h u)] = 0 for all E h and Π 0 [n (u Π h u)] = 0 for all E i h hold for th modifid dfinition of E h. Furthrmor, all th stimats for u Π h u can b carrid out triangl by triangl and T +,1 hnc also hold for partitions with hanging nods. n + n + T+,2 [n v] = (n v T ) + n j=1 T +,2 n

21 6 Numrical xprimnts C967 Of cours, th constants in th stimats now dpnd on th shap rgularity of th partition, which roughly spaking involvs th shap rgularity of th triangls in P h and th distribution of th hanging nods on th dgs in E h. Brnnr [12, 13] dtaild th concpt of shap rgularity of partitions. 6 Numrical xprimnts In this sction w rport th rsults of a sris of numrical xprimnts that corroborat our thortical rsults. Both th L 2 rror u u h L2 (Ω) and th nrgy rror u u h h ar computd for γ = 1 in all th xprimnts. In th first xprimnt w xamin th convrgnc bhavior of our numrical schm on th squar domain (0, 1) 2 with conforming uniform mshs (Figur 4, lft), whr th xact solution is u = [ y(1 y) x(1 x) ]. (57) Tabl 1 givs th rsults for α = 1, 0 and 1. Thy show that th schm (14) is scond ordr accurat in th L 2 norm and first ordr accurat in th nrgy norm, which agrs with th rror stimats in Thorms 13 and 14. In th scond xprimnt w chck th bhavior of th schm (14) on th squar (0, 1) 2 using nonconforming mshs with hanging nods dpictd in Figur 4 (right). Th rsults in Tabl 2 show that th schm also bhavs as prdictd in Thorms 13 and 14. Th goal of th final xprimnt is to dmonstrat th convrgnc bhavior of our schm on th L-shapd domain ( 0.5, 0.5) 2 \[0, 0.5] 2. Th right-hand sid function is chosn to b [ ] 1 f =. (58) 1 Th mshs ar gradd around th r-ntrant cornr (0, 0) using th rfinmnt procdur of Bacuta t al. [7] with th grading paramtr 1/3. Th

22 6 Numrical xprimnts C968 Tabl 1: Errors of th schm on th squar (0, 1) 2 with conforming uniform mshs and xact solution givn by (57). h u u h L2 (Ω) u L2 (Ω) ordr u u h h u h ordr α = 1 1/ / / / α = 0 1/ / / / α = 1 1/ / / / Figur 4: Conforming uniform msh (lft) and nonconforming msh (right) on th squar domain.

23 6 Numrical xprimnts C969 Tabl 2: Errors of th schm on th squar (0, 1) 2 with nonconforming mshs and xact solution givn by (57). h u u h L2 (Ω) u L2 (Ω) ordr u u h h u h ordr α = 1 1/ / / / α = 0 1/ / / / α = 1 1/ / / /

24 7 Concluding rmarks C970 Figur 5: Gradd mshs on th L-shapd domain. first thr lvls of gradd mshs ar dpictd in Figur 5. Th rsults in Tabl 3 dmonstrat that th schm is scond ordr accurat in th L 2 norm and first ordr accurat in th nrgy norm. 7 Concluding rmarks W xtndd th nonconforming mthod of our arlir work [15] to an intrior pnalty mthod that can b applid to nonconforming mshs. This intrior pnalty mthod njoys optimal convrgnc rats without any tuning of pnalty paramtrs. Th condition numbrs of th discrt problms ar worsnd by th wak ovr pnalization. It is thrfor important to hav good prconditionrs in th cas of fin mshs. For th Laplac oprator, fficint prconditionrs for intrior pnalty mthods with wak ovr pnalisation wr dvlopd by Owns t al. [31, 19]. Th dvlopmnt of good prconditionrs for th family of H(curl; Ω) H(div; Ω) mthods in arlir work [16, 15, 17, 20, 18] and this articl is currntly undr invstigation.

25 7 Concluding rmarks C971 Tabl 3: Errors of th schm on th L-shapd domain with gradd mshs and right-hand sid givn by (58). h u u h L2 (Ω) u L2 (Ω) ordr u u h h u h ordr α = 1 1/ / / / α = 0 1/ / / / α = 1 1/ / / /

26 Rfrncs C972 Acknowldgmnts Th work in this articl is supportd in part by th National Scinc Foundation undr Grant No. DMS Rfrncs [1] Th. Apl. Anisotropic Finit Elmnts. Tubnr, Stuttgart, C952 [2] Th. Apl, A.-M. Sändig, and J.R. Whitman. Gradd msh rfinmnt and rror stimats for finit lmnt solutions of lliptic boundary valu problms in non-smooth domains. Math. Mthods Appl. Sci., 19:63 85, C952 [3] F. Assous, P. Ciarlt, Jr., S. Labruni, and J. Sgré. Numrical solution to th tim-dpndnt Maxwll quations in axisymmtric singular domains: th singular complmnt mthod. J. Comput. Phys., 191: , C948 [4] F. Assous, P. Ciarlt, Jr., E. Garcia, and J. Sgré. Tim-dpndnt Maxwll s quations with chargs in singular gomtris. Comput. Mthods Appl. Mch. Engrg., 196: , C948 [5] F. Assous, P. Ciarlt Jr., S. Labruni, and S. Lohrngl. Th singular complmnt mthod. In N. Dbit, M. Garby, R. Hopp, D. Kys, Y. Kuzntsov, and J. Périaux, ditors, Domain Dcomposition Mthods in Scinc and Enginring, pags CIMNE, Barclona, C948 [6] F. Assous, P. Ciarlt, Jr., and E. Sonnndrückr. Rsolution of th Maxwll quation in a domain with rntrant cornrs. M2AN Math. Modl. Numr. Anal., 32: , C950 [7] C. Băcuţă, V. Nistor, and L. T. Zikatanov. Improving th rat of convrgnc of high ordr finit lmnts on polygons and domains with cusps. Numr. Math., 100: , C952, C967

27 Rfrncs C973 [8] K. J. Bath, C. Nitikitpaiboon, and X. Wang. A mixd displacmnt-basd finit lmnt formulation for acoustic fluid-structur intraction. Comput. Struct., 56: , C948 [9] A. Brmúdz and R. Rodríguz. Finit lmnt computation of th vibration mods of a fluid-solid systm. Comput. Mthods Appl. Mch. Engrg., 119: , C948 [10] D. Boffi and L. Gastaldi. On th grad div + s curl rot oprator. In Computational fluid and solid mchanics, Vol. 1, 2 (Cambridg, MA, 2001), pags Elsvir, Amstrdam, C948 [11] A.-S. Bonnt-Bn Dhia, C. Hazard, and S. Lohrngl. A singular fild mthod for th solution of Maxwll s quations in polyhdral domains. SIAM J. Appl. Math., 59: , C948 [12] S. C. Brnnr. Poincaré Fridrichs inqualitis for picwis H 1 functions. SIAM J. Numr. Anal., 41: , C967 [13] S. C. Brnnr. Korn s inqualitis for picwis H 1 vctor filds. Math. Comp., 73: , C967 [14] S. C. Brnnr and C. Carstnsn. Finit Elmnt Mthods. In E. Stin, R. d Borst, and T.J.R. Hughs, ditors, Encyclopdia of Computational Mchanics, pags Wily, Winhim, C952 [15] S. C. Brnnr, J. Cui, F. Li, and L.-Y. Sung. A nonconforming finit lmnt mthod for a two-dimnsional curl-curl and grad-div problm. Numr. Math., 109: , C948, C949, C950, C954, C963, C970 [16] S. C. Brnnr, F. Li, and L.-Y. Sung. A locally divrgnc-fr nonconforming finit lmnt mthod for th tim-harmonic Maxwll quations. Math. Comp., 76: , C949, C957, C958, C964, C970

28 Rfrncs C974 [17] S. C. Brnnr, F. Li, and L.-Y. Sung. A nonconforming pnalty mthod for a two dimnsional curl-curl problm. M3AS, 19: , C949, C970 [18] S. C. Brnnr, F. Li, and L.-Y. Sung. A locally divrgnc-fr intrior pnalty mthod for two dimnsional curl-curl problms. SIAM J. Numr. Anal., 46: , C949, C956, C970 [19] S. C. Brnnr, L. Owns, and L.-Y. Sung. A wakly ovr-pnalizd symmtric intrior pnalty mthod. ETNA, 30: , C970 [20] S. C. Brnnr and L.-Y. Sung. A quadratic nonconforming lmnt for H(curl; Ω) H(div; Ω). Appl. Math. Ltt., 22: , C949, C970 [21] P. Ciarlt, Jr. Augmntd formulations for solving Maxwll quations. Comput. Mthods Appl. Mch. Engrg., 194: , C948 [22] M. Costabl. A corciv bilinar form for Maxwll s quations. J. Math. Anal. Appl., 157: , C948 [23] M. Costabl and M. Daug. Singularitis of lctromagntic filds in polyhdral domains. Arch. Ration. Mch. Anal., 151: , C950 [24] M. Costabl and M. Daug. Wightd rgularization of Maxwll quations in polyhdral domains. Numr. Math., 93: , C948 [25] M. Costabl, M. Daug, and C. Schwab. Exponntial convrgnc of hp-fem for Maxwll quations with wightd rgularization in polygonal domains. Math. Modls Mthods Appl. Sci., 15: , C948 [26] 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, C948, C954

29 Rfrncs C975 [27] M. A. Hamdi, Y. Ousst, and G. Vrchry. A displacmnt mthod for th analysis of vibrations of coupld fluid-structur systms. Intrnat. J. Numr. Mthods Engrg., 13: , C948 [28] C. Hazard and S. Lohrngl. A singular fild mthod for Maxwll s quations: Numrical aspcts for 2D magntostatics. SIAM J. Numr. Anal., 40: , C948 [29] J.-M. Jin. Th Finit Elmnt Mthod in Elctromagntics (Scond Edition). John Wily & Sons, Inc., Nw York, C948 [30] P. Monk. Finit Elmnt Mthods for Maxwll s Equations. Oxford Univrsity Prss, Nw York, C948, C950, C954 [31] L. Owns. Multigrid Mthods for Wakly Ovr-Pnalizd Intrior Pnalty Mthods. PhD thsis, Univrsity of South Carolina, C970 [32] A. Schatz. An obsrvation concrning Ritz Galrkin mthods with indfinit bilinar forms. Math. Comp., 28: , C964 Author addrsss 1. S. C. Brnnr, Dpartmnt of Mathmatics and Cntr for Computation and Tchnology, Louisiana Stat Univrsity, Baton Roug, LA 70803, USA. mailto:brnnr@math.lsu.du 2. J. Cui, Dpartmnt of Mathmatics, Louisiana Stat Univrsity, Baton Roug, LA 70803, USA. 3. L.-Y. Sung, Dpartmnt of Mathmatics, Louisiana Stat Univrsity, Baton Roug, LA 70803, USA.