H(curl; Ω) : n v = n

Transcription

1 A LOCALLY DIVERGENCE-FREE INTERIOR PENALTY METHOD FOR TWO-DIMENSIONAL CURL-CURL PROBLEMS SUSANNE C. BRENNER, FENGYAN LI, AND LI-YENG SUNG Abstract. An intrior pnalty mthod for crtain two-dimnsional curl-curl problms is invstigatd in this papr. This mthod computs th divrgnc-fr part of th solution using locally divrgnc-fr discontinuous P 1 vctor filds on gradd mshs. It has optimal ordr convrgnc (up to an arbitrarily small ɛ) for th sourc problm and th ignproblm. Rsults of numrical xprimnts that corroborat th thortical rsults ar also prsntd. Ky words. curl-curl problm, Maxwll ignproblm, locally divrgnc-fr, intrior pnalty mthods, gradd mshs AMS subjct classification. 65N30, 65N15, 35Q60 1. Introduction. Lt Ω R 2 b a boundd polygonal domain. Considr th following wak curl-curl problm: Find u H 0 (curl; Ω) such that (1.1) ( u, v) + α(u, v) = (f, v) v H 0 (curl; Ω), whr α is a constant, (, ) dnots th innr product of L 2 (Ω) (or [L 2 (Ω)] 2 ), { [ ] v1 H(curl; Ω) = v = [L v 2 (Ω)] 2 : v = v 2 v } 1 L 2 (Ω) 2 x 1 x 2 and H 0 (curl; Ω) = {v = [ n1 [ v1 ] H(curl; Ω) : n v = n v 1 v 2 n 2 v 1 = 0 on Ω}, 2 ] with n = bing th unit outr normal on Ω. n 2 Th curl-curl problm (1.1) is rlatd to lctromagntic problms. For α 0, it is th wak form of th tim-harmonic (frquncy-domain) Maxwll quations. For α > 0, it is rlatd to th spatial problms apparing in implicit smi-discrtizations of th tim-dpndnt (tim-domain) Maxwll quations. It is wll-known [24, 25, 5] that H 1 conforming nodal finit lmnt mthods can lad to a wrong solution of (1.1) if Ω is not convx. Many altrnativ approachs hav bn dvlopd which includ H(curl; Ω) conforming dg lmnt mthods [41, 42, 13, 33, 38, 40], H 1 conforming nodal finit mthods with wightd rgularization [27, 29], th singular complmnt/fild mthod [32, 5], and intrior pnalty mthods [43, 36, 37, 35, 34]. Dpartmnt of Mathmatics and Cntr for Computation and Tchnology, Louisiana Stat Univrsity, Baton Roug, LA (brnnr@math.lsu.du). Th work of this author was supportd in part by th National Scinc Foundation undr Grant No. DMS Dpartmnt of Mathmatical Scincs, Rnsslar Polytchnic Institut, Troy, NY (lif@rpi.du). Th work of this author was supportd in part by th National Scinc Foundation undr Grant No. DMS Dpartmnt of Mathmatics, Louisiana Stat Univrsity, Baton Roug, LA (sung@math.lsu.du). 1

2 2 Susann C. Brnnr, Fngyan Li and Li-yng Sung In this papr w tak a diffrnt approach. By th Hlmholtz dcomposition, th solution u H 0 (curl; Ω) of (1.1) can b writtn uniquly as (1.2) u = ů + φ, whr ů H 0 (curl; Ω) H(div 0 ; Ω), φ H 1 0 (Ω), H(div 0 ; Ω) = It is asy to s that φ satisfis { [ ] v1 v = [L v 2 (Ω)] 2 : v = v 1 + v } 2 = 0. 2 x 1 x 2 (1.3) α( φ, ψ) = (f, ψ) ψ H 1 0(Ω). Sinc th Poisson problm (1.3) (whn α 0) can b solvd by many standard mthods undr th assumption that f H(div; Ω), w will focus on th divrgncfr part ů, which satisfis (1.4) ( ů, v) + α(ů, v) = (f, v) v H 0 (curl; Ω) H(div 0 ; Ω). W shall rfr to (1.4) as th wak form of th rducd curl-curl problm and assum in th cas whr α 0 that (1.4) has a uniqu solution (i.. α is not a Maxwll ignvalu). Not that th strong form of th rducd curl-curl problm is givn by (1.5) ( ů) + αů = Qf, whr Q : [L 2 (Ω)] 2 H(div 0 ; Ω) is th orthogonal projction. Rmark 1.1. Th strong form (1.5) and th wll-posdnss of (1.4) imply immdiatly (1.6) ů L2(Ω) C Ω,α Qf L2(Ω) C Ω,α f L2(Ω). Hr and blow w us C (with or without subscripts) to dnot a gnric positiv constant that can tak diffrnt valus at diffrnt occurrncs. Th advantag of working with (1.4) is that it bhavs lik an lliptic problm, unlik th full curl-curl problm (1.1). In particular, th solution ů of (1.4) njoys lliptic rgularity undr th assumption that f L 2 (Ω), which gratly simplifis th duality argumnt ndd for th analysis. In [15] w dvlopd a numrical schm (in th cas whr α 0) for (1.4) using th Crouzix-Raviart nonconforming P 1 vctor filds [30]. In this papr w invstigat an intrior pnalty mthod that avoids th difficultis involvd in th construction of local bass for th Crouzix-Raviart vctor filds at th cost of a largr numbr of unknowns. Ths two mthods ar vry closly rlatd, which simplifis th analysis of th nw schm. Sinc th solution oprator of th intrior pnalty mthod convrgs uniformly to th solution oprator of (1.4) with rspct to th norm L2(Ω), classical spctral approximation thory [7] implis that w can also apply th intrior pnalty mthod to th following Maxwll ignproblm: Find (λ, ů) R H 0 (curl; Ω) H(div 0 ; Ω) such that (1.7) ( ů, v) = λ(ů, v) v H 0 (curl; Ω) H(div 0 ; Ω).

3 An Intrior Pnalty Mthod for 2D Curl-Curl Problms 3 In particular, th spctral approximation basd on our intrior pnalty mthod is fr of spurious ignvalus. Th rst of th papr is organizd as follows. W introduc th finit lmnt spac and gradd mshs in Sction 2. Th intrior pnalty mthod and som prliminary stimats ar prsntd in Sction 3. Th convrgnc analysis for th sourc problm (1.1) is carrid out in Sction 4. Application to th ignproblm (1.7) is discussd in Sction 5, followd by numrical rsults in Sction 6, and w nd with som concluding rmarks in Sction Finit Elmnt Spacs and Gradd Mshs. In this sction w will introduc discontinuous finit lmnt spacs dfind on mshs gradd around th cornrs c 1,..., c L of Ω. W assum that th triangulation T h of Ω satisfis th following condition: (2.1) h T hφ µ (T) T T h, whr h T = diamt, h is a msh paramtr proportional to max T Th h T, and µ = (µ 1,..., µ L ) is th vctor containing th grading paramtrs. Th constants in th quivalnc (2.1) ar indpndnt of h and dpnd in gnral on th minimum angl of th msh. Th wight Φ µ (T) is dfind by (2.2) Φ µ (T) = Π L l=1 c l c T 1 µ l, whr c T is th cntr of T. Not that (2.3) h T Ch. Th choics of th grading paramtrs ar dictatd by th singularitis [6, 28, 26] of th solution ů of (1.4). In ordr to rcovr optimal convrgnc rats in both th nrgy norm and th L 2 norm (cf. [15]), w tak (2.4) µ l = 1 if ω l π 2, µ l < π 2ω l if ω l > π 2, whr ω l is th intrior angl at th cornr c l. Th construction of T h satisfying (2.1) can b found for xampl in [31, 4, 1, 14]. Not that T h can b constructd so that it satisfis a minimum angl condition for any fixd choic of th grading paramtrs. Rmark 2.1. Th choic of th grading paramtrs in (2.4) indicats that grading is ndd at any cornr whr th angl is largr than π/2. This is diffrnt from problms involving th Laplac oprator whr grading is only ndd at r-ntrant cornrs and is du to th fact that th singularitis of th curl-curl oprator ar on ordr wors than th singularitis of th Laplac oprator. Th finit lmnt spac V h is dfind by V h = {v [L 2 (Ω)] 2 : v T = v T [P 1 (T)] 2 and v T = 0 T T h }. For any s > 1 2 thr is a natural wak intrpolation oprator Π T : [H s (T)] 2 [P 1 (T)] 2 dfind by (2.5) (Π T ζ)(m i ) = 1 ζ ds for i = 1, 2, 3, i i

4 4 Susann C. Brnnr, Fngyan Li and Li-yng Sung whr m i is th midpoint of th dg i of T and dnots th lngth of an dg. It follows immdiatly from (2.5), th midpoint rul and Grn s thorm that (2.6) (Π T ζ)dx = ζ dx ζ [H s (T)] 2, T T (2.7) (Π T ζ)dx = ζ dx ζ [H s (T)] 2. T T Furthrmor, givn s (1/2, 2], w hav th following intrpolation rror stimats [30]: (2.8) 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) for all ζ [H s (T)] 2, whr th positiv constant C T dpnds on th minimum angl of T (and also on s whn s is clos to 1/2). Sinc th solution ů of th rducd curl-curl problm blongs to [H s (Ω)] 2 for som s > 1 2 (cf. [40]), w can dfin a global intrpolant of ů by (Π h ů) T = Π T ů T T T h, whr ů T = ů T. It follows from (2.7) and ů = 0 that Π h ů V h. Rmark 2.2. Not that Π h ů is th sam intrpolation oprator usd in [15] and Π h ů blongs to th spac of locally divrgnc-fr Crouzix-Raviart nonconforming P 1 vctor filds [30]. Sinc th vctor filds in V h ar discontinuous, thir jumps across th dgs of T h play an important rol in th dvlopmnt of intrior pnalty mthods. W will dnot by E h (rsp. Eh i) th st of th dgs (rsp. intrior dgs) of T h. Lt Eh i b shard by th two triangls T 1, T 2 T h (cf. Figur 2.1) and n 1 (rsp. n 2 ) b th unit normal of pointing towards th outsid of T 1 (rsp. T 2 ). W dfin, on, (2.9a) [[n v]] = n 1 v T 1 + n 2 v T 2, (2.9b) [[n v]] = n 1 v T 1 + n 2 v T 2. T 2 n 2 n 1 T 1 Fig Triangls and normals in th dfinitions of [[n v]] and [[n v]] For an dg along Ω, w tak n to b th unit normal of pointing towards th outsid of Ω and dfin (2.10) [[n v]] = n v. W will also dnot th picwis dfind curl and div oprators by h and h, i.., ( h v) T = v T and ( h v) T = v T.

5 An Intrior Pnalty Mthod for 2D Curl-Curl Problms 5 3. Discrtization and Prliminary Error Estimats. In this sction w prsnt th intrior pnalty mthod for th rducd curl-curl problm and som prliminary stimats. Th discrt problm is to find ů h V h such that (3.1) a h (ů h, v) = (f, v) v V h, whr (3.2) a h (w, v) = ( 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 Eh i + 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 whr Π 0 is th orthogonal projction from L 2 () onto P 0 () (th spac of constant functions on ). Th wight Φ µ () in (3.2) is dfind by Φ µ () = Π L l=1 c l m 1 µ l, whr m is th midpoint of. Rmark 3.1. Th wight Φ µ () is closly rlatd to th wight Φ µ (T) in (2.2). In fact, w hav (3.3) Φ µ () Φ µ (T) if T, whr th constants in th quivalnc ar indpndnt of h and dpnd in gnral on th minimum angl of th msh. Th inclusion of Φ µ () in (3.2) is crucial for th drivation of optimal convrgnc rats on gradd mshs. Rmark 3.2. Th wak problm (1.4) for th rducd curl-curl problm can b writtn as a h (ů, v) = (f, v) v H 0 (curl; Ω) H(div 0 ; Ω). Rmark 3.3. For nonconforming Crouzix-Raviart P 1 vctor filds w and v, w hav a h (w, v) = ( 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, E i h

6 6 Susann C. Brnnr, Fngyan Li and Li-yng Sung which is th variational form usd in [15]. Th two additional sums in (3.2) compnsat for th lack of wak continuity for th vctor filds in V h. Rmark 3.4. Th wak ovr-pnalization causs th discrt systm to bcom mor ill-conditiond. For xampl, th condition numbr grows at th rat of O(h 4 ) on quasi-uniform mshs. Howvr, thr xists a simpl block-diagonal prconditionr that can rduc th growth of th condition numbr to O(h 2 ) on quasi-uniform mshs and O(h 2 (1 + lnh ) on gradd mshs [16, 17]. Lt th msh-dpndnt nrgy norm h b dfind by (3.4) v 2 h = h v 2 L 2(Ω) + v 2 L 2(Ω) Not that + [Φ µ ()] 2 [[n v]] 2 L + 2() E h Eh i + h 2( E h 1 Π0 [[n v]] 2 L 2() + [Φ µ ()] 2 [[n v]] 2 L 2() E i h 1 ) Π0 [[n v]] 2 L 2(). (3.5) v L2(Ω) v h v H 0 (curl; Ω) H(div 0 ; Ω) + V h, and from (3.2) and (3.4), (3.6) a h (w, v) ( α + 1) w h v h w, v [H 0 (curl; Ω) H(div 0 ; Ω)] + V h. For α > 0, a h (, ) is corciv with rspct to h, i.., (3.7) a h (v, v) min(1, α) v 2 h v H 0 (curl; Ω) H(div 0 ; Ω) + V h. In this cas th discrt problm is wll-posd and w hav th following abstract rror stimat. Lmma 3.5. Lt α b positiv, β = min(1, α), ů b th solution of (1.4) and ů h satisfy (3.1). It holds that ( 1 + α + β ) (3.8) ů ů h h β inf v V h ů v h + 1 β a h (ů ů h, w) max. w V h \{0} w h Proof. Lt v V h b arbitrary. It follows from (3.6), (3.7) and th triangl inquality that ů ů h h ů v h + v ů h h ů v h + 1 β ( 1 + α + β β a h (v ů h, w) max w V h \{0} w h ) ů v h + 1 β which implis (3.8). For α 0, th following Gårding (in)quality holds: a h (ů ů h, w) max w V h \{0} w h (3.9) a h (v, v) + ( α + 1)(v, v) = v 2 h v [H 0 (curl; Ω) H(div 0 ; Ω)] + V h.

7 An Intrior Pnalty Mthod for 2D Curl-Curl Problms 7 In this cas th discrt problm is indfinit and th following lmma provids an abstract rror stimat for th schm (3.1) undr th assumption that it has a solution. Lmma 3.6. Lt ů H 0 (curl; Ω) H(div 0 ; Ω) satisfy (1.4) and ů h b a solution of (3.1). It holds that (3.10) ů ů h h (2 α + 3) inf ů v h + v V h + ( α + 1) ů ů h L2(Ω). max w V h \{0} a h (ů ů h, w) w h Proof. It follows from (3.5) and (3.9) that, for v V h \ {0}, (3.11) v h a h(v, v) (v, v) + ( α + 1) v h v h max w V h \{0} a h (v, w) w h + ( α + 1) v L2(Ω). Lt v V h b arbitrary. W find, using (3.5), (3.6), (3.11) and th triangl inquality, ů ů h h ů v h + v ů h h a h (v ů h, w) ů v h + max + ( α + 1) v ů h L2(Ω) w V h \{0} w h a h (ů ů h, w) (2 α + 3) ů v h + max + ( α + 1) ů ů h w V h \{0} w L2(Ω), h which implis (3.10). From hr on w considr α to b fixd and drop th dpndnc on α in our stimats. Rmark 3.7. Th first trm on th right-hand sid of (3.8) and (3.10) masurs th approximation proprty of V h with rspct to th norm h. Th scond trm masurs th consistncy rror. Th third trm on th right-hand sid of (3.10) addrsss th indfinitnss of th rducd curl-curl problm whn α 0. As mntiond in Rmark 2.2, th intrpolation oprator Π h is also th on usd in [15]. Thrfor in our analysis w can us th following two rsults from that papr (cf. Lmma 5.1 and Lmma 5.2 of [15]), which wr obtaind using (2.4), (2.8) and a rprsntation of ů as th sum of a rgular part and a singular part. Lmma 3.8. Lt ů H 0 (curl; Ω) H(div 0 ; Ω) b th solution of (1.4). W hav th following intrpolation rror stimat: (3.12) ů Π h ů L2(Ω) C ɛ h 2 ɛ f L2(Ω) for any ɛ > 0. Lmma 3.9. Lt ů H 0 (curl; Ω) H(div 0 ; Ω) b th solution of (1.4). W hav th following intrpolation rror stimat: (3.13) [Φ µ ()]2 [[ů Π h ů]] 2 L C 2() ɛh 2 ɛ f 2 L 2(Ω) E h

8 8 Susann C. Brnnr, Fngyan Li and Li-yng Sung for any ɛ > 0, whr [[ů Π h ů]] is th jump of ů Π h u across th intrior dgs of T h and [[ů Π h ů]] = ů Π h ů on th boundary dgs of T h. Lmma It holds that [Φ µ ()] 2 η η T 2 L 2() Ch2 η 2 H 1 (Ω) η H 1 (Ω), E h whr (3.14) η T = 1 T T η dx is th man of η ovr T, on of th triangls in T h that has as an dg. Proof. This is th consqunc of (2.1), (3.3), th trac thorm (with scaling) and a standard intrpolation rror stimat [22, 18]: [Φ µ ()] 2 η η T 2 L 2() E h C [Φ µ (T)] 2( η η T 2 L + 2(T ) h2 T η η T 2 ) H 1 (T ) E h C E h [Φ µ (T)] 2 h 2 T η 2 H 1 (T ) Ch2 η 2 H 1 (Ω). Rcall that Q is th L 2 orthogonal projction oprator onto H(div 0 ; Ω). Th following rsult will b usful in addrssing th consistncy rror causd by th apparanc of Q in (1.5). Lmma Th following stimat holds: (3.15) v Qv L2(Ω) Ch v h v [H 0 (curl; Ω) H(div 0 ; Ω)] + V h. Proof. Lt v [H 0 (curl; Ω) H(div 0 ; Ω)]+V h b arbitrary. Sinc v Qv blongs to H 1 0 (Ω), th orthogonal complmnt of H(div0 ; Ω) in [L 2 (Ω)] 2, w hav, by duality, (v Qv, η) (3.16) v Qv L2(Ω) = sup = sup η H0 1(Ω)\{0} η L2(Ω) η H0 1(Ω)\{0} (v, η) η L2(Ω) Lt η H 1 0(Ω) b arbitrary. Sinc v = 0 on ach triangl T T h, w find using intgration by parts and th fact that η Ω = 0, (3.17) (v, η) = E i h (η η T )[[n v]] ds + E i h η T ( Π 0 [[n v]] ) ds = S 1 + S 2, whr η T is dfind in (3.14). By th Cauchy-Schwarz inquality, (3.4) and Lmma 3.10, w hav. (3.18) [ ] 1/2 [ S 1 [Φ µ ()] 2 η η T 2 [Φ µ ()] 2 L 2() [[n v]] 2 L 2() Eh i Eh i Ch η L2(Ω) v h. ] 1/2

9 An Intrior Pnalty Mthod for 2D Curl-Curl Problms 9 On th othr hand, from (3.4), th Cauchy-Schwarz inquality, and a Poincaré- Fridrichs inquality, w find ( 1/2 η )( T L 2() 1/2 Π 0 [[n v]] L2()) (3.19) S 2 Eh i [ Ch E i h η T 2 L 2(T ) Ch η L2(Ω) v h Ch η L2(Ω) v h. ] 1/2 [ h 2 1 ] 1/2 Π0 [[n v]] 2 L 2() Eh i Hr w hav also usd th simpl fact that, if is an dg of a triangl T, thn (3.20) q 2 L 2() C T q 2 L 2(T) for any constant function q, whr th positiv constant C T dpnds only on th shap rgularity of T. Th stimat (3.15) follows from (3.16) (3.19). 4. Convrgnc Analysis. W bgin with thr lmmas that provid stimats to th thr trms on th right-hand sid of (3.8) and (3.10). Lmma 4.1. Lt ů H 0 (curl; Ω) H(div 0 ; Ω) b th solution of (1.4). It holds that (4.1) inf v V h ů v h ů Π h ů h C ɛ h 1 ɛ f L2(Ω) for any ɛ > 0. Proof. Sinc Π 0 [[n (ů Π h ů)]] = 0 for all E h and Π 0 [[n (ů Π h ů)]] = 0 for all Eh i, w hav (4.2) ů Π h ů 2 h = h (ů Π h ů) 2 L 2(Ω) + ů Π hů 2 L 2(Ω) + [Φ µ ()] 2 [[n (ů Π h ů)]] 2 L 2() E h + E i h [Φ µ ()] 2 [[n (ů Π h ů)]] 2 L. 2() Th scond trm on th right-hand sid of (4.2) has bn stimatd in Lmma 3.8, and th third and fourth trms can b stimatd using Lmma 3.9. Thrfor it only rmains to stimat th first trm. Obsrv that (2.6) implis (4.3) h (Π h ů) = Π 0 h ( ů), whr Π 0 h is th orthogonal projction from L 2(Ω) onto th spac of picwis constant functions with rspct to T h. It thn follows from (1.6), (2.3), (4.3) and a standard intrpolation rror stimat [22, 18] that (4.4) h (ů Π h ů) 2 L 2(Ω) = ů Π0 h ( ů) 2 L 2(Ω) Ch 2 ů 2 H 1 (Ω) Ch2 f 2 L 2(Ω). Th stimat (4.1) follows from (4.2), (4.4) and Lmmas

10 10 Susann C. Brnnr, Fngyan Li and Li-yng Sung Lmma 4.2. Lt ů H 0 (curl; Ω) H(div 0 ; Ω) b th solution of (1.4) and ů h V h satisfy (3.1). It holds that a h (ů ů h, w) (4.5) max Ch f L2(Ω). w V h \{0} w h Proof. Lt w V h b arbitrary. Using intgration by parts, th discrt problm (3.1), th strong form of th rducd curl-curl problm (1.5), and th fact that h w = 0, w find (4.6) a h (ů ů h, w) = a h (ů, w) (f, w) = (Qf f, w) + ( ů)(n T w T )ds, T T T h whr w T = w T and n T is th unit outr normal along T. W s from (4.6) that thr ar two sourcs for th inconsistncy of th schm dfind by (3.1), namly th projction Q that appars in (1.5) and th discontinuity of th vctor filds in V h. Th quation (4.6) can b rwrittn as a h (ů ů h, w) = (f, Qw w) + ( ů)[[n w]] ds E h (4.7) = (f, Qw w) + ( ů ( ů) T )[[n w]] ds E h + E h ( ů) T ( Π 0 [[n w]] ) ds whr ( ů) T is th man of ů on on of th triangls T T h that has as an dg. Th first two trms on th right-hand sid of (4.7) satisfy th stimat (4.8) (f, Qw w) + ( ů ( ů) T )[[n w]] ds Ch f L2(Ω) w h. E h Th drivation of (4.8), which is basd on Lmma 3.10, Lmma 3.11 and (1.6), can b found in th proof of Lmma 6.2 in [15]. Using th Cauchy-Schwarz inquality, (1.6), (3.4) and (3.20), th third trm on th right-hand sid of (4.7) can b stimatd as follows: ( ( ů) T Π 0 [[n w]] ) ds (4.9) E h ( 1/2 ( ů) T L2())( 1/2 Π 0 ) [[n w]] L2() E h ( ) 1/2 ( Ch ( ů) T 2 L 2(T ) h 2 1 ) 1/2 Π0 [[n w]] 2 L 2() E h E h Ch ů L2(Ω) w h Ch f L2(Ω) w h. Th stimat (4.5) follows from (4.7) (4.9).

11 An Intrior Pnalty Mthod for 2D Curl-Curl Problms 11 Lmma 4.3. Lt ů H 0 (curl; Ω) H(div 0 ; Ω) b th solution of (1.4) and ů h V h satisfy (3.1). It holds that (4.10) ů ů h L2(Ω) C ɛ ( h 2 ɛ f L2(Ω) + h 1 ɛ ů ů h h ) for any ɛ > 0. Proof. W us a duality argumnt. Lt z H 0 (curl; Ω) H(div 0 ; Ω) satisfy th rducd curl-curl problm (4.11) a h (v, z) = (v, ů ů h ) v H 0 (curl; Ω) H(div 0 ; Ω). Th strong form of (4.11) is (4.12) ( z) + α z = Q(ů ů h ), and w hav th following analog of (1.6): (4.13) z H1 (Ω) C Ω ů ů h L2(Ω). It follows from (4.12) and intgration by parts that (4.14) a h (ů h, z) = (ů h, Q(ů ů h )) + ( z)[[n ů h ]] ds. E h From (4.11) and (4.14), w hav (4.15) ů ů h 2 L = (ů, ů ů 2(Ω) h) (ů h, ů ů h ) = a h (ů ů h, z) + a h (ů h, z) (ů h, ů ů h ) = a h (ů ů h, z) (ů h, (I Q)(ů ů h )) + ( z)[[n ů h ]] ds. E h W will stimat th thr trms on th right-hand sid of (4.15) sparatly. Using (4.7) and th fact that [[n (Π h z)]] vanishs at th midpoints of all E h, w can rwrit th first trm as (4.16) a h (ů ů h, z) = a h (ů ů h, z Π h z) + a h (ů ů h, Π h z) = a h (ů ů h, z Π h z) + (f, QΠ h z Π h z) + ( ů ( ů) T )[[n (Π h z)]] ds, E h whr ( ů) T is th man of ů on on of th triangls T T h that has as an dg. Using th xprssion (4.16), th following stimat was obtaind in th proof of Lmma 6.5 of [15]: (4.17) a h (ů ů h, z) C ɛ ( h 2 ɛ f L2(Ω) + h 1 ɛ ů ů h h ) ů ů h L2(Ω). Sinc (I Q)ů = 0, th scond trm on th right-hand sid of (4.15) can b stimatd by Lmma 3.11: (4.18) ( ů h, (I Q)(ů ů h ) ) = ( ů ů h, (I Q)(ů ů h ) )

12 12 Susann C. Brnnr, Fngyan Li and Li-yng Sung C ů ů h L2(Ω)( h ů ůh h ). Finally w stimat th third trm on th right-hand sid of (4.15). W hav ( z)[[n ů h ]] ds = ( ) (4.19) z ( z)t [[n ůh ]] ds E h E h + E h ( z) T ( Π 0 [[n ů h ]] ) ds, whr ( z) T is th man of z on a triangl T T h that has as an dg. Th following stimat can b found in th proof of Lmma 6.5 of [15]: ( ) (4.20) z ( z)t [[n ůh ]] ds Ch ů ů h L2(Ω) ů ů h h. E h On th othr hand, w obtain by th Cauchy-Schwarz inquality, (3.4), (3.20) and (4.13), ( ( z) T Π 0 [[n ů h ]] ) ds (4.21) E h = E h ( z) T ( Π 0 [[n (ů h ů)]] ) ds Ch z L2(Ω) ů ů h h Ch ů ů h L2(Ω) ů ů h h. Th stimat (4.10) follows from (4.15) and (4.17) (4.21). In th cas whr α > 0, th following thorm is an immdiat consqunc of Lmma 3.5 and Lmmas Thorm 4.4. Lt α b positiv. Th following stimats hold for th solution ů h of (3.1): (4.22) (4.23) ů ů h h C ɛ h 1 ɛ f L2(Ω) for any ɛ > 0, ů ů 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 (3.1). Thorm 4.5. Assum that α 0 is not a Maxwll ignvalu. Thr xists a positiv numbr h such that th discrt problm (3.1) is uniquly solvabl for all h h, in which cas th following discrtization rror stimats ar valid: (4.24) (4.25) ů ů h h C ɛ h 1 ɛ f L2(Ω) for any ɛ > 0, ů ů h L2(Ω) C ɛ h 2 ɛ f L2(Ω) for any ɛ > 0. Proof. W follow th approach of Schatz [44] for indfinit problms. Assuming ů h satisfis (3.1), it follows from (3.10) and Lmmas that (4.26) ů ů h h C ɛ h 1 ɛ( ) f L2(Ω) + ů ů h h for any ɛ > 0.

13 An Intrior Pnalty Mthod for 2D Curl-Curl Problms 13 By choosing an ɛ > 0, w dduc from (4.26) that, for h h = 1/(2C ɛ ) 1/(1 ɛ ), ů ů h h C ɛ h 1 ɛ ( f L2(Ω) + ů ů h h ) and hnc C ɛ h 1 ɛ f L2(Ω) + C ɛ h 1 ɛ ů ů h h C ɛ h 1 ɛ f L2(Ω) ů ů h h, (4.27) ů ů h h 2C ɛ h 1 ɛ f L2(Ω). Thrfor, any solution z h V h of th homognous discrt problm (4.28) a h ( z h, v) = 0 v V h, which corrsponds to th spcial cas whr f = 0 = z, will satisfy th following spcial cas of (4.27): z h h 0. Hnc th only solution of (4.28) is th trivial solution and th discrt problm (3.1) is uniquly solvabl for h h. Th nrgy rror stimat (4.24) now follows (4.27), and th L 2 rror stimat (4.25) follows from Lmma 4.3 and (4.24). 5. Application to th Maxwll Eignproblm. Givn any f [L 2 (Ω)] 2, w dfin Tf H 0 (curl; Ω) H(div 0 ; Ω) by th condition that (5.1) ( (Tf), v) + (Tf, v) = (f, v) v H 0 (curl; Ω) H(div 0 ; Ω). Clarly T is a boundd linar oprator from [L 2 (Ω)] 2 into H 0 (curl; Ω) H(div 0 ; Ω). Sinc H 0 (curl; Ω) H(div 0 ; Ω) is a compact subspac of [L 2 (Ω)] 2 (cf. [23]), th oprator T : [L 2 (Ω)] 2 [L 2 (Ω)] 2 is symmtric, positiv and compact. Morovr, (λ, ů) satisfy th Maxwll ignproblm (1.7) if and only if (5.2) Tů = λů. Similarly, lt T h f V h b dfind by th condition (5.3) a h,1 (T h f, v) = (f, v) v V h, whr a h,1 (, ) is th bilinar form a h (, ) in (3.2) with α = 1, thn (5.4) T h ů h = is quivalnt to λ h ů h (5.5) a h,0 (ů h, v) = λ h (ů h, v) v V h, whr a h,0 (, ) is th bilinar form a h (, ) in (3.2) with α = 0. From Thorm 4.4 w hav (5.6) (T T h )f L2(Ω) C ɛ h 2 ɛ f L2(Ω)

14 14 Susann C. Brnnr, Fngyan Li and Li-yng Sung for all f [L 2 (Ω)] 2 and (5.7) (T T h )f h C ɛ h 1 ɛ f L2(Ω) C ɛ h 1 ɛ f h for all f H 0 (curl; Ω) H(div 0 ; Ω)+V h. Thus th symmtric finit rank oprator T h convrgs uniformly to th symmtric positiv compact oprator T as h 0, which implis that th classical thory of spctral approximation [39, 7] can b applid to th approximation of th ignvalus and ignfunctions of T by th ignvalus and ignfunctions of T h. Hnc w hav th following rsult through th connctions btwn (1.7) and (5.2) and btwn (5.5) and (5.4). Thorm 5.1. Lt 0 λ 1 λ 2... b th ignvalus of (1.7), λ = λ j = λ j+1 = = λ j+m 1 b an ignvalu with multiplicity m, and V λ H 0 (curl; Ω) H(div 0 ; Ω) b th corrsponding m dimnsional ignspac. Lt λ h,1 λ h,2... b th ignvalus of (5.5). Thn, as h 0, w hav (5.8) λ h,l λ C λ,ɛ h 2 ɛ l = j,..., j + m 1. Furthrmor, if V h,λ V h is th spac spannd by th ignfunctions corrsponding to λ h,j,...,λ h,j+m 1, thn th gap btwn V λ and V h,λ gos to zro at th rat of C ɛ h 2 ɛ in th L 2 norm and at th rat of C ɛ h 1 ɛ in th norm h. Proof. Lt E λ (rsp. E h,λ ) b th L 2 orthogonal projction onto V λ (rsp. V h,λ ). It follows from (5.6) (5.7) and th classical thory of spctral approximation of compact oprators [39, 7] that (5.9) (E λ E h,λ )w L2(Ω) C ɛ h 2 ɛ w L2(Ω) for all w [L 2 (Ω)] 2 and (5.10) (E λ E h,λ )w h C ɛ h 1 ɛ w h for all w H 0 (curl; Ω) H(div 0 ; Ω) + V h. Lt x V h,λ b a unit ignfunction of λ h,l for som l btwn j and j + m 1. Thn w hav (5.11) (T h x, x) = µ h,l and x L2(Ω) = 1, whr (cf. (5.4)) (5.12) µ h,l = (1 + λ h,l ) 1. Lt ˆx = E λ x and ŷ = ˆx x. Thn ˆx and ŷ ar orthogonal with rspct to both th L 2 innr product and th innr product (T, ), and th stimat (5.9) implis that (5.13) ŷ L2(Ω) = (E λ E h,λ )x L2(Ω) C ɛ h 2 ɛ. In particular, it follows from Pythagoras thorm with rspct to th L 2 innr product that (5.14) 1 ˆx 2 L 2(Ω) = x 2 L 2(Ω) ˆx 2 L 2(Ω) = ŷ 2 L 2(Ω) C ɛh 4 ɛ. Lt ê = ˆx/ ˆx L2(Ω). Thn ê is a unit ignfunction of T corrsponding to th ignvalu (cf. (5.2)) (5.15) µ = (1 + λ) 1,

15 An Intrior Pnalty Mthod for 2D Curl-Curl Problms 15 and w hav (5.16) µ µ h,l = (Tê, ê) (Th x, x) ((T Th )x, x) + (Tê, ê) (Tx, x). (5.17) From (5.6) w hav ((T T h )x, x) C ɛ h 2 ɛ, and it follows from Pythagoras thorm with rspct to th innr product (T, ) that (5.18) (Tê, ê) (Tx, x) = (Tê, ê) (T ˆx, ˆx) (Tŷ, ŷ) (1 ˆx 2 L 2(Ω) )(Tê, ê) + (Tŷ, ŷ). Combining (5.13), (5.14) and (5.18), w find (Tê, ê) (Tx, x) Cɛ h 4 ɛ, which togthr with (5.16) (5.17) implis (5.19) µ µ h,l C ɛ h 2 ɛ. Th stimat (5.8) follows from (5.12), (5.15) and (5.19). Rcall that th gap ˆδ(M, N) btwn two subspacs M and N of a normd linar spac (X, X ) is dfind by (cf. [39]) whr ˆδ(M, N) = max(δ(m, N), δ(n, M)), δ(m, N) = sup x M x X=1 inf x y X. y N Thrfor th statmnts about th gap btwn V λ and V h,λ follow immdiatly from (5.9) and (5.10). Rmark 5.2. Th compactnss of th solution oprator T and th xistnc of th uniform stimats (5.6) and (5.7) gratly simplify th analysis of th Maxwll ignproblm. Ths ingrdints ar absnt from Maxwll spctral approximations basd on th full curl-curl problm and hnc thir justifications ar much mor involvd [10, 11, 8, 20, 21, 9, 19, 12]. 6. Numrical Rsults. In this sction, w rport a sris of numrical xampls that corroborat our thortical rsults. Bsids th L 2 rror ů ů h L2(Ω) and th nrgy rror ů ů h h, w also rport th rrors masurd in th smi-norm curl dfind by v curl = h v L2(Ω) Sourc Problms. W first dmonstrat th prformanc of our schm for th sourc problms whr w tak α to b k 2 for all th computations in this subsction. In th first xprimnt, w chck th convrgnc bhavior of our numrical schm (3.1) on th squar (0, 0.5) 2 with uniform mshs, whr th xact solution is (6.1) ů = [y(y 0.5)sin(ky), x(x 0.5)cos(kx)]

16 16 Susann C. Brnnr, Fngyan Li and Li-yng Sung Tabl 6.1 Convrgnc of th schm (3.1) on th squar (0, 0.5) 2 with uniform mshs and xact solution ů givn by (6.1) h ů ů h L2(Ω) ů L2(Ω) ordr ů ů h h ů h ordr ů ů h curl ů curl ordr k = 0 1/ E E E 01 1/ E E E / E E E / E E E / E E E k = 1 1/ E E E 01 1/ E E E / E E E / E E E / E E E k = 10 1/ E E E 01 1/ E E E / E E E / E E E / E E E for k = 0, 1 and 10. Th rsults tabulatd in Tabl 6.1 confirm th stimats (4.24) and (4.25). Obsrv also that, as xpctd, finr mshs ar ndd for computing satisfactory approximat solutions whn th wav numbrs bcom largr. Sinc our numrical schm is dsignd for solving th rducd curl-curl problm (1.5), its prformanc should not b affctd by th addition to th right-hand sid of a gradint trm G whr G H 1 0(Ω). This is dmonstratd by our scond xprimnt. Hr w add G with (6.2) G(x, y) = xy(x 0.5)(y 0.5)sin(x + y) to th right-hand sid (6.3) f = ( ů) ů of th problm in th first xprimnt (with k = 1) and compar thir prformanc. Th rsults ar rportd in Tabl 6.2. Not that th schm (3.1) diffrs from th on proposd in [15] bcaus of th last two stabilizing trms. In th third xprimnt, w show that ths two trms ar ncssary for th convrgnc of our mthod du to th lack of th continuity of th approximating functions across th lmnt intrfacs. S Tabl 6.3 for th numrical vidnc. Th last xampl in this subsction dmonstrats th convrgnc bhavior of our schm on th L-shapd domain ( 0.5, 0.5) 2 \[0, 0.5] 2 with cornrs (0.5, 0), (0, 0), (0, 0.5), ( 0.5, 0.5), ( 0.5, 0.5) and (0.5, 0.5). W tak k = 1 and th xact solution

17 An Intrior Pnalty Mthod for 2D Curl-Curl Problms 17 Tabl 6.2 Comparison of th prformanc of th schm (3.1) on (0, 0.5) 2 with k = 1 for th right-hand sids f and f + G, whr f is givn by (6.3) and G is givn by (6.2) h rhs = f rhs = f + G ů ů h L2(Ω) 1/ / / / / ů ů h h 1/ / / / / Tabl 6.3 Loss of convrgnc without th last two stabilizing trms in th schm (3.1). Th computation is on th squar (0, 0.5) 2 with uniform mshs and th xact solution ů is givn by (6.1) with k = 1 h ů ů h L2(Ω) ů L2(Ω) ordr ů ů h curl ů curl ordr 1/ E E 01 1/ E E / E E / E E is chosn to b (6.4) ů = ( ( 2 r 2/3 cos 3 θ π ) ) φ(r/0.5), 3 whr (r, θ) ar th polar coordinats at th origin and th cut-off function is givn by 1 r (r 0.75) φ(r) = 3 [ (r 0.75) + 12(r 0.75) 2] 0.25 r r 0.75 Th mshs ar gradd around th r-ntrant cornr with th grading paramtr qual to 1/3. Th rsults ar tabulatd in Tabl 6.4 and thy agr with th stimats (4.24) and (4.25) Maxwll Eignproblm. In this subsction, w rport th numrical rsults for th Maxwll ignproblm. In th first xprimnt, th computation is carrid out on th squar domain (0, π) 2 with uniform mshs. In this cas, th xact ignvalus ar r 2 + s 2, r, s = 0, 1, 2, 3, 4, with r 2 +s 2 0. For instanc, th first tn ignvalus ar 1, 1, 2, 4, 4,

18 18 Susann C. Brnnr, Fngyan Li and Li-yng Sung Tabl 6.4 Convrgnc of th schm (3.1) with gradd mshs on th L-shapd domain ( 0.5, 0.5) 2 \ [0,0.5] 2 with k = 1 and xact solution ů givn by (6.4) h ů ů h L2(Ω) ů L2(Ω) ordr ů ů h h ů h ordr ů ů h curl ů curl ordr 1/4 1.77E E E 00-1/8 4.22E E E / E E E / E E E / E E E , 5, 8, 9, 9. In Figur 6.1, w plot th first twnty numrical ignvalus vrsus th paramtr n = π/h. Th symbol o on th right sid dnots th xact ignvalu, and (2) indicats that th multiplicity of th ignvalu is 2. All th numrical approximations convrg at a scond ordr rat. To sav spac, only th first fiv numrical ignvalus ar includd in Tabl 6.5. Furthrmor, thr is no spurious ignvalu in our rsults. ignvalu (2) (2) (2) (2) (2) ig1 ig2 ig3 ig4 ig5 ig6 ig7 ig8 ig9 ig10 ig11 ig12 ig13 ig14 ig15 ig16 ig17 ig18 ig19 ig20 xact 6 4 (2) (2) n=pi/h (2) Fig First twnty numrical ignvalus vrsus n = π/h for Maxwll oprator on (0, π) 2 In th scond xprimnt, w comput th ignvalus for th L-shapd domain ( 0.5, 0.5) 2 \[0, 0.5] 2 and th mshs ar gradd around th r-ntrant cornr with th grading paramtr qual to 1/3. Tabl 6.6 contains th first fiv numrical ignvalus,

19 An Intrior Pnalty Mthod for 2D Curl-Curl Problms 19 Tabl 6.5 Convrgnc of th first fiv numrical ignvalus on (0, π) 2, with γ = ordr of convrgnc and λ = xact ignvalu h 1 st γ 2 nd γ 3 rd γ 4 th γ 5 th γ π/ π/ π/ π/ π/ π/ λ which show scond ordr convrgnc of our mthod. This agrs with our analysis. In this cas th xact ignvalus ar drivd from numrical rsults of Daug, which can b found at In Figur 6.2, w plot th first tn numrical ignvalus vrsus th paramtr n = 1/2h. Again, thr is no spurious ignvalu ig1 ig2 ig3 ig4 ig5 ig6 ig7 ig8 ig9 ig10 xact ignvalu (2) n=1/(2h) Fig First tn numrical ignvalus vrsus n = 1/2h for th Maxwll oprator on ( 0.5, 0.5) 2 \ [0,0.5] Concluding Rmarks. W hav analyzd an intrior pnalty mthod with wak ovr-pnalization for th two-dimnsional rducd curl-curl problm and th

20 20 Susann C. Brnnr, Fngyan Li and Li-yng Sung Tabl 6.6 Convrgnc of th first fiv numrical ignvalus on th L-shapd domain ( 0.5, 0.5) 2 \[0, 0.5] 2 with gradd mshs, µ = 1/3 at th r-ntrant cornr, γ = ordr of convrgnc and λ = xact ignvalu h 1 st γ 2 nd γ 3 rd γ 4 th γ 5 th γ 1/ / / / / λ Maxwll ignproblm. This mthod is inconsistnt but stabl without th nd of tuning any pnalty paramtr. W xpct othr discontinuous Galrkin schms will b discovrd along this lin. An advantag of this nw approach is that fast solvrs for th rsulting schms can b constructd using tchniqus dvlopd for lliptic problms. Th schm dvlopd in this papr can b xtndd to thr dimnsional domains. In th cas of uniform mshs, our two dimnsional analysis can b asily gnralizd to stablish convrgnc at a sub-optimal rat. It is likly that quasioptimal convrgnc rats can b rcovrd using gradd mshs, which is of cours much mor complicatd in thr dimnsions [2, 3]. This will b on of th topics of our ongoing rsarch. Acknowldgmnt. Th first author would 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 and third authors wr visiting th Humboldt Univrsität zu Brlin and thy would lik to thank th mmbrs of th Institut für Mathmatik for thir hospitality. REFERENCES [1] Th. Apl. Anisotropic Finit Elmnts. Tubnr, Stuttgart, [2] Th. Apl and S. Nicais. Th finit lmnt mthod with anisotropic msh grading for lliptic problms in domains with cornrs and dgs. Math. Mthods Appl. Sci., 21: , [3] Th. Apl, S. Nicais, and J. Schöbrl. Crouzix-Raviart typ finit lmnts on anisotropic mshs. Numr. Math., 89: , [4] 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, [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, [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: , [7] I. Babuška and J. Osborn. Eignvalu Problms. In P.G. Ciarlt and J.L. Lions, ditors, Handbook of Numrical Analysis II, pags North-Holland, Amstrdam, [8] D. Boffi. Fortin oprators and discrt compactnss for dg lmnts. Numr. Math., 87: , [9] D. Boffi. Approximation of ignvalus in mixd form, discrt compactnss proprty, and application to hp mixd finit lmnts. prprint, 2005.

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