Convergence analysis of a discontinuous Galerkin method with plane waves and Lagrange multipliers for the solution of Helmholtz problems

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1 Convrgnc analysis of a discontinuous Galrkin mtod wit plan wavs and Lagrang multiplirs for t solution of Hlmoltz problms Moamd Amara Laboratoir d Matématiqus Appliqués Univrsité d Pau t ds Pays d l Adour & CNRS-UMR542 BP 55, 6403 Pau cdx, FRANCE Rabia Djllouli Dpartmnt of Matmatics, California Stat Univrsity Nortridg 8 Nordoff Strt, Nortridg, CA , USA Carbl Farat Dpartmnt of Mcanical Enginring & Institut for Computational and Matmatical Enginring Building 500, Mailcod 3035, Stanford Univrsity, Stanford, CA 94305, USA Sptmbr 22, 2008 Abstract W analyz t convrgnc of a discontinuous Galrkin mtod (DGM) wit plan wavs and Lagrang multiplirs tat was rcntly proposd by Farat t al. [3] for solving twodimnsional Hlmoltz problms at rlativly ig wav numbrs. W prov tat t undrlying ybrid variational formulation is wll-posd. W also prsnt various a priori rror stimats tat stablis t convrgnc and ordr of accuracy of t simplst lmnt associatd wit tis mtod. W prov tat, for k (k ) 2 3 sufficintly small, t rlativ rror in t L 2 -norm (rsp. in t H smi-norm) is of ordr k (k ) 4 3 (rsp. of ordr (k ) 2 3 ) for a solution bing in H 5 3 (Ω). In addition, w stablis an a postriori rror stimat tat can b usd as a practical rror indicator wn rfining t partition of t computational domain. y words: acoustic scattring, discontinuous Galrkin, Hlmoltz problms, ybrid finit lmnt, inf sup condition, plan wavs. Corrsponding autor: Rabia.Djllouli@csun.du

2 Introduction T discontinuous nricmnt mtod (DEM) was dvlopd in [, 2] for t solution of multiscal boundary valu problms wit sarp gradints and rapid oscillations. Ts ar problms for wic t standard finit lmnt mtod (FEM) can bcom proibitivly xpnsiv. DEM can b dscribd as a discontinuous Galrkin mtod (DGM) wit Lagrang multiplir dgrs of frdom (dofs), in wic t standard finit lmnt polynomial fild is nricd witin ac lmnt by fr-spac solutions of t omognous partial diffrntial quation to b solvd. Usually, ts ar asily obtaind in analytical form and ar discontinuous across t lmnt intrfacs. T Lagrang multiplir dofs ar introducd at ts intrfacs to nforc a wak continuity of t solution. For t Hlmoltz quation, t nricmnt fild can b constructd wit plan wavs as ts ar fr-spac solutions of tis quation. In [3], it was sown tat for a larg class of Hlmoltz problms, t polynomial fild is not ncssary for capturing fficintly t solution. Hnc, for ts applications, t polynomial fild was droppd and DEM was transformd into a DGM wit plan wav basis functions. Similar xponntial functions wr prviously introducd in t wak lmnt mtod (WEM) [4], t partition of unity mtod (PUM) [5], t ultra wak variational mtod (UWM) [7], and t last-squars mtod (LSM) prsntd in [8], for t solution of t Hlmoltz quation. Howvr, unlik WEM, t DGM proposd in [3] is basd on a variational framwork, and unlik PUM, it is discontinuous. Furtrmor, in contrast to LSM, t continuity of t solution at t intr-lmnt boundaris is nforcd in DEM by Lagrang multiplirs ratr tan pnalty paramtrs, wic incrass t robustnss and accuracy of t undrlying framwork of approximation. In [3], two lowr-ordr rctangular DGM lmnts wit four and igt plan wavs, rspctivly, wr constructd and applid to t solution of two-dimnsional wavguid problms wit 0 kl 00, wr k dnots t wavnumbr and l is a caractristic lngt of t wavguid. T discrtization by ts lmnts of suc Hlmoltz problms was found to rquir fiv to svn tims fwr dofs tan tir discrtization by t standard Q2 lmnt, dpnding on t dsird lvl of accuracy. In [9], tis DGM was xtndd to xtrior Hlmoltz problms and was coupld wit a scond-ordr absorbing boundary condition. A lowr-ordr quadrilatral lmnt wit igt Lagrang multiplir dofs was dsignd and igligtd wit t solution on unstructurd mss of sampl acoustic scattring problms wit 20 kl 40, wr l dnots a caractristic lngt of t scattrr. Tis lmnt was sown to dlivr significant improvmnt ovr t prformanc of t standard and comparabl Q2 lmnt. In [0], two igr-ordr quadrilatral DGM lmnts wit 6 and 32 plan wavs, rspctivly, wr prsntd. T DGM lmnt wit 6 plan wavs as a computational complxity tat is comparabl to tat of t standard Q4 lmnt and was sown numrically to av t sam convrgnc rat wit rspct to t ms siz. Howvr, tis DGM lmnt was also sown numrically in [9] to dlivr t sam lvl of accuracy as Q4 using 6 tims fwr dofs. All of ts prformanc rsults igligt t potntial of t DGM introducd in [3] and xpandd in [9] and [0]. Howvr, no matmatical analysis of tis mtod as bn prformd yt. T objctiv of tis papr is to fill tis gap in t spcific contxt of t two-dimnsional low-ordr lmnt wit four plan wavs in ordr to st tis DGM mtod on a firm tortical basis. T proposd study assums tat t computational domain Ω is a polygonal-sapd domain tat can b partitiond into rctangular lmnts. Not tat t computational domain Ω may av rntrant cornrs, and trfor t considrd acoustic scattrd fild is in H 5 3 (Ω) only. W partition t computational domain into rctangular-sapd lmnts and considr t cas of t so-calld R-4- lmnt, tat is w approximat locally t primal variabl by four plan wavs and t dual variabl by constants 2

3 on t dgs of intrior lmnts. W must point out tat tis study cannot b xtndd at tis tim to igr ordr lmnts bcaus it assums tat t normal drivativ of t primal variabl is constant along t intrior dgs. Tis crucial proprty is valid only in t cas of R-4- lmnt. W prov tat for k (k ) 2 3 small noug, t rlativ rror in t L 2 -norm (rsp. in t H sminorm) is of ordr k (k ) (rsp. (k ) 3 ). W rcall tat in t cas of t standard finit lmnt mtod using P lmnt (s [, 2]), it as bn stablisd tat for k 2 small noug, t rlativ rror in t L 2 -norm (rsp. in t H smi-norm) is of ordr k 3 2 (rsp. k ). Morovr, if w assum tat k is small noug, it as bn stablisd in Rfrnc [2], tat t rlativ rror for bot t L 2 -norm and t H smi-norm ar boundd by k (k ) 2. Howvr, all ts rror stimats av bn stablisd assuming tat t scattrd fild is in H 2 (Ω) wic is not a ralistic assumption for most applications. W must also point out tat, to t bst of our knowldg, no rror stimats av bn drivd yt in t particular cas of Q4 finit lmnt wn applid to Hlmoltz problms. W also driv a postriori rror stimat tat can b usd as a practical rror indicator wn rfining t partition of t computational domain. Tis rror stimat rvals tat t rlativ rror in t L 2 norm dpnds on t rrors in t approximation of t intrior and xtrior boundary conditions as wll as on t jump across t lmnts of t partition. T rmaindr of tis papr is organizd as follows. In Sction, w spcify t notations and assumptions usd in tis papr, stat t formulation of a two-dimnsional acoustic scattring problm in a boundd domain, and prov tat t ybrid problm obtaind by applying t DGM introducd abov to t solution of t focus Hlmoltz problm is wll posd in t sns of Hadamard [3]. Mor spcifically, w introduc Torm to addrss t issus of xistnc, uniqunss, and stability of t DGM formulation. Nxt, w dvot Sction 3 to t analysis of t discrt solution obtaind wit a DGM lmnt wit four plan wavs. Mor spcifically, w rcall in Sction 3.2 t discrt DGM formulation and announc t main rsults of tis papr. Ts ar xistnc and uniqunss rsults, a priori rror stimats tat ar statd in Torm 2, and a postriori stimat tat is statd in Torm 3. T proofs of ts tr sts of fundamntal rsults ar dtaild in Sction 3.3 and Sction 3.4. Finally, Sction 4 concluds tis papr. Prliminaris W considr trougout tis papr t acoustic scattring problm by a sound-ard scattrr [4] formulatd in a boundd domain as follows Find u H (Ω) suc tat (BVP) u + k 2 u = 0 in Ω, n u = n ikx d on Γ, () n u = iku on Σ, wr u is t scattrd fild and Ω is t computational domain. Ω is a boundd polygonal-sapd domain tat can b partitiond into rctangular lmnts. Γ is its intrior boundary and Σ is t xtrior boundary. n is t unitary outward normal vctor to t boundaris Γ and Σ and n is t normal drivativ. k is a positiv numbr rprsnting t wavnumbr. d is a unit vctor 3

4 rprsnting t dirction of t incidnt plan wav. T quation on Γ is t Numann boundary condition tat caractrizs t sound-ard proprty of t scattrr. W must point out tat t intrior Numann boundary condition on Γ and t xtrior condition on Σ ar usd only for simplicity. T rsults prsntd rin apply to all typs of admissibl boundary conditions. In addition, as it is wll-known, on sould us igr ordr local absorbing boundary conditions for solving practical problms. 2 T continuous ybrid variational formulation 2. Nomnclatur and proprtis W us trougout tis papr t following notations and proprtis. 4 is a rctangular-sapd lmnt of Ω and is its boundary. = T j wr T j is t jt dg of wit vrtics (s j, s j+ ), and n j j= its outward unitary normal vctor. j is t lngt of t dg T j and = max j 4 j. (T ) is a rgular triangulation of t computational domain Ω into lmnts i.. ĉ > 0 /, T ; 2 ĉ wr dnots t ara of t lmnt [5]. Not tat (T ) is a quasi-uniform triangulation sinc its lmnts ar rctangls. = max. W also assum tat k π. Tis condition mans tat tr is at last two lmnts pr wavlngt. X is t spac of t primal variabl. X is givn by: X = { v L 2 (Ω); T, v = v H () } H () and is quippd wit t following norm: v X = v 2 X() T 2, v X, wr v X() = ( v 2, + ) v 2 2 0, 0, (rsp., ) is t L 2 -norm (rsp. smi-nom) on t lmnt. 4

5 ,T is t smi-norm in t spac X dfind by: v,t = v 2, T 2, v X. H 2 ( ) is t spac of t tracs of lmnts of H () and H 2 ( ) is t dual spac of H 2 ( ). H 2 ( ) is quippd wit t following norm: λ, = inf w X() = Λ X() (2) 2 w W (λ) wr W (λ) = { w H () ; w = λ } and Λ is t uniqu lmnt in W (λ) satisfying Λ + Λ = 0 a.. in. It follows from t dfinition of t norm X and Eq. (2) tat M is t spac of t dual variabl dfind by: M = µ v 2, v X(), v H (). (3) H 2 ( ) ; λ T, < µ, λ > 2 2, = 0 wr µ = µ and t spac T is givn by: T = λ 2 ( ); T, λ = λ on H T spac M is quippd wit t following norm: µ M = µ 2 2, T 2, µ M wr µ, = sup 2 λ H 2 ( ) < µ, λ > 2 2, < µ, v > 2 = sup 2, (4) λ 2, v H () v X() and <.,. > 2 2, is t duality product btwn H 2 ( ) and H 2 ( ) [6]. M is a subspac of M dfind by: M = µ L 2 ( ); µ = 0 on Ω and T, } µ + µ = 0 on. Trfor, w av M = M L 2 ( ). 5

6 2.2 Formulation and matmatical rsults W adopt t following ybrid-typ variational formulation (VP) for solving t boundary valu problm (BVP). Not tat t VP is quivalnt to BVP as indicatd in Rmark. (VP) Find (u, λ) X M suc tat a(u, v) + b(v, λ) = F (v) v X, b(u, µ) = 0 µ M, wr t bilinar forms a(, ) and b(, ) and t function F ar givn by: a(u, v) = ( ) u v dx k 2 uv dx ik uv dt u, v X, T Σ b(v, µ) = < µ, v > 2 2, (v, µ) X M, F (v) = Γ v n ikx d dt v X. Not tat t bilinar form b(, ) also satisfis b(v, µ) = µ v dt (v, µ) X M. (5) In addition, t bilinar forms a(, ) and b(, ) satisfy t following important proprtis. Proprty T bilinar forms a(.,.) and b(, ) ar continuous on X X and X M rspctivly. Furtrmor, w av: i. a(.,.) satisfis t Gärding inquality in H (Ω) wr R dsignats t ral part. Ra(v, v) + k 2 v 2 0,Ω = v 2,T ; v X (6) ii. T null spac N corrsponding to t bilinar form b(.,.) is givn N = {v X ; b(v, µ) = 0 µ M} = H (Ω) (7) iii. T bilinar form b(.,.) satisfis t so-calld inf-sup condition [22]: b(v, µ) b(φ, µ) µ M, φ X : sup = = µ M (8) v X v X φ X 6

7 Proof of Proprty. W prov only t tird point sinc t proof of Eq. (6) and Eq. (7) is straigtforward. From t continuity of t bilinar form b(.,.) w dduc tat b(v, µ) sup µ M µ M. (9) v X v X Nxt, for a fixd µ M, w considr t function φ X suc tat, for vry T, φ = φ is t uniqu solution of t following variational problm: φ v dx + φ v dx = < µ, v > 2 2, v H (). (0) Hnc, using Eq. (3) and Eq. (0), w av φ 2 X() = < µ, φ > 2 2, µ 2, φ 2, µ 2, φ X(). Tus, w dduc tat φ X() µ 2,, and tn φ X µ M. Morovr, from Eq. (4) and Eq. (0), w av µ 2, φ X(). Trfor, it follows tat φ X = µ M. On t otr and, from Eq. (0) and t dfinition of t bilinar form b(, ), w also av b(φ, µ) = φ 2 X() = φ 2 X = φ X µ M wic concluds t proof of t inf-sup condition givn by Eq. (8). Rmark T problms BVP and VP ar quivalnt in t following sns: i. If t pair (u, λ) is a solution of VP, tn it follows from t scond quation of VP tat u is in H (Ω). Morovr, using t first quation of VP wit tst functions v D(Ω), w dduc tat u is t solution of t first quation of BVP. Last, t us of tst functions v H (Ω) allows to vrify tat u is satisfis t boundary conditions on Γ and Σ. ii. If u is t solution of BVP, tn from t standard rgularity rsults for Laplac s oprator [23] and du to t possibl rntrant cornrs (wit a masur angl of 3π 2 ), it follows tat u H 5 3 (Ω). Tus, n u L 2 ( ) for all T ( n u is vn in H 6 ( )). Tn w st n u on \ Ω, λ = () 0 on Ω. Trfor, t dual variabl λ satisfis () in t L 2 ( ) sns, wic is t classical sns. Having tat in mind, on can multiply BPV by tst functions v X and dduc tat t pair (u, λ) satisfis VP. 7

8 Nxt, w prov tat t variational problm (VP) is wll-posd in t sns of Hadamard [3]. Tis is main rsult of tis sction. It is statd in t following torm. Torm T variational problm (VP) admits a uniqu solution (u, λ) X M. In addition, u blongs to H 5 3 (Ω), and for all θ [0, 5 3 ] tr is a positiv constant C (C dpnds on Ω and θ only) suc tat u θ,ω C ( + k) θ. T proof of tis torm is basd on t following intrmdiat stability rsult: Lmma Lt f b in L 2 (Ω). Tn, t following boundary valu problm U + k 2 U = f in Ω, n U = 0 on Γ, n U = ik U on Σ (2) as on and only on solution U in H 5 3 (Ω). Morovr, for all θ [0, 5 3 ] tr is a positiv constant C (C dpnds on Ω and θ only) suc tat U θ,ω C ( + k) θ f 0,Ω. (3) Proof of Lmma. First, obsrv tat t variational formulation corrsponding to t boundary valu problm (2) is givn by Find U H (Ω) suc tat (4) a(u, v) = f v dx v H (Ω). Ω From Eq. (6), it follows tat t bilinar form a(.,.) satisfis t Frdolm altrnativ on H (Ω). Hnc, t uniqunss nsurs t xistnc of t solution U in H (Ω). Trfor, w nd only to prov t uniqunss of t solution of t boundary valu problm (2). Lt w b t solution of t corrsponding omognous boundary valu problm. T function w satisfis a(w, w) = 0 tn w = 0 on Σ and w dduc tat n w = 0 on Γ and w = n w = 0 on Σ. Trfor, using t continuation torm [7, 8], w obtain tat w = 0 in Ω. From t standard rgularity rsults for scond-ordr lliptic boundary valu problms [23] and du to t possibl rntrant cornrs ( wit a masur angl of 3π 2 ), it follows tat t solution of 8

9 problm (2) satisfis U H 5 3 (Ω), and tr is a positiv constant C (C dpnds on Ω only) suc tat: ( ) U 5 3,Ω C U 3,Ω + nu 6, Ω. (5) Morovr, using t rsults stablisd in Rfrncs [9] and [20], w dduc t xistnc of a positiv constant C (C dpnds on Ω only) suc tat: U 0,Ω C + k f 0,Ω and U,Ω C f 0,Ω. (6) Nxt, w stablis t stimat (3). To do tis, w will us t spac intrpolation rsults in Rfrnc [2]. First, using boundary conditions in t boundary valu problm (2), w dduc tat tr is a positiv constant C (C dpnds on Ω only) suc tat: n U 6, Ω = nu 6,Σ = k U 6,Σ C k U 2 3,Ω. Trfor, it follows from t spac intrpolation rsults in [2] tat tr is a positiv constant C (C dpnds on Ω only) suc tat: n U 6, Ω C k U 3 0,Ω U 2 3,Ω. Finally, it follows from Eq. (6) tat tr xists a positiv constant C (C dpnds on Ω only) suc tat: n U 6, Ω C ( + k) 2 3 f 0,Ω (7) Furtrmor, from t first quation of t boundary valu problm (2), w dduc tat U 0,Ω k 2 U 0,Ω + f 0,Ω. Hnc, it follows from Eq. (6) tat tr is a positiv C (C dpnds on Ω only) suc tat U 0,Ω C ( + k) f 0,Ω In addition, from t norms proprtis and Eq. (6), tr is a positiv C (C dpnds on Ω only) suc tat U,Ω U,Ω U,Ω C f 0,Ω. Consquntly, it follows from ts quations and t intrpolation spac rsults torm (s [2]) tat tr is a positiv constant C (C dpnds on t domain Ω only) suc tat U 3,Ω C ( + k) 2 3 f 0,Ω (8) Estimat (3) is tn a dirct consqunc of Eq. (5), Eq. (7), and Eq. (8). Proof of Torm. Sinc H (Ω) is t null spac of t bilinar form b(.,.) (s Eq. (7)), t VP is rducd to t variational problm a(u, v) = F (v) v H (Ω) From Eq. (6), it follows tat t bilinar form a(.,.) satisfis t Frdolm altrnativ on H (Ω). Hnc, t uniqunss nsurs t xistnc of t solution u in H (Ω). On t otr and, t uniqunss rsults radily from t solution of t boundary valu problm (2). Trfor, t 9

10 solution u of t rducd variational problm in t null spac H (Ω) of t bilinar form b(.,.) xists and is uniqu. Trfor, bot xistnc and uniqunss of t solution of t complt variational problm VP ar standard consquncs (For xampl, s Rfrnc [22]) of t inf-sup condition givn by Eq. (8). To prov t stability stimats, w first obsrv tat t pair (u, λ) solution of t variational formulation (VP) satisfis t following mixd boundary valu problm: and T, w av Consquntly, if w st wr φ D(Ω) satisfis u + k 2 u = 0 in Ω, n u = n ikx d on Γ, n u = iku on Σ. n u on \ Ω, λ = 0 on Ω. U = u + ikx d φ (9) φ = on Γ, n φ = 0 on Γ, φ = n φ = 0 on Σ, tn, it is asy to vrify tat U is t uniqu solution of boundary valu problm (2) wit t rigt and-sid f givn by f = (2ik d φ + φ) ikx d and tr is a positiv constant C (C dpnds on Ω only) suc tat f 0,Ω C ( + k). Trfor, t proof of Torm s stimat is an immdiat consqunc of stimat (3) in Lmma wic concluds t proof of Torm. 3 T discrt formulation 3. Assumptions, notations, and proprtis W adopt trougout tis sction t following notations and proprtis. T, φ j = ik n j (x s j ) ; j 4. X is t discrt spac for t primal variabl. X is givn by: X = { v X; T, v X () } 0

11 wr X () = v H () ; v = 4 αj φ j wr αj C Not tat X X, and trfor X is also quippd wit t norm. X. M is t discrt spac of t dual variabl. M is dfind as follows: { } M = µ M ; T and Tj : µ j = µ T C, j 4 j ( ) For vry T, T matrix B = Blj rprsnts t lmntary matrix corrsponding to t bilinar form b(, ). Hnc, t ntris of t matrix B ar givn l,j 4 by: B lj = l T l j= φ j dt, l, j 4. (20) Ĉ dsignats a gnric positiv constant. Ĉ is indpndnt of k, Ω, and t triangulation T. For a givn T and v H (), w av t following two classical inqualitis [5]: v 0, Ĉ( v 2 0, + v 2,) 2, (2) v v dx 0, Ĉ v,. (22) In addition, it follows from combining Eq. (2) (wn applid to v v dx) and Eq. (22) tat: v v dx 0, Ĉ 2 v,. (23) 3.2 Discrt formulation and announcmnt of t main rsults T discrt variational problm (DVP) corrsponding to t variational formulation (VP) can b formulatd as follows: Find (u, λ ) X M suc tat (DVP) a(u, v ) + b(v, λ ) = F (v ) v X, (24) b(u, µ ) = 0 µ M. T nxt two torms summariz t main rsults of tis sction.

12 Torm 2 T discrt variational problm (DVP) admits a uniqu solution (u, λ ) X M. Morovr, for 0 > 0 suc tat k ( + k) is sufficintly small and k 0 π, tr is a positiv constant C (C dpnds on Ω only) suc tat for all 0, w av u u 0,Ω C( + k) u u,t + λ λ M C( + k) wr (u, λ) is t solution of t continuous variational problm VP(5). (25) Torm 3 Lt u b t solution of t continuous variational problm VP(5) and u b t solution of t discrt variational problm (DVP). W assum tat k π, tn tr xists a constant C > 0 (C dpnds on Ω only) suc tat ( u u 0,Ω Ĉ ( Σ n u iku 2 0,) 2 + ( n u + n ikx d 2 0,) 2 + ( Γ intrior [u ] 2 0,) 2 wr is an dg of T, [u ] is t jump of u across t dg and is t lngt of. ) (26) Rmark 2 W must point out tat it as bn rportd in [, 2] tat for ig frquncy rgim, t us of P finit lmnt mtod lads to t following stimats: u u,ω C k 2 and u u 0,Ω C k 3 2 wn k 2 is small noug. Ts stimats wr drivd assuming tat u H 2 (Ω) wic is not owvr valid for most problms. T a postriori stimat givn by Eq. (26) is a practical tool for a ms adaptiv stratgy. Tis stimat rvals tat t L 2 rror dpnds on ow wll t jump of t primal variabl as wll as t intrior and xtrior boundary conditions ar approximatd at t lmnt lvl. In ordr to prov Torm 2 and Torm 3, w nd first to stablis intrmdiat intrpolation rsults. Tis is accomplisd in Sction 3.3. Tn, w prov in Sction 3.4. t xistnc and t uniqunss of t solution of t discrt variational problm. Tis rsult is stablisd as a dirct consqunc of Proposition and Proposition 2. Sction is dvotd to t proof of (25) and (26). T rror stimat givn by Eq. (25) is stablisd in four stps, ac stp is formulatd as a lmma (s Lmma 7 to Lmma 0). T a postriori rror stimat givn by Eq. (26) is stablisd at t nd of Sction T nxt rsult, tat can b asily stablisd, sows wy t xistnc and t uniqunss of t solution of (DVP) is not a dirct consqunc of t xistnc and t uniqunss of t solution of (VP). Lmma 2 T null spac N corrsponding to t bilinar form b(, ) dfind by satisfis N = { v X ; N = {v X : b(v, µ ) = 0 ; µ M } v dt = v dt, T } (27) 2

13 Rmark 3 Lmma 2 stats tat N is not a subspac of N = H (Ω) wic is t null spac of t bilinar form b(.,.). Indd, t trac of an lmnt of N on an dg of an lmnt is wakly continuous in t sns givn by (27), wil t trac of an lmnt of N on an dg of an lmnt is continuous almost vrywr. Trfor, t inf-sup condition givn by Eq. (8) and tn Torm ar no longr valid if w simply rplac X and M by X and M rspctivly. 3.3 Matmatical analysis of t intrpolation oprators W stablis in tis sction intrmdiat intrpolation rsults tat summariz t main proprtis of t projction oprator Π from X onto X and t projction oprator P from M onto M. Ts rsults ar obtaind in t cas of a rctangular-sapd partition of t computational domain Ω Intrpolation oprator in X Lmma 3 For a fixd T, w av t following two proprtis: i. T normal drivativ n φ j is constant on vry dg T l ( l, j 4). ii. If k π tn t matrix B is invrtibl and tr is a positiv constant Ĉ suc tat (B ) 2 Ĉ k 2 2. (28) Proof of Lmma 3. It follows from t dfinition of φ j (S Sction 3.) tat n φ j = ik n j n l φ j on T l ( l, j 4). Trfor, sinc is a rctangular-sapd lmnt, a simpl calculation sows tat n φ j = ik on T j, n φ j = ik on T j+2 and n φ j = 0 on T j+ T j+3. In addition, it follows from t dfinition of t lmntary matrix B (S Eq. (20)) tat B = b a 2 b b 2 b 2 a a 2 b b b 2 a b 2 wr a j = ik j and b j = ik j ik, j 4. j W st = ( + a )( + a 2 ) 4b b 2. Tn, it is asy to vrify tat 0 for k π (wic is in 3

14 fact a sufficint but not ncssary condition). Tis nsurs tat t matrix B is invrtibl, and w av +a + a 2 2 b +a a 2 2 b [B ] = 2 b 2 +a 2 + a 2 b 2 +a 2 a 2 +a a 2 2 b +a + a 2 2 b 2 b 2 +a 2 a 2 b 2 +a 2 Finally, on can vrify tat tr is a positiv constant Ĉ and k suc tat a [B ] 2 Ĉ k 2 2. Nxt, w introduc t squnc of linar oprators (π ) T dfind as follows: π : H () C 4 v π v wr (π v ) j = j T j v dt, j 4. (29) Tn, it follows from Eq.(2) tat, for any indpndnt vctorial norm. in C 4, tr is a positiv constant Ĉ suc tat In addition, w av π v Ĉ v X(), v H (). (30) 4 v X (), v = αj φ j wr αj = ([B ] π v ) j, j 4. (3) j= T nxt rsults stats tat, for a givn T, t st of dgrs of frdom associatd to t planar wavs (φ j )4 j= is unisolvnt. Lmma 4 For a givn T and for any v X (), w av t following quivalnc: ( ) v dt = 0, l 4 ( v = 0 on ) T l Proof of Lmma 4. Using Eq. (29) and Eq. (3), it follows tat for a givn T, w av v dt = 0, l 4 π v = 0 v = 0 T l 4

15 wic provs Lmma 4. Consquntly, on can construct a squnc of local linar oprator (Π ) T Π : H () X () v Π v as follows: wit T j v dt = T j Π v dt, j 4. (32) Nxt, w stat tr proprtis of t oprator Π. Ts proprtis ar immdiat consquncs of t dfinition of Π, t inqualitis (2)-(22), proprty (32) of t oprator Π, and t caractrization of lmnts of X () wit t lmntary matrix B (s Eq. (3)). Not tat T scond idntity of (33) is obtaind by Grn s formula using t rctangular sap of. Proprty 2 T oprator Π satisfis t following tr proprtis: i. T and v H (), w av (v Π v ) dt = 0, (v Π v ) dx = 0. (33) ii. Tr is a positiv constant Ĉ suc tat T, v Π v 0, Ĉ 2 v Π v,, v H (). (34) iii. For a givn v H (), w av π v = π o Π v and Π v = 4 αj φ j wit αj = ([B ] π v ) j. (35) j= Proof of Proprty 2. W prov only t scond proprty sinc t two otrs ar immdiat. Using Eq. (33) and t dfinition of t norm. 0,, w av v Π v 0, = v Π v (v Π v ) dt 0, W tn conclud using Eq. (23). inf β C v Π v β 0, v Π v (v Π v ) dt 0,. In t nxt two lmmas, w stablis a priori stimats on t oprator Π. 5

16 Lmma 5 Assum k π. Tn, tr is a positiv constant Ĉ suc tat T and v H (), w av v Π v 0, Ĉ v Π v, (36) k Π v 0, + Π v X() Ĉ v X() (37) Proof of Lmma 5. W stablis t stimat givn by Eq. (36) using Aubin-Nitsc argumnt [24, 25, 26]. Mor spcifically, considr t following auxiliary boundary valu problm Find ϕ H0 () suc tat ϕ = v Π v on. Sinc is a rctangular-sapd lmnt, tn ϕ is in fact in H 2 () H0 () and w av It follows tat v Π v 2 0, = Using Eq. (33), w dduc tat ( v Π v ) ϕ dx = Tn, ϕ 2, = ϕ 0, = v Π v 0,. ( v Π v ) ( ϕ dx v Π v ) n ϕ dt ( v Π v ) ( ϕ ) ϕ dx ( v Π v ) ϕ dx v Π v, ϕ It follows from Eq. (22), tat tr is a positiv constant Ĉ suc tat ( v Π v ) ϕ dx Ĉ v Π v, ϕ 2, Morovr, using Eq. (32) w obtain tat ( v Π v ) n ϕ dt = ϕ dx 0,. ( v Π v ) ( ϕ ) ϕ dx Hnc, w av ( v Π v ) n ϕ dt v Π v 0, ϕ dx. n dt ϕ dx 0, Finally, using t inquality (23) and Eq. (34), it follows tat tr is positiv constant Ĉ suc tat ( v Π v ) n ϕ dt Ĉ v Π v, ϕ 2, 6

17 Trfor, Eq. (36) rsults from : v Π v 2 0, Ĉ v Π v, ϕ 2, = Ĉ v Π v, v Π v 0,. Nxt, w stablis t stimat givn by Eq. (37). To do tis, w first not tat it follows from Eq. (35) tat 4 v H (), Π v αj φ j, wr. is any norm in X (). Hnc, using Eq. (3), Eq. (30), and Eq. (28), tr is a positiv constant Ĉ suc tat j= v H (), Π v Ĉ k 2 2 v X() max j 4 φ j. On t otr and, it is asy to vrify tat φ j 0, and φ j, k. Consquntly, tr is a positiv constant Ĉ suc tat Π v 0, Furtrmor, using Eq. (36), w dduc tat Tus, Ĉ k 2 v X() and Π v, Ĉ v k X(). v Π v 0, Ĉ( v, + Π v, ) Ĉ( v, + C k v X() ) k v Π v 0, Ĉ ( k v, + v X() ) and trfor, using t dfinition of t norm X(), it follows tat k Π v 0, Ĉ v X() wic concluds t proof of t first part of Eq. (37). Finally, w stablis t scond part of t stimat givn by Eq. (37). To do tis, w obsrv tat v H (), w av (v Π v ). v dx + (v Π v ) Π v dx, v Π v 2, = = (v Π v ). v dx k 2 (v Π v )Π v dx, v Π v, v, +k 2 v Π v 0, Π v 0,. Not tat tr is no boundary trms in t prvious qualitis bcaus of Lmma 3 and Eq. (32). Using again Eq. (36), w dduc t xistnc of a positiv constant Ĉ suc tat v Π v, v, + Ĉk2 Π v 0, 7

18 Trfor, using t first part of Eq. (37), w dduc tat v Π v, v, + Ĉk v X() Consquntly, tr is a positiv constant ĉ suc tat Π v, 2 v, + Ĉk v X() ĉ v X() Morovr, using Eq. (36), w dduc tat tr is a positiv constant Ĉ suc tat Π v 0, v 0, + Ĉ v Π v, and tus, Π v 0, Ĉ v X() wic concluds t proof of Eq. (37). Lmma 6 Assum k π. Tn for vry s [0, ], tr is a positiv constant all T, w av Ĉ suc tat for v Π v, Ĉ2( s v +s, + k 2 v 0, + k 2 2 v, ), v H +s (). (38) Proof of Lmma 6. First, lt ϕ b in P () wr P () is t spac of t affin polynomial functions. Tn, using first Eq. (33) and t fact tat ϕ is constant in ac triangl, nxt tat functions in X satisfy t omognous Hlmoltz quation in ac triangl, w can writ: ϕ Π ϕ 2, = (ϕ Π ϕ). (ϕ Π ϕ) dx = (ϕ Π ϕ). Π ϕ dx = = From rlation (36), w obtain Morovr, quation (37) givs Hnc, (ϕ Π ϕ). Π ϕ dx (ϕ Π ϕ). n Π ϕ dt (ϕ Π ϕ). Π ϕ dx = k 2 (ϕ Π ϕ).π ϕ dx k 2 ϕ Π ϕ 0, Π ϕ 0,. ϕ Π ϕ 0, Ĉ ϕ Π ϕ, Π ϕ 0, Ĉ( ϕ 0, + ϕ, ). ϕ Π ϕ, Ĉk2 ( ϕ 0, + ϕ, ). On t otr and, it follows from quation (37) tat for v H () and ϕ P (), w av Π (ϕ v ), Ĉ( v ϕ 0, + v ϕ, ) 8

19 and tn v Π v, v ϕ, + ϕ Π ϕ, + Π (ϕ v ), Furtrmor, sinc k π, w dduc tat Ĉ( v ϕ 0, + v ϕ, + k 2 ( ϕ 0, + ϕ, )). v Π v, Ĉ( v ϕ 0, + v ϕ, + k 2 v 0, + k 2 2 v, ). Sinc v H +s () wit s [0, ], w cos ϕ to b t P -polynomial approximation (t Lagrang polynomial intrpolation) of v on if s 0, and ϕ = v dx if s = 0. Trfor, it follows from t standard P intrpolation rsults on (s [5]) tat v Π v, Ĉ(s v +s, + k 2 v 0, + k 2 2 v, ). Nxt, w introduc t global intrpolation linar oprator Π as follows Π : X X wit v Π v (Π v) = Π (v ) X (), T. Proprty 3 T global intrpolation oprator Π : X X satisfis t following four proprtis: i. v H +s (Ω) wit s [0, ], w av v Π v 0,Ω Ĉ(+s v +s,ω + k 2 3 v,ω + k 2 2 v 0,Ω ) (39) v Π v,t Ĉ(s v +s,ω + k 2 2 v,ω + k 2 v 0,Ω ) (40) ii. v H (Ω), Π v N wr N is t null spac of b(.,.). iii. v X and v X, w av a(v Π v, v ) = ik a(v, v Π v) = ik Σ Σ (v Π v) v dt v (v Π v) dt (4) iv. v X and µ M, w av b(v, µ ) = b(π v, µ ) (42) Not tat Eqs. (39) (40) ar immdiat consquncs of Lmma 6, wil t two qualitis givn by Eq. (4) ar obtaind by Grn s formula and using t fact tat t plan wavs ar solutions of t Hlmoltz quation. 9

20 3.3.2 Intrpolation oprator in M W introduc r t projction oprator P for t dual variabl λ. P is dfind as follows: P : M M wr Tn, t oprator P satisfis T, P µ T j = j T, µ M, µ P µ T j µdt = µ dt, j 4. P µdt. (43) 3.4 Proof of Torm 2 W first prov tat t discrt variational problm (DVP) admits a uniqu solution (u, λ ) in X M, and tn w stablis t rror stimat givn by Eq. (25) Existnc and uniqunss First, w prov tat t bilinar form b(, ) satisfis t inf-sup condition [22]. Tis rsult is statd in Proposition. Tn, w prov in Proposition 2 t uniqunss of t solution of t omognous problm corrsponding to t variational problm (DVP). T xistnc and uniqunss of t discrt variational problm (DVP) is tn a dirct consqunc of Proposition and Proposition 2. Proposition Assum k π. Tn, tr is a positiv constant γ indpndnt of k and suc tat b(v, µ ) γ µ M sup µ M µ M. v X v X Proof of Proposition. From Eq. (9), w dduc tat µ M, b(v, µ ) sup µ M v X v X In addition, it follows from Eq. (8) tat µ M, φ X, b(v, µ ) sup = b(φ, µ ) = µ M v X v X φ X 20

21 Trfor, it follows from Eq. (42) tat µ M = b(π φ, µ ) Π φ X Π φ X φ X Sinc k π, it follows from Eq. (37) tat tr is a positiv constant Ĉ suc tat wic concluds t proof of Proposition. µ M Ĉ sup b(v, µ ) v X v X Proposition 2 Assum k π. Tn, t only solution of t following omognous discrt variational problm Find u N suc tat a(u, v ) = 0, v N. is t trivial on. Proof of Proposition 2. Lt u N suc tat a(u, v ) = 0 v N, tn a(u, u ) = 0 wic implis: u = 0 on Σ and k u 0,Ω = u,t. In addition, sinc u X, tn u + k 2 u = 0 in vry T. Trfor, using t intgration by parts, it follows tat: a(u, v ) = v n u dt = 0 v N tn, w also av n u = 0 on Γ Σ and [ n u ] = 0 on T wr [ n u ] = n u + nu is t jump of t normal drivativ of u across. To conclud t proof of tis proposition, w us a discrt continuation rsult. W considr first t following proprty (P): Lt T and Tl and Tm two adjacnt dgs of suc tat n u Tl = n u Tm = u dt = u dt = 0 tn u = 0 in. T l Not tat proprty (P) is asy to stablis sinc u X (a sum of four plan wavs), and trfor u satisfis t Hlmoltz quation at t lmnt lvl. T m Now sinc, tr is at last on lmnt T wit two adjacnt dgs blonging to t boundary Σ, tn using proprty (P) lads to u = 0 in. Tn, w obtain squntially tat u = 0 in all t quadrilatrals blonging to t first layr adjacnt to t boundary Σ. W rpat tis procss on t scond layr of t quadrilatrals and so on, until t boundary Γ is racd, wic provs t uniqunss of t solution u. 2

22 3.4.2 A priori rror stimats In t nxt lmmas, w stablis a priori stimats in ordr to prov t rror stimat (25) givn in Torm 2 btwn t xact solution (u, λ) and t discrt solution (u, λ ). W considr t following notations: κ = ( + k) and z = u Π u. (44) Lmma 7 Tr is a positiv constant variational problm VP(5) satisfis Ĉ indpndnt of k and suc tat t solution λ of t λ P λ M Ĉκ 2 3 ( + k). Proof of Lmma 7. First, rcall tat n u on \ Ω, λ = 0 on Ω. Trfor, using t dfinition of t oprator P along wit t fact t normal unit vctor n is constant on ac dg of, w dduc tat T, w av λ P λ 2 0, = u.n u.n dt 2 0,, intrior, intrior, intrior u u dt 2 0, =, intrior inf u β C β 2 2 0, u u dx 2 0, u u dx 2 0,. Finally, using t classical intrpolation rsults [5], tr is a positiv constant Ĉ suc tat In addition, w av from quation (4) tat T, λ P λ 0, Ĉ 6 u 5,. (45) 3 λ P λ H 2 ( ) = sup v H () (λ P λ)v dt. v X() On t otr and, from quation (43), w dduc tat (λ P λ)vdt = (λ P λ)(v v dx) dt, v H (). 22

23 Hnc, (λ P λ)v dt λ P λ 0, v v dx 0,, v H (). Using t following classical intrpolation rsults [5], it follows tat tr is a positiv constant Ĉ suc tat v v dx 0, Ĉ 2 v, Ĉ 2 v X(). W tn dduc t xistnc of a positiv constant Ĉ suc tat T, λ P λ H 2 ( ) Ĉ 2 λ P λ 0,, µ M. (46) Lmma 7 is t consqunc of quations (45)-(46) and Torm. T nxt lmma can b viwd as a consistncy rsult. Lmma 8 Assum k π. Tn, tr is a positiv constant Ĉ indpndnt of k and suc tat v X and v H (Ω): a(z, v ) + b(v, λ P λ) Ĉ ( + k)κ 2 3 [ κ v,t + v v,t ]. Proof of Lmma 8. W av a(z, v ) = a(u Π u, v ) = a(u Π u, v ) a(u u, v ). Morovr, sinc u satisfis VP, w av a(u, v ) + b(v, λ) = F (v ) and sinc u satisfis DVP, w av a(u, v ) + b(v, λ ) = F (v ) Consquntly, w obtain wic lads to a(u u, v ) = b(v, λ λ ) a(z, v ) + b(v, λ P λ) = a(u Π u, v ) + b(v, λ P λ) Hnc, it follows from quation (4) tat a(u Π u, v ) + b(v, λ P λ) = ik (u Π u)v dt + b(v, λ P λ) v X. (47) Σ 23

24 Nxt, using (32) and following t sam proof of Eq. (45) in Lmma 7, w obtain (u Π u)v dt u Π u v v dt dt Σ Σ u Π u 0, v v dx 0, Σ Hnc, using Eq. (23), it follows tat tr is a positiv constant Ĉ suc tat (u Π u)v dt Ĉ u Π u, v, Σ Tn, it follows from using Torm and Lmma 6, tat tr is a positiv constant Ĉ suc tat (u Π u)v dt Ĉ (κ κ 2 + κ3 ) v,t Σ wic implis (assuming k π) tat (u Π u)v dt Ĉ κ 5 3 v,t. (48) On t otr and, w av v H (Ω): Σ b(v, λ P λ) = = intrior intrior [v ] (λ P λ) dt = intrior (λ P λ).[(v v ) (v v )] dt λ P λ 0, (v v ) [v v ] (λ P λ) dt (v v ) dx 0, Trfor, it follows from using using Eq. (23), tat tr is a positiv constant Ĉ suc tat b(v, λ P λ) Ĉ 2 v v, λ P λ 0, Hnc, from Eq. (45) and Torm, w obtain tat tr is a positiv constant Ĉ suc tat b(v, λ P λ) Ĉ κ 2 3 ( + k) v v,t (49) W conclud t proof of Lmma 8 by substituting Eq. (48) and Eq. (49) into Eq. (47). 24

25 Rmark 4 W dduc from Lmma 8 tat, wn k π, tr is a positiv constant Ĉ suc tat v N and v H (Ω), a(z, v ) Ĉ ( + k) κ 2 3 [ κ v,t + v v,t ]. (50) Lmma 9 Assum k π. Tn, tr is a positiv constant C (C dpnds on Ω only) suc tat, z 0,Ω C κ 2 3 [( + k)κ z,t ] (5) Proof of Lmma 9. First obsrv tat z blongs to N and lt φ b t solution of t following boundary valu problm (s Lmma ): and φ k 2 φ = z in Ω, n φ = 0 on Γ, n φ = ik φ on Σ. Hnc, it follows from Lmma tat φ H 5 3 (Ω) and (s Eq. (3)) tr is constant C > 0 (C dpnds on Ω only) suc tat, for vry s [0, 5 3 ], w av In addition, w av φ s,ω C ( + k) s z 0,Ω. (52) z 2 0,Ω = a(z, φ) intrior [z ] n φ dt. (53) Eq. (53) rsults from multiplying t boundary valu problm introducd in Lmma 9, intgrating by parts on Ω, and using t dfinition of t bilinar form a. T scond trm of tis quality is du to t discontinuity of z along t intrior dgs. Rcall tat t jump [φ] along is givn by [φ] = φ φ. On t otr and, w av a(z, φ) a(z, Π φ) + a(z, φ Π φ). It follows from Eq. (4) tat a(z, φ) a(z, Π φ) + k Σ z (φ Π φ) dt. (54) Sinc Π φ N (s proprty ii in Proprty 3), tn it follows from Rmark 4 tat tr is a positiv constant Ĉ suc tat a(z, Π φ) Ĉ ( + k) κ 2 3 [κ Π φ,t + φ Π φ,t ]. Morovr, it follows from Lmma 6, tat tr is a positiv constant Ĉ suc tat } φ Π φ,t { Ĉ 2 3 φ 53,Ω + k2 φ 0,Ω + k 2 2 φ,ω tn using rlation (52) and t assumption k π, w obtain φ Π φ,t Ĉκ 2 3 z 0,Ω and Π φ,t Ĉ z 0,Ω. 25

26 W obtain tn: a(z, Π φ) Ĉ ( + k)κ 4 3 z 0,Ω. For t scond part of Eq. (54), w av z (φ Π φ) dt Ĉ φ Π φ,t z,t Ĉκ 2 3 z,t z 0,Ω. Σ Not tat t prvious inquality was obtaind using t sam mtodology to prov Lmma 5. Hnc, w first, w us Eq. (32) wn w add t constant ( z dt) to z. Tn, w apply Caucy-Scwartz along wit inqualitis (2) and ( 23). Finally, it follows tat tr is a positiv constant C (C dpnds on Ω only) suc tat Nxt, w stimat t trm and a(z, φ) C [( + k)κ κ 5 3 z,t ] z 0,Ω. (55) intrior z dt = (z z ) nφ dt = [z ] n φ dt in Eq. (53). First, obsrv tat z dt, and T + ( ) ( ) z z dt φ φ dx n dt ( ) ( z z dt φ ) φ dx n dt Trfor, [z ] n φ dt intrior z z dt φ φ dx dt. Hnc, it follows tat intrior [z ] n φ dt Ĉ 2 3 z,t φ 5 3,Ω Cκ 2 3 z,t z 0,Ω. (56) W conclud t proof of Lmma 9 by substituting Eq. (55) and Eq.(56) into Eq.(53). Lmma 0 Lt 0 b a positiv numbr suc tat k ( + k) 2 3 is sufficintly small. Tn, tr is a positiv constant C (C dpnds on Ω only) suc tat for all 0, w av u Π u 0,Ω Ĉ( + k)κ 4 3 and u Π u,t Ĉ( + k)κ

27 Proof of Lmma 0. It follows from t dfinition of t bilinar form a(.,.) tat a(z, z ) 2 = z 2,T k 2 z 2 2 0,Ω + k 2 z 4 0,Γ Morovr, using Rmark 4 wit v = z and v = 0 along wit t fact tat k π w obtain Trfor, w dduc tat a(z, z ) Ĉ ( + k)κ 2 3 z,t. z 2,T k 2 z 2 0,Ω + Ĉ ( + k)κ 2 3 z,t. Tn, using Eq. (5) along wit Young s inquality, w obtain Consquntly, w av z 2,T C[k 2 ( + k) 2 κ k 2 κ 4 3 z 2,T + ( + k)κ 2 3 z,t ]. z 2,T C[k 2 ( + k) 2 κ k 2 κ 4 3 z 2,T + ( + k) 2 κ 4 3 ]. Lt us considr 0 suc tat Ck 2 ( + k) tn for vry 0, w av Ck 2 κ W dduc tat z 2,T C[k 2 ( + k) 2 κ ( + k) 2 κ 4 3 ] tn z,t Ĉ ( + k)κ 2 3. In addition, w obtain from using Eq. (5), tat wic concluds t proof of Lmma 0. z 0,Ω Ĉ( + k)κ 4 3. Proof of t a priori rror stimat of Torm 2. W ar now rady to prov t stimat givn by Eq. (25). From Lmma 6 and Lmma 0, it follows tat tr is a positiv constant C (C dpnds on Ω only) suc tat and Hnc, w dduc tat u u 0,Ω u Π u 0,Ω + u Π u 0,Ω C [κ ( + k)κ 4 3 ] u u,t u Π u,t + u Π u,t C [κ kκ + ( + k)κ 2 3 ]. u u 0,Ω C ( + k)κ 4 3 and u u,t C ( + k)κ 2 3. Morovr, w dduc from Lmma 8 tat tr is a positiv constant Ĉ suc tat b(v, λ P λ) Ĉ ( + k)κ 2 3 v,t + a(z, v ) v X. 27

28 On t otr and, it follows from t dfinition of t bilinar form a(.,.) tat a(z, v ) z,t v,t + k 2 z.v dx + k z 0,Σ v 0,Σ v X. Ω Trfor, using t dfinition of t norm X and invrs inquality rsults, w dduc tat tr is a positiv constant Ĉ suc tat a(z, v ) ( z 2,T + k 2 2 z 2 ) 2 0,Ω v X + Ĉ k z 0,Σ 2 v X v X. In addition, it follows from t dfinition of t bilinar form a(.,.) and from using Eq. (50) wit v = z and v = 0 (s Rmark 4) tat tr is a positiv constant Ĉ suc tat k z 2 0,Σ a(z, z ) Ĉ ( + k)κ 2 3 z,t Trfor, using Lmma 0, w dduc tat tr is a positiv constant C (C dpnds on Ω only) suc tat k 2 z 0,Σ ( + k)κ 2 3. Hnc, w dduc tat tr is a positiv constant C (C dpnds on Ω only) suc tat a(z, v ) C ( + k)κ 2 3 v X v X. Consquntly, it follows Proposition tat tr is a positiv constant C (C dpnds on Ω only) suc tat λ P λ M C ( + k)κ 2 3. Finally, w dduc from Lmma 7 tat tr is a positiv constant C (C dpnds on Ω only) suc tat λ λ M C ( + k)κ 2 3. wic concluds t proof of t rror stimat of Torm 2. Proof of t a postriori rror stimat (26) in Torm 3. Lt φ b t solution of t boundary valu problm (2) (s Lmma ) wit f = u u. Tn tis solution φ blongs to H 5 3 (Ω) and for vry s [0, 5 3 ], tr xists a constant C > 0 dpnding only on s and Ω suc tat φ s,ω C( + k) s u u 0,Ω. Using intgration by parts, on can asily vrify tat u u 2 0,Ω = + intrior On t otr and, w also av φ( n u ik u ) dt + [ n u ]φ dt [u ] n φ dt Σ a(u, Π φ) = intrior Γ Γ n ikx d Π φ dt φ( n u + n ikx d ) dt 28

29 Trfor, using intgration by parts along wit t fact tat u satisfis t Hlmoltz quation at t lmnt lvl, w av n u Π φ dt + ( n u iku ) Π φ dt + [ n u ] Π φ dt Γ Σ = Γ n ikx d Π φ dt intrior Consquntly, using t fact tat for vry intrior dg, w av [ nu ] φ dt = [ nu ] Π φ dt, w dduc tat u u 2 0,Ω = ( φ Π φ ) ( n u ik u ) dt T Σ + ( φ Π φ ) ( n u + n ikx d ) dt (57) [u ] n φ dt Γ intrior Nxt, w stimat ac intgral in t rigt-and sid of Eq. (57) to dduc t a postriori stimat givn by Eq. (26) in Torm 3. ( First, w stimat: I = φ Π φ ) ( n u ik u ) dt T Σ. W av ( ) ( ) ( ) I n u ik u 2 0, φ Π φ 2 0, Ĉ n u ik u 2 0, φ Π φ,t. Σ Σ Σ Trfor, assuming tat k π, it follows from t proprtis of t oprator Π (s Eq. (40) in Proprty 3) tat tr is a positiv constant Ĉ suc tat: ( ) 2 ) I Ĉ n u ik u 2 0, ( 2 3 φ 53,Ω + φ,ω + k φ 0,Ω. Σ W dduc from t a priori stimat on φ s,ω tat tr is a positiv constant Ĉ suc tat: ( I Ĉ n u ik u 2 0, Σ ) 2 u u 0,Ω. Similarly, tr is also a positiv constant Ĉ2 suc tat: ( I 2 = φ Π φ ) ( ( n u + n ikx d ) dt Γ Ĉ n u + n ikx d 2 0, Tn, tr is tr is a positiv constant dnotd again by Ĉ2 suc tat ( ) 2 I 2 Ĉ2 n u + n ikx d 2 0, u u 0,Ω. Γ 29 Γ ) 2 φ Π φ,t

30 Last, w stimat: I 3 = intrior [u ] n φ dt. Considr an intrior dg = () (), tn [u ] n φ dt = [u ] φ.n dt = [u ] ( φ β).n dt β C 2. W tn obtain [u ] n φ dt [u ] 0, inf φ β 0,. β C 2 On t otr and, sinc tr is a positiv constant Ĉ suc tat it follows tat I 3 Ĉ intrior inf β C 2 φ β 0, Ĉ 6 φ 5 3,(), 6 [u ] 0, φ 5 3,() Ĉ Tn, tr is a positiv constant Ĉ3 suc tat ( intrior [u ] 2 0, ) φ 5 3,Ω. I 3 Ĉ3 ( intrior [u ] 2 0, ) 2 u u 0,Ω. 4 Conclusion A discontinuous Galrkin mtod (DGM) wit plan wavs and Lagrang multiplirs was rcntly proposd by Farat t al. [3] for solving two-dimnsional Hlmoltz problms at rlativly ig wav numbrs. In many prvious paprs, tis mtod was sown numrically to offr a significant potntial for wav propagation problms including acoustic scattring. Howvr, it lackd a formal convrgnc tory. Tis papr is a first stp toward filling tis gap. Indd, it is provd tat t ybrid variational formulation undrlying tis DGM is wll-posd in t sns of Hadamard. In addition, a priori rror stimats provd for t so-calld R-4- lmnt, tat is t simplst two-dimnsional lmnt associatd wit tis discrtization mtod, stablis t convrgnc of tis lmnt and rval its formal ordr of accuracy. Furtrmor, a postriori rror stimat was drivd and tat can b usd as a practical rror indicator wn rfining t partition of t computational domain. Higr ordr lmnts will b analyzd in futur rsarc. Acknowldgmnt T autors ar gratful to t Rfrs for tir constructiv suggstions and rmarks. T scond and t tird autors acknowldg partial support by t National Scinc Foundation (NSF) undr 30

31 Grant No. DMS , and partial support by t Offic of Naval Rsarc (ONR) undr Grant N Any opinions, findings, and conclusions or rcommndations xprssd in tis matrial ar tos of t autors and do not ncssarily rflct t viws of t NSF or t ONR. 3

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