MsFEM à la Crouzeix-Raviart for highly oscillatory elliptic problems

Transcription

1 MsFEM à la Crouzix-Raviart for highly oscillatory lliptic problms Claud L Bris 1, Frédéric Lgoll 1, Alxi Lozinski 2 1 Écol National ds Ponts t Chaussés, 6 t 8 avnu Blais Pascal, Marn-La-Vallé Cdx 2, FRANCE and INRIA Rocquncourt, MICMAC projct-tam, L Chsnay Cdx, FRANCE lbris@crmics.npc.fr, lgoll@lami.npc.fr 2 Institut d Mathématiqus d oulous, Univrsité Paul Sabatir, 118 rout d Narbonn, oulous Cdx 9, FRANCE alxi.lozinski@math.uni-toulous.fr Sptmbr 20, 2012 Abstract W introduc and analyz a multiscal finit lmnt (MsFEM typ mthod in th vin of th classical Crouzix-Raviart finit lmnt mthod that is spcifically adaptd for highly oscillatory lliptic problms. W illustrat numrically th fficincy of th approach and compar it with diffrnt variants of MsFEM. 1 Introduction Lt Ω R d b a boundd domain and f L 2 (Ω (mor rgularity on th right-hand sid will b ndd latr on. W considr th problm div [A (x u ] = f in Ω, u = 0 on Ω, (1 1

2 whr A is a highly oscillatory, uniformly lliptic and boundd matrix. o fix th idas (and this will in fact b a ncssary assumption for th analysis w provid blow to hold tru, on might think of A as th matrix A (x = A pr (x/ whr A pr is Z d priodic. h approach w introduc hr to addrss this problm is a multiscal finit lmnt typ mthod (hncforth abbrviatd as MsFEM. As any such mthod, it is not rstrictd to th priodic stting. Only our analysis is. Likwis, w will assum for simplicity of our analysis that th matrics A w manipulat ar symmtric matrics. Our purpos is to propos and study a spcific multiscal finit lmnt mthod for th problm (1, whr th Galrkin approximation spac is constructd from idas similar to thos by Crouzix and Raviart in thir construction of a classical FEM spac [14]. Rcall that th gnral ida of MsFEM approachs is to construct an approximation spac using prcomputd, local functions, that ar solutions to th quation of intrst with simpl (typically vanishing right hand sids. his is in contrast to standard finit lmnt approachs, whr th approximation spac is basd on picwis polynomials. o construct our spcific multiscal finit lmnt mthod for th problm (1, w rvisit th classical work of Crouzix and Raviart [14]. W prsrv th main fatur of thir nonconforming FEM spac, i.. that th continuity accross th dgs of th msh is nforcd only in a wak sns by rquiring that th avrag of th jump vanishs on ach dg. As shown in Sction 2.1 blow, this wak continuity condition lads to som natural boundary conditions for th multiscal basis functions. Our motivation for th introduction of such finit lmnt functions stms from our wish to addrss svral spcific multiscal problms, most of thm in a nonpriodic stting, for which implmnting flxibl boundary conditions on ach msh lmnt is of particular intrst. A prototypical situation is that of a prforatd mdium, whr inclusions ar not priodically locatd and whr th accuracy of th numrical solution is xtrmly snsitiv to an appropriat choic of valus of th finit lmnt basis functions on th boundaris of lmnts whn th lattr intrsct inclusions. h Crouzix- Raviart typ lmnts w construct thn provid an advantagous flxibility. Additionally, whn th problm undr considration is not (as abov a simpl scalar lliptic Poisson problm but a Stoks typ problm, it is wll known that th Crouzix-Raviart approach also allows in th classical stting for ncoding th incomprssibility constraint dirctly in th finit lmnt 2

3 spac. his proprty will b prsrvd for th drivation w prsnt hr in th multiscal contxt. W will not procd furthr in this dirction and rfr th intrstd radr to our forthcoming publication [25] for mor dtails on this topic and rlatd issus. Of cours, our approach is not th only possibl on to addrss th catgory of problms w considr. Snsitivity of th numrical solution upon th choic of boundary conditions st for th multiscal finit lmnt basis functions is a classical issu. Formally it may b asily undrstood on a on-dimnsional situation (s for instanc [26] for a formalization of this argumnt: th rror committd using a multiscal finit lmnt typ approach coms thn ntirly from th rror committd in th bulk of ach lmnt, bcaus it is asy to mak th numrical solution agr with th xact solution on nods. In dimnsions highr than on, howvr, it is impossibl to match th finit dimnsional approximation on th boundary of lmnts with th xact, infinit dimnsional trac of th xact solution on this boundary. A scond sourc of numrical rror thus follows from this. And th drivation of variants of MsFEM typ approachs can b sn as th qust to solv th issu of inappropriat boundary conditions on th boundaris of msh lmnts. Many tracks hav bn followd to addrss th issu, ach of thm lading to a spcific variant of th gnral approach. h simplst choic [21, 22] is to us linar boundary conditions, as in th standard P1 finit lmnt mthod. his yilds a multiscal finit lmnt spac consisting of continuous functions. h us of nonconforming finit lmnts is an attractiv altrnativ, lading to mor accurat and mor flxibl variants of th mthod. h work [12] uss Raviart-homas finit lmnts for a mixd formulation of an highly oscillatory lliptic problm similar to that considrd in th prsnt articl. Many contributions such as [1, 2, 5, 7] prsnt variants of and follow-up on this work. For non mixd formulations, w mntion th wll known ovrsampling mthod (giving birth to nonconforming finit lmnts, s.g. [16, 21, 20]. W also mntion th work [11], whr a variant of th classical MsFEM approach (i.. without ovrsampling is prsntd. Basis functions again satisfy Dirichlt linar boundary conditions on th boundaris of th finit lmnts, but continuity accross th dgs is only nforcd at th midpoint of th dgs, as in th approach suggstd by Crouzix and Raviart [14]. Not that this approach, although also inspird by th work [14], diffrs from ours in th sns that w do not impos any Dirichlt boundary conditions whn constructing th basis functions (s Sction 2.1 blow for mor dtails. 3

4 In th contxt of a HMM-typ mthod, w mntion th works [3, 4] for th computation of an approximation of th coars scal solution. An xcllnt rviw of many of th xisting approachs is prsntd in [6], and for th gnral dvlopmnt of MsFEM w rfr to [15]. Our purpos hr is to propos yt anothr possibility, which may b usful in spcific contxts. Rsults for problms of typ (1, although good, will not b spctacularly good. Howvr, th ingrdints w mploy hr to analyz th approach and th structur of our proof will b xtrmly usful whn studying th sam Crouzix-Raviart typ approach for a spcific stting of particular intrst: th cas for prforatd domains. In that cas, w will show in [25] that th approach w introduc along ths lins outprforms all th xisting approachs w ar awar of. Our articl is articulatd as follows. W outlin our approach in Sction 2 and stat th corrsponding rror stimat, for th priodic stting, in Sction 3 (horm 3. h subsqunt two sctions ar dvotd to th proof of th main rror stimat. W rcall som lmntary facts and tools of numrical analysis in Sction 4 and turn to th actual proof of horm 3 in Sction 5. Our final sction, Sction 6, prsnts som numrical comparisons btwn th approach w introduc hr and som xisting MsFEM typ approachs. 2 Prsntation of our MsFEM approach hroughout this articl, w assum that th ambint dimnsion is d = 2 or d = 3 and that Ω is a polygonal (rsp. polyhdral domain. W dfin a msh H on Ω, i.. a dcomposition of Ω into polygons (rsp. polyhdra ach of diamtr at most H, and dnot E H th st of all th intrnal dgs (or facs of H. W assum that th msh dos not hav any hanging nods. Othrwis statd, ach intrnal dg (rsp. fac is shard by xactly two lmnts of th msh. In addition, H is assumd a rgular msh in th following sns: for any msh lmnt H, thr xists a smooth on-toon and onto mapping K : whr R d is th rfrnc lmnt (a polygon, rsp. a polyhdron, of fixd unit diamtr and K L CH, K 1 L CH 1, C bing som univrsal constant indpndnt of, to which w will rfr as th rgularity paramtr of th msh. o avoid som tchnical complications, w also assum that th mapping K corrsponding to ach H is affin on vry dg (rsp. fac of. In th following and to 4

5 fix th idas, w will hav in mind th two-dimnsional situation and a msh consisting of triangls, which satisfis th minimum angl condition to nsur th msh is rgular in th sns dfind abov (s.g. [10, Sction 4.4]. W will rpatdly us th notation and trminology (triangl, dg,... of this stting, although th analysis carris ovr to quadrangls if d = 2 or to ttrahdra and paralllpipda if d = 3. h bottom lin of our multiscal finit lmnt mthod à la Crouzix- Raviart is, as for th classical vrsion of th mthod, to rquir th continuity of th (hr highly oscillatory finit lmnt basis functions only in th sns of avrags on th dgs, rathr than to rquir th continuity at th nods (which is for instanc th cas in th ovrsampling variant of th MsFEM. In doing so, w xpct mor flxibility, and thrfor bttr approximation proprtis in dlicat cass. 2.1 Construction of th MsFEM basis functions Functional spacs W introduc th functional spac u L 2 (Ω such that u H 1 ( for any H, W H = [[u]] = 0 for all E H and u = 0 on Ω whr [[u]] dnots th jump of u ovr an dg. W nxt introduc its subspac } WH {u 0 = W H such that u = 0 for all E H and dfin th MsFEM spac à la Crouzix-Raviart V H = { u W H such that a H (u, v = 0 for all v W 0 H as th orthogonal complmnt of WH 0 in W H, whr by orthogonality w man orthogonality for th scalar product dfind by a H (u, v = ( v A (x u. W rcall that for simplicity w assum all matrics ar symmtric. Notation: For any u W H, w hncforth dnot by u E := a H (u, u th nrgy norm associatd with th form a H. } 5

6 Strong form o gt a mor intuitiv grasp on th spac V H, w not that any function u V H satisfis, on any lmnt H, ( v A u = 0 for all v H 1 ( s.t. v = 0 for all i = 1,..., N Γ, Γ i whr Γ i (with i = 1,...,N Γ ar th N Γ dgs composing th boundary of (not that, if Γ i Ω, th condition v = 0 is rplacd by v = 0 on Γ i Γ i ; this is a convntion w will us throughout our articl without xplicitly mntioning it. his can b rwrittn as N Γ ( v A u = λ i v for all v H Γ 1 ( i i=1 for som scalar constants λ 1,...,λ NΓ. Hnc, th rstriction of any u V H to is a solution to th boundary valu problm div [A (x u] = 0 in, n A u = λ i on ach Γ i. h flux along ach dg intrior to Ω is thrfor a constant. his of cours dfins u only up to an additiv constant, which is fixd by th continuity condition [[u]] = 0 for all E H and u = 0 on Ω. (2 Rmark 1. Obsrv that, in th cas A = Id, w rcovr th classical nonconforming finit lmnt spacs: Crouzix-Raviart lmnt [14] on any triangular msh: on ach, u Span{1, x, y}. Rannachr-urk lmnt [29] on any rctangular Cartsian msh: on ach, u Span{1, x, y, x 2 y 2 }. 6

7 Basis functions W can associat th basis functions of V H with th intrnal dgs of th msh as follows. Lt b such an dg and lt 1 and 2 b th two msh lmnts that shar that dg. h basis function φ associatd to, th support of which is 1 2, is constructd as follows. Lt us dnot th dgs composing th boundary of k (k = 1 or 2 by Γ k i (with i = 1,...,N Γ, and without loss of gnrality suppos that Γ 1 1 = Γ 2 1 =. On ach k, th function φ is th uniqu solution in H 1 ( k to div [A (x φ ] = 0 in k, φ = δ i1 for i = 1,...,N Γ, Γ k i n A φ = λ k i on Γ k i, i = 1,..., N Γ, whr δ i1 is th Kronckr symbol. Not that, for th dg Γ 1 1 = Γ2 1 = shard by th two lmnts, th valu of th flux may b diffrnt from on sid of th dg to th othr on: λ 1 1 may b diffrnt from λ2 1. h xistnc and uniqunss of φ follow from standard analysis argumnts. Dcomposition proprty A spcific dcomposition proprty basd on th abov finit lmnt spacs will b usful in th squl. Considr som function u W H, and introduc v H V H such that, for any lmnt H, w hav v H H 1 (, and div [A (x v H ] = 0 in, v H = u for i = 1,...,N Γ, Γ i Γ i n A v H = λ i on Γ i, i = 1,...,N Γ. Considr now v 0 = u v H W H. W s that, for any dg, v 0 = u v H = 0, thus v 0 WH 0. W can hnc dcompos (in a uniqu way any function u W H as th sum u = v H + v 0, with v H V H and v 0 WH 0. 7

8 2.2 Dfinition of th numrical approximation Using th finit lmnt spacs introducd abov, w now dfin th MsFEM approximation of th solution u to (1 as th solution u H V H to a H (u H, v = fv for any v V H. (3 Ω Not that (3 is a nonconforming approximation of (1, as V H H 1 0 (Ω. h problm (3 is wll posd. Indd, it is finit dimnsional so that it suffics to prov that f = 0 implis u H = 0. But f = 0 implis, taking v = u H in (3 and using th corcivity of A, that u H = 0 on vry H. h continuity condition (2 thn shows that u H = 0. 3 Main rsult h main purpos of our articl is to prsnt th numrical analysis of th mthod outlind in th prvious sction. o this nd, w nd to rstrict th stting of th approach (statd abov for, and indd applicabl to, gnral matrics A to th priodic stting. h ssntial rason for this rstriction is that, in th cours of th proof of our main rror stimat (horm 3 blow, w nd to us an accurat dscription of th asymptotic bhaviour (as 0 of th oscillatory solution u. Schmatically spaking, our rror stimat is stablishd using a triangl inquality of th form u u H u u,1 + u,1 u H, whr u,1 is an accurat dscription, for small, of th xact solution u to (1. Such an accurat dscription is not availabl in th compltly gnral stting whr th mthod is applicabl. In th priodic stting, howvr, w do hav such a dscription at our disposal. It is providd by th two-scal xpansion of th homognizd solution to th problm. his is th rason why w rstrict ourslvs to this stting. Som othr spcific sttings could prhaps allow for th sam typ of analysis but w will not procd in this dirction. On th othr hand, in th prsnt stat of our undrstanding of th problm and to th bst of our knowldg of th xisting litratur, w ar not awar of any stratgy of proof that could accommodat th fully gnral oscillatory stting. 8

9 Priodic homognization W hncforth assum that, in (1, A (x = A pr, (4 whr A pr is Z d priodic (and of cours boundd and uniformly lliptic. It is thn wll known (s.g. th classical txtbooks [8, 13, 24], and also [17] for a gnral, numrically orintd prsntation that th solution u to (1 convrgs, wakly in H 1 (Ω and strongly in L 2 (Ω, to th solution u to div ( A pr u = f in Ω, u = 0 on Ω, (5 with th homognizd matrix givn by, for any 1 i, j d, (A pr ij = ( i + w i (y A pr (y( j + w j (y dy, (0,1 d whr, for any p R d, w p is th uniqu (up to th addition of a constant solution to th corrctor problm associatd to th priodic matrix A pr : div [A pr (p + w p ] = 0, w p is Z d -priodic. (6 h corrctor functions allow to comput th homognizd matrix, and to obtain a convrgnc rsult in th H 1 strong norm. Indd, introduc hn, w hav: u,1 (x = u (x + d u w i (x. (7 x i i=1 Proposition 2. Suppos that th dimnsion is d > 1, that th solution u to (5 blongs to W 2, (Ω and that, for any p R d, th corrctor w p solution to (6 blongs to W 1, (R d. hn u u,1 H 1 (Ω C u W 1, (Ω (8 for a constant C indpndnt of and u. W rfr.g. to [24, p. 28] for a proof of this rsult. 9

10 Error stimat W ar now in position to stat our main rsult. horm 3. Lt u b th solution to (1 for a matrix A givn by (4. W furthrmor assum that A pr is Höldr continuous (9 and that th solution u to (5 blongs to C 2 (Ω. Lt u H b th solution to (3. W hav ( u u H E CH f L 2 (Ω + C + H + u H C 1 (Ω, (10 whr th constant C is indpndnt of H,, f and u. wo rmarks ar in ordr, first on th ncssity of our assumption (9, and nxt on th comparison with othr, wll stablishd variants of MsFEM. Rmark 4. (On th rgularity of A pr W rcall that, undr assumption (9, th solution w p to (6 (with, say, zro man satisfis, for any p R d, w p C 1,δ (R d for som δ > 0. (11 W rfr.g. to [19, horm 8.22 and Corollary 8.36]. Rmark 5. (Comparison with othr approachs It is usful to compar our rror stimat (10 with similar stimats for som xisting MsFEM-typ approachs in th litratur. h classical MsFEM from [22] (by classical, w man th mthod using basis functions satisfying linar boundary conditions on ach lmnt yilds an xactly similar majoration in trms of + H + /H. It is claimd in [22] that th sam majoration also holds for th MsFEM-O variant. his variant (in th form prsntd in [22] is rstrictd to th two-dimnsional stting. It uss boundary conditions providd by th solution to th oscillatory ordinary diffrntial quation obtaind by taking th trac of th original quation (1 on th dg considrd. h famous variant of MsFEM using ovrsampling (s [21, 16] givs a slightly bttr stimation: + H + /H. h bst stimation w ar awar of is obtaind using a Ptrov-Galrkin variant of MsFEM with ovrsampling (s [23]. It bounds th rror from abov by + H + but this only holds in th rgim /H C t and for a sufficintly (possibly prohibitivly larg ovrsampling ratio. All ths comparisons show that th mthod w prsnt 10

11 hr is guarantd to b accurat, although not spctacularly accurat, for th quation (1 considrd. An actually much bttr bhaviour will b obsrvd in practic, in particular for th cas of a prforatd domain that w study in [25]. A comparison with othr, rlatd but slightly diffrnt in spirit approachs can also b of intrst. h approachs [27] and [28] yild an rror stimat bttr than that obtaind with th ovrsampling variant of MsFEM. h computational cost is howvr largr, owing to th larg siz of th ovrsampling domain mployd. 4 Som classical ingrdints for our analysis Bfor w gt to th proof of our main rsult, horm 3, w first nd to collct hr som standard rsults. hs includ rac thorms, Poincaré-typ inqualitis, rror stimats for nonconforming finit lmnts and vntually convrgncs of oscillating functions. With a viw to nxt using ths rsults for our proof, w actually nd not only to rcall thm but also, for som of thm, to mak xplicit th dpndncy of th constants apparing in th various stimats upon th siz of th domain (which will b takn, in practic, as an lmnt of th msh, of diamtr H. Of cours, ths rsults ar standard, and thir proof is rcalld hr only for th sak of compltnss. First w rcall th dfinition, borrowd from.g. [18, Dfinition B.30], of th H 1/2 spac. Dfinition 6. For any opn domain ω R n and any u L 2 (ω, w dfin th norm u 2 H 1/2 (ω := u 2 L 2 (ω + u 2 H 1/2 (ω, whr and dfin th spac u 2 H 1/2 (ω := ω ω u(x u(y 2 x y n+1 dxdy, H 1/2 (ω := { u L 2 (ω, u H 1/2 (ω < }. 4.1 Rfrnc lmnt W first work on th rfrnc lmnt, with dgs (w rcall that our trminology and notation suggst that, to fix th idas, w hav in mind 11

12 triangls in two dimnsions. By th standard trac thorm, w know that thr xists C such that v H 1 (,, v H 1/2 ( C v H 1 (. (12 In addition, w hav th following rsult: Lmma 7. hr xists C (dpnding only on th rfrnc msh lmnt such that v H 1 ( with v = 0 for som, v H 1 ( C v L 2 (. h proof follows from th following rsult (s.g. [18, Lmma A.38]: (13 Lmma 8 (Ptr-artar. Lt X, Y and Z b thr Banach spacs. Lt A L(X, Y b an injctiv oprator and lt L(X, Z b a compact oprator. If thr xists c such that c x X Ax Y + x Z, thn Im (A is closd. Equivalntly, thr xists α > 0 such that x X, α x X Ax Y. Proof of Lmma 7. Considr. W apply Lmma 8 with Z = L 2 (, Y = ( L 2 ( d, { } X = v H 1 ( with v = 0 quippd with th H 1 ( norm, Av = v (which is indd injctiv on X, and v = v (which is indd compact from X to Z. Lmma 8 radily yilds th bound (13 aftr taking th maximum ovr all dgs. 4.2 Finit lmnt of siz H W will rpatdly us th following Poincaré inquality: Lmma 9. hr xists C (dpnding only on th rgularity of th msh indpndnt of H such that, for any H, v H 1 ( with v = 0 for som, v L 2 ( CH v L 2 (. 12 (14

13 Proof. o convy th ida of th proof in a simpl cas, w first assum that th actual msh lmnt considrd in th msh is homothtic to th rfrnc msh lmnt with a ratio H. W introduc v H (x = v(hx dfind on th rfrnc lmnt. W hnc hav v(x = v H (x/h, thus v 2 L 2 ( = v 2 (xdx = vh(x/hdx 2 = H d vh(ydy 2 and v 2 L 2 ( = v(x 2 dx = H 2 W now us Lmma 7, and conclud that v H (x/h 2 dx = H d 2 v 2 L 2 ( = Hd v H 2 L 2 ( CHd v H 2 L 2 ( = CH2 v 2 L 2 (, v H (y 2 dy. which is (14 in this simpl cas. o obtain (14 in full gnrality, w hav to slightly adapt th abov argumnt. W shall us hr and throughout th proof of th subsqunt lmma th notation A B whn th two quantitis A and B satisfy c 1 A B c 2 A with th constants c 1 and c 2 dpnding only on th rgularity paramtr of th msh. Lt us rcall that for all H thr xists a smooth on-to-on and onto mapping K : satisfying K L CH and K 1 L CH 1. W now introduc v H (x = v(k(x dfind on th rfrnc lmnt. W hnc hav v 2 L 2 ( = v 2 (xdx = vh(k 2 1 (xdx H d vh(ydy 2 and v 2 L 2 ( = v(x 2 dx H 2 v H (K 1 (x 2 dx H d 2 v H (y 2 dy. Using Lmma 7 (not that v H (ydy = 0 sinc th mapping K is affin on th dgs, hnc of constant jacobian on, w obtain v 2 L 2 ( Hd v H 2 L 2 ( CHd v H 2 L 2 ( CH2 v 2 L 2 (, which is th bound (14. 13

14 W also hav th following trac rsults: Lmma 10. hr xists C (dpnding only on th rgularity of th msh such that, for any H and any dg, w hav v H 1 (, v 2 L 2 ( (H C 1 v 2 L 2 ( + H v 2 L 2 (. (15 Undr th additional assumption that v = 0, w hav If v 2 L 2 ( CH v 2 L 2 (. (16 v = 0 and H 1, thn v 2 H 1/2 ( C v 2 L 2 (. (17 hs bounds ar classical rsults (s.g. [10, pag 282]. W provid hr a proof for th sak of compltnss. Proof of Lmma 10. W procd as in th proof of Lmma 9 and us th sam notation. W us v H (x = v(k(x dfind on th rfrnc lmnt. W hav v 2 L 2 ( = v 2 (xdx = vh(k 2 1 (xdx H d 1 vh(ydy 2 = H d 1 v H 2 L 2 (. By a standard trac inquality, w obtain v 2 L 2 ( CH ( v d 1 H 2 + v L 2 ( H 2 L 2 ( ( 1 CH d 1 H d v 2 L 2 ( + 1 H d 2 v 2 L 2 (, whr w hav usd som ingrdints of th proof of Lmma 9. his shows (15. W now turn to (16: v 2 L 2 ( Hd 1 v H 2 L 2 ( CHd 1 v H 2 H 1 ( CH d 1 v H 2 L 2 ( CH v 2 L 2 (, whr w hav usd (12 and (13. his provs (16. 14

15 W vntually stablish (17. W first obsrv, using Dfinition 6 with th domain ω R d 1, that v 2 v(x v(y 2 H 1/2 ( = dxdy x y d 1 v H (K 1 (x v H (K 1 (y 2 dxdy H d K 1 (x K 1 (y d H d 2 v H (x v H (y 2 dxdy x y d H d 2 v H 2 H 1/2 (. Hnc, using (12 and (13 and sinc H 1, v 2 H 1/2 ( = v 2 L 2 ( + v 2 H 1/2 ( Hd 1 v H 2 L 2 ( + Hd 2 v H 2 H 1/2 ( CH d 2 v H 2 H 1/2 ( CHd 2 v H 2 H 1 ( CHd 2 v H 2 L 2 ( C v 2 L 2 (. his provs (17 and concluds th proof of Lmma 10. h following rsult is a dirct consqunc of (16 and (17: Corollary 11. Considr an dg E H, and lt H dnot all th triangls sharing this dg. hr xists C (dpnding only on th rgularity of th msh such that v W H, [[v]] 2 L 2 ( CH v 2 L 2 (. (18 If H 1, thn v W H, [[v]] 2 H 1/2 ( C v 2 L 2 (. (19 Proof. W introduc c = 1 v, which is wll dfind on th dg sinc [[v]] = 0. On ach sid of th dg, th function v c has zro avrag on 15

16 that dg. Hnc, using (16, [[v]] 2 L 2 ( = [[v c ]] 2 L 2 ( = (v 1 c (v 2 c 2 L 2 ( 2 v 1 c 2 L 2 ( + 2 v 2 c 2 L 2 ( ( v CH 1 2 L 2 ( 1 + v 2 2 L 2 ( 2 = CH v 2 L 2 (, whr w hav usd th notation v 1 = v 1. h proof of (19 follows a similar pattrn, using ( Error stimat for nonconforming FEM h rror stimat w stablish in th nxt sction is ssntially basd on a Céa-typ (or Strang-typ lmma xtndd to nonconforming finit lmnt mthods. W stat this standard stimat in th actual contxt w work in (but again mphasiz it is of cours compltly gnral in natur. Lmma 12. (.g. [10, Lmma ] Lt u b th solution to (1 and u H b th solution to (3. hn u u H E inf u a H (u u H, v v E + sup. (20 v V H v V H \{0} v E h first trm in (20 is th usual bst approximation rror alrady prsnt in th classical Céa Lmma. his trm masurs how accuratly th spac V H (or, in gnral, any approximation spac approximats th xact solution u. h scond trm of (20 masurs how th nonconforming stting affcts th rsult. his trm would vanish if V H wr a subst of H 1 0 (Ω. 4.4 Intgrals of oscillatory functions W shall also nd th following rsult. Lmma 13. Lt E H, 1 and 2 b th two lmnts adjacnt to and τ R d, τ 1, b a vctor tangnt (i.. paralll to. hn, for any 16

17 function u that is H 1 in 1 and 2, any v W H and any J C 1 (R d, w hav u(x [[v(x]] τ J C H J C 1 (R d = 1, 2 v H 1 ( ( u L 2 ( + H u H 1 ( (21 with a constant C which dpnds only on th rgularity of th msh. As will b clar from th proof blow, th fact that w considr in th abov lft-hand sid th jump of v, rathr than v itslf, is not ssntial. A similar stimat holds for th quantity u(x v(x τ J, whr u and v ar any functions of rgularity H 1 ( for som H and is an dg of. Proof of Lmma 13. Lt c b th avrag of v ovr and dnot v j = v j. Sinc [[v]] = (v 1 c (v 2 c, w obviously hav u(x [[v(x]] τ J j=1 2 u(x (v j (x c τ J. (22 Lt us fix j. W first rcall that thr xists a on-to-on and onto mapping K : j from th rfrnc lmnt onto j satisfying K L CH and K 1 L CH 1. In particular, thr xists an dg of such that K( =. W introduc th functions u H (x = u(k(x, v H (x = v j (K(x c dfind on th rfrnc lmnt, and obsrv that ( ( x K(y u(x (v j (x c τ J H dx d 1 u H (yv H (y τ J dy. (23 W now claim that ( K(y u H (yv H (y τ J dy C H J C 1 (R u d H H ( v 1/2 H H (. 1/2 (24 his inquality is obtaind by intrpolation. Suppos indd, in a first stp, that u H and v H blong to H 1 (. Using that th mapping K is affin on th 17

18 dgs and thus of constant gradint, w first s that ( K(y u H (yv H (y τ J dy = C u H (yv H (y τ H By intgration by parts, w nxt obsrv that [ ( ] K(y u H (yv H (y τ J dy H = ( K(y u H (yv H (y τ ν J H ( K(y J H dy [ J ( K(y ] dy. (25 τ (u H (yv H (ydy, (26 whr ν is th outward normal unit vctor to tangnt to. Collcting (25 and (26, and using th Cauchy-Schwarz inquality, w obtain that ( K(y u H (yv H (y τ J dy C H J [ ] C 0 (R d uh L 2 ( v H L 2 ( + 2 u H H 1 ( v H H 1 ( C H J C 0 (R d u H H 1 ( v H H 1 (, (27 whr th last inquality abov follows from th trac inquality which is valid with a constant C dpnding only on. On th othr hand, for u H and v H that only blong to L 2 (, w obviously hav ( K(y u H (yv H (y τ J dy J C 0 (R u d H L 2 ( v H L 2 (. (28 By intrpolation btwn (27 and (28 (s [9, horm 4.4.1], w obtain (24. h squl of th proof is asy. Collcting (23 and (24, w dduc that u(x (v j (x c τ J dx CH d 3/2 J C 1 (R d u H H 1/2 ( v H H 1/2 ( CH d 3/2 J C 1 (R d u H H 1 ( v H L 2 ( (29 18

19 whr w hav usd in th last lin th trac inquality (12 and Lmma 7 for v H (rcall that v H = 0, sinc, on th on hand, v j c = 0 and, on th othr hand, th mapping K is affin on, and hnc of constant gradint. o rturn to norms on th actual lmnt j rathr than on th rfrnc lmnt, w us th following rlations, alrady stablishd in th proof of Lmma 9: u L 2 ( j H d/2 u H L 2 (, u H 1 ( j H d/2 1 u H H 1 (, W thn infr from (29 that u(x(v j (x c τ J v j H 1 ( j H d/2 1 v H H 1 (. CH d 3/2 [ J C 1 (R d H d/2 u L 2 ( j + H d/2+1 u H 1 ( j ] H d/2+1 v j H 1 ( j C H J [ ] C 1 (R d u L 2 ( j + H u H 1 ( j vj H 1 ( j. Insrting this bound in (22 for j = 1 and 2 yilds th dsird bound (21. 5 Proof of th main rror stimat Now that w hav rviwd a numbr of classical ingrdints, w ar in position, in this sction, to prov our main rsult, horm 3. As announcd abov, our proof is basd on th stimat (20 providd by Lmma 12. o bound both trms in th right-hand sid of (20, w will us th following rsult, th proof of which is postpond until Sction 5.2: Lmma 14. Undr th sam assumptions as thos of horm 3, w hav, for any v W H, ( ( v A pr u n C + H + v E u H C 1 (Ω, whr th constant C is indpndnt of H,, f, u and v. (30 Rmark 15. A mor prcis stimat is givn in th cours of th proof, s (52. 19

20 5.1 Proof of horm 3 Momntarily assuming Lmma 14, w now prov our main rsult. W argu on stimat (20 providd by Lmma 12. In th right-hand sid of (20, w first bound th nonconforming rror (scond trm. Lt v V H. W us th dfinition of a H and (3 to comput: u fv Ω a H (u u H, v = ( v A pr = ( v A pr u n H v div ( A pr u fv Ω using th Grn formula, = ( v A pr u n, using (1 and th rgularity of u. Obsrving that, by dfinition, v V H W H, w can us Lmma 14 to majoriz th abov right-hand sid. W obtain ( a H (u u H, v sup C + H + u v V H \{0} v E H C 1 (Ω. (31 W now turn to th bst approximation rror (first trm of th righthand sid of (20. As shown at th nd of Sction 2.1, w can dcompos u H 1 0 (Ω W H as u = v H + v 0, v H V H, v 0 W 0 H. W may comput, again using (1 and th rgularity of u, that u v H 2 E = a H(u v H, u v H = a H (u v H, v 0 (by dfinition of v 0 = a H (u, v 0 (by orthogonality of V H with WH 0 = ( v 0 A pr u = ( v 0 A pr u n + v 0 f. (32 20

21 Sinc v 0 WH 0, w may us (14 and bound th scond trm of th right-hand sid of (32 as follows: v 0 f v 0 L 2 ( f L 2 ( (Cauchy Schwarz inquality H CH v 0 L 2 ( f L 2 ( CH v 0 E f L 2 (Ω, (33 whr w hav usd in th last lin th Cauchy Schwarz inquality and an quivalnc of norms. h first trm of th right-hand sid of (32 is boundd using Lmma 14 (sinc v 0 WH 0 W H, which yilds ( v 0 A pr u n C ( + H + v 0 E u H C 1 (Ω. (34 Insrting (33 and (34 in th right-hand sid of (32, w dduc that ( u v H 2 E CH v0 E f L 2 (Ω + C + H + v 0 E u H C 1 (Ω. Sinc v 0 = u v H w may factor out v 0 E and obtain ( u v H E CH f L 2 (Ω + C + H + u H C 1 (Ω. By dfinition of th infimum, w of cours hav inf u v E u v H E, v V H thus inf u v E CH f L v V 2 (Ω + C H ( + H + u H C 1 (Ω. (35 Insrting (31 and (35 in th right-hand sid of (20, w obtain th dsird bound (10. his concluds th proof of horm Proof of Lmma 14 W now stablish Lmma 14, actually th ky stp of th proof of horm 3. 21

22 Lt v W H. Using (1 and (5, and insrting in th trm w ar stimating th approximation u,1 dfind by (7 of th xact solution u, w writ ( v A pr u n = vf + H ( v A pr = v div ( A pr u + + = + ( v A pr u,1 v ( A pr u n + ( v ( A pr u ( v A pr ( v A pr u,1 A pr u (u u,1 (u u,1 = A + B + C. (36 W now succssivly bound th thr trms A, B and C in th right-hand sid of (36. Loosly spaking, th first trm A is macroscopic in natur and would b prsnt for th analysis of a classical Crouzix-Raviart typ mthod. It will vntually contribut for O(H to th ovrall stimat (30 (and thus (10; th scond trm B is indpndnt from th discrtization: it is an infinit dimnsional trm, th siz of which, namly O(, is ntirly controlld by th quality of approximation of u by u,1. It is th trm for which w spcifically nd to put ourslvs in th priodic stting; th third trm C would likwis go to zro if th siz of th msh wr much largr than th small cofficint ; it will contribut for th ( O /H trm in th stimat (30. Stp 1: bound on th first trm of (36: W first not that v ( A pr u n = [[v]] ( A pr u n. E H 22

23 W now us argumnts that ar standard in th contxt of Crouzix-Raviart finitlmnts (s [10, pag 281]. Introducing, for ach dg, th constant ( c = A pr u n, and using that, sinc v W H, w hav [[v]] = 0, w writ v ( A pr u n H = [[v]] ( A pr u n E H [[v]] (( A pr u n c E H E H [[v]] L 2 ( ( A pr u n c L 2 ( [ E H [[v]] 2 L 2 ( ] 1/2 [ E H ( A pr u n c 2 L 2 ( ] 1/2, succssivly using th continuous and discrt Cauchy-Schwarz inqualitis in th last two lins. W now us (18 and (16 to rspctivly stimat th two factors in th abov right-hand sid. Doing so, w obtain v ( A pr u n H [ ] 1/2 [ ] 1/2 C v 2 L 2 ( H 2 u 2 L 2 (. E H ; choos on E H H W hnc hav that v ( A pr u n C [ H ] 1/2 [ ] 1/2 v 2 L 2 ( H 2 u 2 L 2 ( CH v E 2 u L 2 (Ω. (37 23

24 Stp 2: bound on th scond trm of (36: W not that ( v A pr (u u,1 A pr L v L 2 ( (u u,1 L 2 ( vntually using (8. C v E (u u,1 L 2 (Ω using th discrt Cauchy-Schwarz inquality C v E u W 1, (Ω (38 Stp 3: bound on th third trm of (36: o start with, w diffrntiat u,1 dfind by (7: u,1 (x = d i=1 ( i u (x i + w i + d w i i u (x. i=1 h third trm of (36 thus writs ( ( v A pr u,1 A pr u = + d i=1 d i=1 ( v A pr ( v i u (x G i whr w hav introducd th vctor filds i u (x w i, (39 G i (x = A pr (x ( i + w i (x A pr i, 1 i d, which ar all Z d priodic, divrgnc fr and of zro man. In addition, in viw of th assumptions (9 and (11, w s that G i is Höldr continuous. (40 24

25 W now succssivly bound th two trms of th right-hand sid of (39. h first trm is quit straightforward to bound. Using Cauchy-Schwarz inqualitis and that w p L (s (11, w simply obsrv that d ( v A pr i u (x w i i=1 d A pr L max w i L i v L 2 ( 2 u L 2 ( C v E 2 u L 2 (Ω. (41 h rst of th proof is actually dvotd to bounding th scond trm of th right-hand sid of (39, a task that rquirs svral stimations. W first us a classical argumnt alrady xposd.g. in [24, p. 27]. h vctor fild G i bing Z d priodic, divrgnc fr and of zro man, thr xists (s [24, p. 6] a skw symmtric matrix J i R d d such that, 1 α d, [G i ] α = d β=1 [J i ] βα x β (42 and J i ( H 1 loc(r d d d, J i is Z d -priodic, (0,1 d J i = 0. In th two-dimnsional stting, an xplicit xprssion can b writtn. W indd hav ( J i 0 τ (x 1, x 2 = i (x 1, x 2 τ i, x = (x (x 1, x 2 0 1, x 2 R 2, with τ i (x 1, x 2 = τ i ( ( x 2 [G i ] 1 (tx x 1 [G i ] 2 (tx dt whr τ i (0 is such that τ i = 0. In viw of (40, w in particular hav (0,1 2 that J i ( C 1 (R d d d. (43 A bttr rgularity (namly J i ( C 1,δ (R d d d for som δ > 0 actually holds, but w will not nd it hncforth. 25

26 h sam rgularity (43 can b also provn in any dimnsion d 3, although in a lss straightforward mannr. Indd, th componnts of J i constructd in [24, p. 7] using th Fourir sris can b sn to satisfy th quation [J i ] βα = [G i] β x α [G i] α x β, complmntd with priodic boundary conditions. Hnc th function [J i ] βα, as wll as its gradint, is continuous du to th rgularity (40 and gnral rsults on lliptic quations (s.g. [19, Sction 4.5]. ( In viw of (42, w s that th α-th coordinat of th vctor i u ( G i rads [ ] i u (x G i α = d β=1 [J i ] βα x β i u (x d d = ([J i ] βα i u (x [J i ] βα iβ u (x x β β=1 β=1 [ ] = B i (x [Bi (x] α, (44 α whr th vctor filds B i (x Rd and B i (x Rd ar dfind, for any 1 α d, by [B i (x] α = d [J i ] βα iβ u (x β=1 and [ ] B i (x = α h vctor fild B i is divrgnc fr as J i is a skw symmtric matrix. d β=1 h scond trm of th right-hand sid of (39 thus rads = = d i=1 ( v i u (x G i d ( ( v(x B i (x Bi (x i=1 d i=1 v(x B i (x n 26 d i=1 ([J i ] βα i u (x. x β ( v(x Bi (x, (45

27 succssivly using (44 and an intgration by parts of th formr trm and th divrgnc-fr proprty of B i. An uppr bound for th scond trm can asily b obtaind, givn that J i ( L (R d d d (s (43: d i=1 ( v(x B i (x = d i,α,β=1 d 3 max J i L i α v(x[j i ] βα iβ u (x v L 2 ( 2 u L 2 ( C v E 2 u L 2 (Ω. (46 W ar now lft with bounding th first trm of th right-hand sid of (45, which rads d i=1 v(x B i (x n = d [[v(x]] B i (x n E H i=1 = E H d i,α,β=1 = E H d i,α,β=1 + E H d i,α,β=1 [[v(x]] n α ([J i ] βα i u (x x β [[v(x]] n α [J i ] βα x β [[v(x]] n α [J i ] βα i u (x iβ u (x. (47 Our final task is to succssivly bound th two trms of th right-hand sid of (47. W bgin with th first trm. Considring an dg, w rcast th contribution of that dg to th first trm of th right-hand sid of (47 as follows, 27

28 using th skw symmtry of J: d i,α,β=1 = = [[v(x]] n α [J i ] βα x β d i,α,β=1 β>α d i,α,β=1 β>α i u (x ( [J i ] βα [J i ] βα [[v(x]] n α n β x β x α (x i u (x [[v(x]] ( ( τ αβ [J i x ] βα i u (x, (48 whr τ αβ R d is th vctor with α-th componnt st to n β, β-th componnt st to n α, and all othr componnts st to 0. Obviously, τ αβ is paralll to. W can thus us Lmma 13, and infr from (48 that d i,α,β=1 C H [[v(x]] n α [J i ] βα x β d i,α,β=1 [J i ] βα C 1 (R d i u (x v H 1 ( ( i u L 2 ( + H i u H 1 (. Using th rgularity (43 of J i, w dduc that th first trm of th right-hand sid of (47 satisfis d [J i ] ( βα x [[v(x]] n α i u (x E H i,α,β=1 x β [ ] 1/2 C v 2 L H 2 ( E H [ ] 1/2 u 2 L 2 ( + H2 2 u 2 L 2 ( E H C H v E u L 2 (Ω + C H v E 2 u L 2 (Ω. (49 28

29 W nxt turn to th scond trm of th right-hand sid of (47, which satisfis d n α [J E H i,α,β=1 [[v(x]] i ] βα iβ u (x ( d 3 max J i C 0 (R d 2 u C 0 (Ω [[v(x]] L 1 ( i E H [ ] 1/2 [ ] 1/2 C 2 u C 0 (Ω [[v]] 2 L 2 ( E H [ C 2 u C 0 (Ω H v 2 L 2 ( E H [ H L 2 ( E H ;choos on ] 1/2 1 2 L 2 ( E H ] 1/2 C 2 u C 0 (Ω v E Ω 1/2, (50 whr w hav usd (18 of Corollary 11 and (15 of Lmma 10. Collcting th stimats (39, (41, (45, (46, (47, (49 and (50, w bound th third trm of (36: ( ( v A pr u,1 A pr u H C H v E 2 u L 2 (Ω+C H v E u L 2 (Ω+C 2 u C 0 (Ω v E, whr C is indpndnt of, H, v and u (but dpnds on Ω. Stp 4: conclusion: Insrting (37, (38 and (51 in (36, w obtain ( v A pr u n H C v E ( u W 1, (Ω + 2 u C 0 (Ω + C(H + H v E 2 u L 2 (Ω + C H v E u L 2 (Ω, (52 29 (51

30 which yilds th dsird bound (30. his concluds th proof of Lmma Numrical illustrations For our numrical tsts, w considr (1 on th domain Ω = (0, 1 2, with th right-hand sid f(x, y = sin(x sin(y. First tst-cas W first choos th highly oscillatory matrix A (x, y = a (x, y Id 2, a (x, y = cos 2 (150x sin 2 (150y (53 in (1. his matrix cofficint is priodic, with priod = π h 150 rfrnc solution u (computd on a fin msh of Ω is shown on Figur 1. W show on Figur 2 th rlativ rrors btwn th fin scal solution u and its approximation providd by various MsFEM typ approachs, as a function of th coars msh siz H. Our approach is systmatically mor accurat than th standard (maning, without th ovrsampling tchniqu MsFEM approach. In addition, w s that, for larg H, our approach yilds a smallr rror than th othr mthods. Likwis, whn H is small (but not sufficintly small for th standard FEM approach to b accurat, our approach is again mor accurat than th othr approachs. For intrmdiat valus of H, our approach is howvr lss accurat than approachs using ovrsampling (for which w usd an ovrsampling ratio qual to 2. Not that this will no longr b th cas for th problm on a prforatd domain considrd in [25]. Not also that our approach is slightly lss xpnsiv than th approachs using ovrsampling (in trms of computations of th highly oscillatory basis functions, and, mor intrstingly, has no adjustabl paramtr. A comparison with th MsFEM-O variant (s Rmark 5 has also bn prformd but is not includd in th figurs blow. On th particular cas considrd in this articl, w hav obsrvd that this approach sms to prform vry wll. Howvr, it is not clar, in gnral, whthr this approach yilds systmatically mor accurat rsults than th othr MsFEM variants. A mor comprhnsiv comparison of this variant with ours will b mad for th cas of prforatd domains in [25]. 30

31 Figur 1: Rfrnc solution for (1 with th choic (53. Figur 2: st-cas (53: rlativ rrors (in L 2 (lft and H 1 -brokn (right norms with various approachs: FEM th standard Q1 finit lmnts, lin MsFEM with linar boundary conditions, OS MsFEM with ovrsampling, OSPG Ptrov-Galrkin MsFEM with ovrsampling, CR th MsFEM Crouzix-Raviart approach w propos. 31

32 Highr contrast W now considr th cass and A (x, y = a (x, y Id 2, a (x, y = cos 2 (150x sin 2 (150y (54 A (x, y = a (x, y Id 2, a (x, y = cos 2 (150x sin 2 (150y (55 in (1. In comparison with (53, w hav incrasd th contrast by a factor 10 or 100, rspctivly. Rsults ar shown on Figur 3, top and bottom rows rspctivly. W s that th rlativ quality of th diffrnt approachs is not snsitiv to th contrast (at last whn th lattr dos not xcd Of cours, ach mthod provids an approximation of u that is lss accurat than in th cas (53. Howvr, all mthods sm to qually suffr from an highr contrast. Acknowldgmnts. h work of th first two authors is partially supportd by ONR undr Grant N and by EOARD undr Grant FA C h third author acknowldgs th hospitality of INRIA. W thank William Minvill for his suggstions on a prvious vrsion of this articl. Rfrncs [1] J. Aarns, On th us of a mixd multiscal finit lmnt mthod for gratr flxibility and incrasd spd or improvd accuracy in rsrvoir simulation, SIAM MMS, 2(3: , [2] J. Aarns and B.-O. Himsund, Multiscal discontinuous Galrkin mthods for lliptic problms with multipl scals, in Multiscal Mthods in Scinc and Enginring, B. Engquist, P. Lötstdt and O. Runborg, ds., Lctur Nots in Computational Scinc and Enginring, vol. 44, Springr, pp. 1-20, [3] A. Abdull, Multiscal mthod basd on discontinuous Galrkin mthods for homognization problms, C.R. Acad. Sci. Paris, 346(1-2:97-102,

33 Figur 3: st-cass (54 (top row and (55 (bottom row for highr contrasts: rlativ rrors (in L 2 (lft and H 1 -brokn (right norms with various approachs: FEM th standard Q1 finit lmnts, lin MsFEM with linar boundary conditions, OS MsFEM with ovrsampling, OSPG Ptrov- Galrkin MsFEM with ovrsampling, CR th MsFEM Crouzix-Raviart approach w propos. 33

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