A ROBUST NONCONFORMING H 2 -ELEMENT

Transcription

1 MAHEMAICS OF COMPUAION Volum 70, Numbr 234, Pags S (00) Articl lctronically publishd on Fbruary 23, 2000 A ROBUS NONCONFORMING H 2 -ELEMEN RYGVE K. NILSSEN, XUE-CHENG AI, AND RAGNAR WINHER Abstract. Finit lmnt mthods for som lliptic fourth ordr singular prturbation problms ar discussd. W show that if such problms ar discrtizd by th nonconforming Morly mthod, in a rgim clos to scond ordr lliptic quations, thn th rror dtriorats. In fact, a countrxampl is givn to show that th Morly mthod divrgs for th rducd scond ordr quation. As an altrnativ to th Morly lmnt w propos to us a nonconforming H 2 -lmnt which is H 1 -conforming. W show that th nw finit lmnt mthod convrgs in th nrgy norm uniformly in th prturbation paramtr. 1. Introduction Lt Ω R 2 b a boundd polygonal domain and lt Ω dnot th boundary. h purpos of this papr is to discuss finit lmnt mthods for lliptic singular prturbation problms of th form { ε 2 2 u u = f in Ω (1.1) u =0, on Ω. u n =0 Hr is th Laplac oprator, / n dnots th normal drivativ on Ω, and ε is a ral paramtr such that 0 <ε 1. In particular, w ar intrstd in th rgim whn ε is clos to zro. W obsrv that if ε tnds to zro th diffrntial quation in (1.1) formally dgnrats to Poisson s quation. Hnc, w ar studying a plat modl which may dgnrat toward an lastic mmbran problm. Whn fourth ordr problms lik (1.1) ar discrtizd by a finit lmnt mthod th standard variational formulation will rquir function spacs which ar subspacs of th Sobolv spac H 2 (Ω). Hnc, w nd picwis smooth functions which ar globally C 1. Howvr, it is wll known that in ordr to construct C 1 - functions, which ar picwis polynomials with rspct to a givn triangulation of Ω, w ar forcd to us polynomials of dgr fiv or highr. Altrnativly, w can us a macrolmnt tchniqu lik in th Hsih Clough ochr mthod. W rfr to [3, Chaptr 6] for a discussion of ths issus. In ordr to avoid high ordr polynomials or macrolmnts, a common approach is to us nonconforming finit lmnts for such problms, i.., th C 1 -continuity Rcivd by th ditor March 9, 1999 and, in rvisd form, Jun 8, Mathmatics Subjct Classification. Primary 65N12, 65N30. Ky words and phrass. singular prturbation problms, nonconforming finit lmnts, uniform rror stimats. his work was partially supportd by th Rsarch Council of Norway (NFR), undr grant /431, and by ELF Ptrolum Norway AS. 489 c 2000 Amrican Mathmatical Socity

2 490 RYGVE K. NILSSEN, XUE-CHENG AI, AND RAGNAR WINHER rquirmnt is violatd. h simplst nonconforming lmnt for fourth ordr problms is th Morly lmnt. h Morly spac consists of picwis quadratic functions with rspct to a givn triangulation of Ω. h lmnts of this function spac ar not vn continuous functions, but still this spac lads to a convrgnt nonconforming finit lmnt mthod. For discussions on proprtis of th Morly mthod w rfr, for xampl, to [1], [4], [6], and [9]. Howvr, if th Morly mthod is applid to a narly scond ordr problm of th form (1.1) with ε clos to zro, thn th convrgnc rat of th mthod will dtriorat. In fact, if th Morly lmnt is applid to a scond ordr quation lik Poisson s quation thn th mthod will divrg. h main rason for this dgnracy of th Morly mthod is th fact that th finit lmnt spac is not a subspac of H 1 (Ω), or mor prcisly, th Morly spac is not a propr nonconforming finit lmnt spac for scond ordr lliptic quations. his is of cours in contrast to th conforming cas, sinc any subspac of H 2 (Ω) is also a subspac of H 1 (Ω). W will discuss th dgnracy dscribd abov for th Morly mthod in 3. hn, in 4, w will propos an altrnativ nonconforming finit lmnt mthod which is robust with rspct to th paramtr ε. h nw function spac consists of continuous functions which locally blongs to a nin dimnsional subspac of quartic polynomials, constructd by th us of th cubic bubbl function. h global dimnsion of th nw spac, corrsponding to a fixd triangulation and th boundary conditions givn in (1.1), is th sum of th numbr of intrior vrtics and twic th numbr of intrior dgs. As a comparison, th dimnsion of th Morly spac is th th sum of intrior vrtics and dgs. In 5 w will driv propr a priori bounds for th solution of th modl (1.1). hs bounds lad to an rror stimat for th proposd nonconforming mthod which is uniform with rspct to paramtr ε. W should finally mntion that thr is som similarity btwn th study prsntd hr and th rsults of [8], whr finit lmnt mthods for scond ordr singular prturbation problms, dgnrating to a zro ordr problm, ar discussd. 2. Prliminaris h innr product on L 2 = L 2 (Ω) will b dnotd by (, ). For m 0w shall us H m = H m (Ω) to dnot th usual Sobolv spac of functions with partial drivativs of ordr lss than or qual m in L 2, and th corrsponding norm by m. Furthrmor, th notation m,k is usd to indicat that th norm is dfind with rspct to a domain K, diffrnt from Ω. h sminorm drivd from th partial drivativs of ordr qual m is dnotd by m, i.., 2 m = 2 m 2 m 1. h spac H0 m is th closur in Hm of C0 (Ω). Altrnativly, w hav H0 1 = {v H1 : v Ω =0} and H0 2 = {v H2 H0 1 : v n =0on Ω}, whr th rstrictions to Ω ar takn in th sns of tracs. Finally, H m L 2 is th dual of H0 m with rspct to th L2 -innr poduct. W lt Du b th gradint of u and D 2 u =( 2 u/ x i x j ) i,j th 2 2-tnsor of scond ordr partials. In ordr to dfin wak solutions of (1.1) w introduc th bilinar forms (2.1) a(u, v) = D 2 u : D 2 vdx, Ω

3 A ROBUS NONCONFORMING H 2 -ELEMEN 491 whr th colon dnots th scalar product of tnsors, and (2.2) b(u, v) = Du Dv dx. A function u H0 2 is dfind to b a wak solution of (1.1) if (2.3) ε 2 a(u, v)+b(u, v) =(f,v) v H0 2. In fact, in this wak formulation w may simplify th bilinar form a and us a(u, v) = u vdx instad of th on givn by (2.1), sinc (2.4) u vdx= (tracd 2 u)(tracd 2 v) dx = Ω Ω Ω Ω Ω D 2 u : D 2 vdx for all u, v H0 2. Howvr, sinc w will considr nonconforming finit lmnt mthods, this idntity may not hold on th propr finit lmnt spacs, and thrfor w us th form givn by (2.1). It is a consqunc of th rgularity thory for lliptic problms in nonsmooth domains (cf. [5, Corollary ]) that if f H 1 and Ω is convx, thn u H 3, i.., thr is a constant c indpndnt of f such that th corrspondingwak solution u of (1.1) satisfis u 3 c f 1. Howvr, th constant c will in gnral dpnd on ε and will blow up as ε tnds to zro. his issu will b discussd furthr in h Morly mthod Assum that { h } is a quasi-uniform and shap-rgular family of triangulations of Ω, whr th discrtization paramtr h is a charactristic diamtr. W lt X h b th st of vrtics and E h th st of dgs corrsponding to h. h corrsponding Morly finit lmnt spac, M h, consists of all picwis quadratics which ar continuous at ach vrtx of h and such that th normal componnt of th gradint is continuous at th midpoint of ach dg. Furthrmor, in ordr to approximat th boundary conditions in (1.1) th functions in M h ar zro at boundary vrtics and hav zro normal drivativs at th midpoints of all boundary dgs. A function w M h is uniquly dtrmind by th valu of w at ach intrior vrtx and by th valu of th normal componnt of Dw at th midpoint of ach intrior dg; sfigur1. Figur 1. h six dgrs of frdom of th Morly lmnt

4 492 RYGVE K. NILSSEN, XUE-CHENG AI, AND RAGNAR WINHER h finit lmnt approximation u h M h of u is now dtrmind by th linar systm (3.1) ε 2 a h (u, v)+b h (u, v) =(f,v) v M h. Hr th bilinar forms a h and b h ar obtaind from a and b by summing ovr all triangls in h, i.. a h (u, v) = D 2 u : D 2 vdx and h b h (u, v) = h Du Dv dx. Associatd with th bilinar form ε 2 a h + b h, w dfin a sminorm ε,h by w 2 ε,h = ε2 a h (w, w)+b h (w, w). Obsrv that this sminorm is a norm on H0 2 + M h, and, as ε tnds to zro, this nrgy norm approachs a picwis H0 1 -norm. If u H0 2 is th corrsponding wak solution of (1.1), thn th rror u u h can b stimatd in th nrgy norm using th scond Strang lmma (cf. [3, horm 4.2.2]) which stats that E ε,h (u, w) (3.2) u u h ε,h inf u v ε,h + sup, v M h w M h w ε,h whr th consistncy rror E ε,h (u, w) isgivnby E ε,h (u, w) =ε 2 a h (u, w)+b h (u, w) (f,w). Assum that f L 2 and u H 3 H0 2. From th basic stimat (3.2) and by following th approach of [9] to stimat E ε,h (u, w) it is straightforward to obtain an rror stimat of th form ( h 2 u u h ε,h C ε f 0 + h ) (3.3) ε u 3. Not that if ε = 1 this stimat prdicts linar convrgnc with rspct to th msh paramtr h. Howvr, if h is fixd and ε approachs zro, thn th stimat for th rror bhavs lik O(1/ε) (undr th assumption that u 3 is uniformly boundd). h following numrical xampl indicats that this dgnracy is in fact ral. Exampl 3.1. W considr th problm (1.1) with Ω takn as th unit squar and f = ε 2 2 u u, whru(x) =(sin(πx 1 )sin(πx 2 )) 2. h domain is triangulatd by first dividing it into h h squars. hn, ach squar is dividd into two triangls by th diagonal with a ngativ slop. In abl 1 w hav computd th rlativ rror in th nrgy norm, u I h u h ε,h / u I h ε,h, for diffrnt valus of ε and h. Hr u I h dnot th intrpolant of u on M h dfind from th valus of u at ach vrtx and from th valu of th normal componnt of Du on th midpoint of ach dg. For a comparison w also considr th cas ε = 0, i.., th Poisson problm with Dirichlt boundary conditions, and th biharmonic problm 2 u = f. Whn ε is larg, th convrgnc appars to b linar with rspct to h, whil th convrgnc dtriorats as ε approachs zro.

5 A ROBUS NONCONFORMING H 2 -ELEMEN 493 abl 1. h rlativ rror masurd by th nrgy norm ε\h Poisson Biharmonic h rducd problm. Whn ε tnds to zro th problm (1.1) formally approachs a Poisson quation with Dirichlt boundary conditions. Blow w shall giv an analytical argumnt which shows that for such problms th Morly spac will in fact lad to a divrgnt numrical mthod. Hnc, this suggst onc mor that th Morly mthod is not suitabl for problms of th form (1.1) whn th paramtr ε is sufficintly small. W should mntion hr that th divrgnc of th Morly mthod for scond ordr problms has also bn discussd in [7]. In ordr to simplify som calculations blow w shall modify th rducd problm slightly. Instad of th pur Dirichlt problm w considr mixd boundary conditions. W assum that Ω =Γ D Γ N,whrΓ D and Γ N ar disjoint substs of Ω and considr th problm u = f, in Ω, (3.4) u =0 onγ D, u n = g on Γ N. his problm can b considrd to b th formal limit of th fourth ordr problms ε 2 2 u u = f in Ω, u =0 onγ D, (3.5) n (u ε2 u) = g on Γ N u =0 on Ω. As abov lt h b a triangulation of Ω and lt M h b th Morly spac corrsponding to th boundary conditions of (3.4), i.., w assum that th functions in M h ar zro on X h Γ D. h approximation u h M h of th solution u of problm (3.4) is dtrmind by th linar systm (3.6) b h (u h,w)=(f,w)+ g, w for all w M h. Hr g, w = gw ds, Γ N whr s dnots th arc lngth along Ω. Furthrmor, th xact solution u of (3.4) satisfis (3.7) b h (u, w) =(f,w)+ g, w + E h (u, w) for all w M h,

6 494 RYGVE K. NILSSEN, XUE-CHENG AI, AND RAGNAR WINHER whr E h (u, w) =b h (u, w) (f,w) g, w. Lt h = 0,h b th nrgy norm for th rducd problm. From (3.6) and (3.7) w obtain th scond Strang lmma E h (u, w) (3.8) u u h h inf u v h + sup. v M h w M h w h Howvr, th lowr bound E h (u, w) (3.9) u u h h sup w M h w h is also valid. his follows from th Cauchy Schwarz inquality sinc E h (u, w) = b h (u u h,w) u u h h w h. h basic lowr bound (3.9) can b usd to prov th divrgnc of th mthod if w can stablish that th right hand sid of (3.9) dos not tnd to zro with h. W will do this blow for a suitabl choic of th solution u. Howvr, first w will discuss a dcomposition of th Morly spac M h. h function spac M h can naturally b writtn as a sum of two spacs M h = Mh v + M h associatd with th vrtx valus and th dg valus. Mor prcisly, Mh v = {w M w h : n ds =0 forall E h} and Mh = {w M h : w(x) = 0 for all x X h }. For a givn w M h,ltwh v b th corrsponding intrpolant of w on M h v. Sinc this intrpolation procss prsrvs constants locally, w hav (3.10) w wh v L ch Dw L, whr th constant c is indpndnt of h. Obsrv that sinc w w n is linar on ach sid of an dg w must hav that n quals zro at th midpoint of ach dg if w Mh v. Similarly, if w M h,thn w rstrictd to a sid of an dg is a quadratic function which is zro at th two ndpoints. W can thrfor conclud that th tangntial drivativ w s is zro at th midpoint. Also, for any w M h th function w is a constant on ach triangl. In fact, if w Mh v,thn w = 0. his follows sinc w wdx= ds =0. n Furthrmor, th dcomposition of M h is in fact orthogonal with rspct to th bilinar form b h, i.., (3.11) M h = Mh v M h. o s this, not that if w Mh v and φ M h, thn, sinc w is harmonic on ach, w Dw Dφ dx = n φds.

7 A ROBUS NONCONFORMING H 2 -ELEMEN 495 Howvr, on ach dg th function w n φ is a cubic function which is zro at th ndpoints and th midpoint. Hnc, all th boundary intgrals ar zro, which implis b h (w, φ) =0. In fact, th dcomposition is orthogonal locally on ach triangl h. Exampl 3.2. o construct th countrxampl, w lt Ω b th unit squar. h triangulations h ar constructd by dividing Ω into n 2 squars of siz h h, whr h =1/n, and thn dividing ach squar into two triangls using th ngativ slopd diagonals. W assum that Γ D consists of th intrsction of Ω with th coordinat axis, whil w assum Numann boundary conditions on x 1 =1andx 2 = 1. Hnc, th spac M h is assumd to consist of th Morly functions which ar zro at th vrtics on th coordinat axis. W assum that th solution is givn by u(x) =x 1 x 2. Hnc, g = x 1 on x 2 =1 and g = x 2 on x 1 =1. Furthrmoru is harmonic (i.., f = 0). hrfor, (3.12) E h (u, w) =b h (u, w) g, w. Not also that u M h.ltu v h b th intrpolant of u onto M h v. Hnc, u v u v h h(x) =u(x) for all x X h and n ds =0 forall E h. W shall show that lim h 0 E h (u, u v h ) / uv h h is strictly positiv. Du to th lowr bound (3.9) this implis that th mthod divrgs. Lt u h = u uv h.hnu h M h and u h is xactly th intrpolant of u onto M h. Furthrmor, from (3.11) w hav (3.13) b h (u, u) =b h (u v h,u v h)+b h (u h,u h). Sinc Du v h 2 is picwis quadratic, it follows that for any h Du v h 2 dx = Du v h 3 (m) 2, m M( ) whr is th ara of and M( ) dnots th st of th thr dg midpoints (cf. [3, pag 183]). Howvr, th tangntial drivativ u h s at ach dg midpoint is zro. hus u v h / s ar xactly qual to th corrsponding valus for u, whil th normal componnt of Du v h at ach dg midpoint is zro. hrfor, w obtain b h (u v h,uv h2 h )= u 6 s (m) 2. h m M( ) Howvr, sinc u(x) =x 1 x 2, w can vrify that h 2 lim u h 0 6 s (m) 2 h m M( ) = lim { h2 h 0 6 n i=1 j=1 n (ih jh) 2 + h2 3 n i=1 j=1 From this xprssion w obtain that (3.14) lim b h(u v h h 0,uv h )=1/4. n ((ih) 2 +(jh) 2 )}.

8 496 RYGVE K. NILSSEN, XUE-CHENG AI, AND RAGNAR WINHER It is also straightforward to chck that b h (u, u) = Ω u 2 dx =2/3, and hnc (3.13) implis (3.15) lim b h(u h h 0,u h )=5/12. From (3.12) w obtain E h (u, u v h )=b h(u, u v h ) g, uv h = b h(u v h,uv h ) g, uv h (3.16) = b(u, u) g, u v h b h (u h,u h) = g, u u v h b h(u h,u h ). Hnc, w driv from (3.10) and (3.16) that lim E h(u, u v h)= lim b h (u h 0 h 0 h,u h)= 5/12. By using (3.14) this implis that E h (u, u v h lim ) b h (u h h 0 u v h = lim,u h ) h h 0 b h (u v =5/6. h,uv h )1/2 h divrgnc of th mthod is thrfor a consqunc of th basic lowr bound (3.9). 4. Modifications of th Morly lmnt As w hav sn in th discussion abov, th Morly mthod is not wll suitd for solving problms of th form (1.1) whn th positiv paramtr ε is small. h purpos of this sction is to propos an altrnativ nonconforming finit lmnt spac which will lad to a numrical mthod that is robust with rspct to th paramtr ε. For a rviw of othr conforming and nonconforming finit lmnt mthods for fourth ordr problms w rfr to [3, Chaptr 6]. Lt R 2 b a triangl and considr th polynomial spac on givn by W ( )={w P 4 : w P 2 E( )}. Hr P k dnots th st of polynomials of dgr k and E( ) dnots th st of th dgs of. It is a consqunc of Lmma 4.1 that th spac W ( ) can altrnativly b dfind as all functions (4.1) w = q + pb, whr q P 2, p P 1 and b is th cubic bubbl function. W rcall that th cubic bubbl function b is dfind by b = λ 1 λ 2 λ 3,whrλ i (x) ar th barycntric coordinats of x with rspct to th thr cornrs X ( )of. Associatd with an x i X( ), th function λ i P 1 is uniquly dtrmind by λ i (x i )=1, λ i (x) =0 forx X( ),x x i. Furthrmor, 3 λ i (x) 1. i=1 A basis for th spac W ( ), which is usful blow, can b drivd from th following rsult.

9 A ROBUS NONCONFORMING H 2 -ELEMEN 497 Figur 2. h nin dgrs of frdom of th modifid Morly lmnt Lmma 4.1. h spac W ( ) is a linar spac of dimnsion nin. Furthrmor, an lmnt w W ( ) is uniquly dtrmind by th following dgrs of frdom: th valus of w at th cornrs and dg midpoints; w n ds for all E( ). Proof. Sinc any function of th form (4.1) is in W ( )wmusthavdimw() 9. Furthrmor, it is consqunc of th standard Langrangian basis for P 4 ( )(cf. for xampl [2, Chaptr 3]) that if w P 4 ( ), with w 0, thn (4.2) w = pb, whr p P 1 and b is th cubic bubbl function. Assum that w W ( ) is such that th nin dgrs of frdom spcifid in Lmma 4.1 ar all zro. h proof will b compltd if w can show that w 0. Sinc w P 2, with thr roots, w must hav w = 0. hrfor, w is of th form (4.2). Lt b a fixd dg of. Hnc, w = pb = pλ + λ λ, whr λ P 1 is th barycntric coordinat function such that λ 0on, and λ + and λ ar th two othr barycntric coordinats. Not that th quadratic bubbl, b = λ + λ, is strictly positiv in th intrior of. Furthrmor, (Dw) =(pb Dλ ). Not also that λ / n < 0, whr n is th outward unit normal on. hrfor, th condition w n ds = λ pb ds =0 n implis that p must hav a root in th intrior of. Hnc, p P 1 has a root in th intrior of all thr dgs. his implis that p 0, or quivalntly, w 0. As bfor lt h b a quasi-uniform and shap-rgular family of triangulations of Ω. h nw finit lmnt spac W h, associatd with th triangulation h, will consist of continuous functions which ar zro at th boundary, i.., W h C 0 (Ω). In addition w W ( ) for all h, w n ds is continuous for all intrior dgs and zro for boundary dgs. It follows from Lmma 4.1 that any function w W h is uniquly dtrmind by th valus of w at all intrior vrtics and dg midpoints, and by th man valu of w/ n for all intrior dgs (s Figur 2).

10 498 RYGVE K. NILSSEN, XUE-CHENG AI, AND RAGNAR WINHER + Figur 3. Ω = + hs dgrs of frdom also dfins a local intrpolation oprator I h : H 2 W h. Furthrmor, sinc this oprator prsrvs quadratics locally, it follows from a standard scaling argumnt, using th Brambl Hilbrt lmma, that thr is a constant c indpndnt of h such that (4.3) v I h v j, ch k j v k for v H0 2 H k, h whr j =0, 1, 2andk =2, 3. In fact, if ˆ is a rfrnc triangl and Î : H2 ( ˆ ) W ( ˆ ) is th intrpolation oprator with rspct to ˆ, thn for all v H 2 ( ˆ ) Îv 1, ˆ c 1( v L ( ˆ ) + v n 0, ˆ ) c 2 v 1/2 v 1/2 1, ˆ 2, ˆ, whr th constants c 1 and c 2 only dpnd on ˆ. Hr w hav usd th standard trac inquality v 0, ˆ c v 1/2 v 1/2 0, ˆ 1, ˆ (cf. [5, horm ]). Hnc, again from a Brambl Hilbrt argumnt, w obtain (4.4) v I h v 1 ch 1/2 v 1/2 1 v 1/2 2 for v H 2 0. Compard to th stimat (4.3), with j = 1, th nw stimat (4.4) prdicts a lowr ordr convrgnc with rspct to h, but th dpndnc on th function v is wakr. his will b usful blow. Sinc th lmnts of W h ar continuous functions which vanish on th boundary, th inclusion W h H0 1 holds. Howvr, W h is not a subspac of H 2. hrfor, th spac W h again lads to a nonconforming finit lmnt mthod for th fourth ordr problm (1.1). If w W h and is an intrior dg, w lt [ w/ n] dnot th jump of th normal drivativ on. W lt Ω dnot th union of th two triangls + and which hav as common dg (s Figur 3). Obsrv that if w W h and w 2, + + w 2, =0, thn w is linar on Ω.Furthrmor,ifw is linar on Ω,thn[ w/ n] 0. Also, th continuity rquirmnt on W h implis that for any w W h th function [ w/ n] is a cubic polynomial such that (4.5) [ w n ] ds =0.

11 A ROBUS NONCONFORMING H 2 -ELEMEN 499 Hnc, by a standard scaling argumnt (s, for xampl, [2, Sction 8.3]), this implis that thr is a constant c indpndnt of h such that (4.6) φ[ w n ] ds ch φ 1,Ω ( w 2, + + w 2, ) for all φ H 1 and w W h. h stimat (4.6) can b gnralizd such that it is also valid for boundary dgs if w is takn to b zro outsid Ω. Blow w shall nd this stimat whn φ = ψ 2 ψ/ s 2,whrsisaunit tangnt vctor on. In this cas w obtain from (4.6) that ( ψ 2 ψ/ s 2 )[ w n ] ds ch ψ 3,Ω ( w 2, + + w 2, ) for all ψ H 3 and w W h. By summing this stimat ovr all dgs w driv (4.7) ( ψ 2 ψ/ s 2 )[ w n ] ds c h ε ψ 3 w ε,h E h for all ψ H 3,w W h. As abov, w can also obtain a variant of this stimat which is lowr ordr with rspct to h, but which rquirs a wakr dpndnc on th function u. Ifˆ is a rfrnc triangl, and ê is an dg of ˆ,thn φ 0,ê c φ 1/2 φ 1/2 0, ˆ 1, ˆ. hrfor, th stimat (4.6) can b rplacd by φ[ w n ] ds ch 1/2 φ 1/2 0,Ω φ 1/2 1,Ω ( w 2, + + w 2, ) for all φ H 1,w W h. his again lads to th bound (4.8) ( ψ 2 ψ/ s 2 )[ w n ] ds c h1/2 ε ψ 1/2 2 ψ 1/2 3 w ε,h E h for all ψ H 3,w W h, as an altrnativ to (4.7). h finit lmnt solution u h W h is dfind as th solution of (4.9) ε 2 a h (u h,w)+b(u h,w)=(f,w), for all w W h. h following thorm shows that for any fixd ε (0, 1] th nw nonconforming finit lmnt mthod convrgs linarly with rspct to h. Furthrmor, this convrgnc is uniform with rspct to ε if th quantity u 2 + ε u 3 is uniformly boundd. horm 4.2. Assum that th wak solution u of (1.1) is in H0 2 H 3 for a givn f L 2. Furthrmor, lt u h W h b th corrsponding solution of (4.9). hn thr is a constant c, indpndnt of ε and h, such that { (h u u h ε,h c 2 + εh) u 3, h( u 2 + ε u 3 ). Proof. h scond Strang lmma (3.2) is still valid, i.. E ε,h (u, w) (4.10) u u h ε,h inf u v ε,h + sup, v W h w W h w ε,h

12 500 RYGVE K. NILSSEN, XUE-CHENG AI, AND RAGNAR WINHER whr E ε,h (u, w) isgivnby E ε,h (u, w) =ε 2 a h (u, w)+b(u, w) (f,w). Furthrmor, th intrpolation stimat (4.3) implis that { (h inf u v ε,h c 2 + εh) u 3, (4.11) v W h h( u 2 + ε u 3 ). Hnc, it rmains to stimat E ε,h (u, w). Sinc u H 3 it follows from th wak formulation (2.3) and th idntity (2.4) that ( ε 2 D( u)+du) Dw dx =(f,w) w H0 1. Ω In particular, this idntity holds for w W h. h consistncy rror E ε,h (u, w) can thrfor b xprssd as E ε,h (u, w) =ε (4.12) 2 (D 2 u : D 2 w + D u Dw) dx. h h tnsor D 2 u can b writtn as D 2 u =( u)i + C 2 u, whr I is th idntity tnsor and ( 2 u C 2 x u = curl curl u = 2 2 Furthrmor, 2 u x 1 x 2 ( u)i : D 2 w =( u) w 2 u x 1 x 2 2 u x 2 1 ). and thrfor ( u)i : D 2 wdx= u w (4.13) n ds D( u) Dw dx. On th othr hand, sinc ach row of C 2 u is divrgnc fr, w hav div(c 2 u Dw) =C 2 u : D 2 w, and from this w obtain C 2 u : D 2 wdx= n C 2 u Dw ds. Howvr, by combining this idntity with (4.12) and (4.13), and by using th fact that th tangntial componnt of Dw is continuous on ach dg, w obtain E ε,h (u, w) =ε 2 ( u + n C 2 u n)[ w n ] ds E h (4.14) = ε 2 E h ( u 2 u s 2 )[ w n ] ds. It thrfor follows from (4.7) that E ε,h (u, w) chε u 3 w ε,h, and togthr with (4.10) and (4.11) this implis th dsird stimats.

13 A ROBUS NONCONFORMING H 2 -ELEMEN 501 abl 2. h rlativ rror masurd by th nrgy norm ε\h Poisson Biharmonic Not that in th limit as ε tnds to zro th first stimat in horm 4.2 givs th bound u u h 1 ch 2 u 3. In fact, sinc W h is a subst of H0 1, this stimat follows dirctly from (4.3) for th rducd problm { u = f in Ω (4.15) u =0 on Ω. Howvr, if f L 2 and Ω is convx, thn w can only xpct that th solution of th rducd problm (4.15) is in H 2. h scond stimat in horm 4.2 is consistnt with th propr rsult for th rducd problm in this cas. Exampl 4.1. W will rdo th computations w did in Exampl 3.1, th only diffrnc bing that w us th nw finit lmnt spac W h instad of th Morly spac M h. h rsults for th rlativ rror ar givn in abl 2. hs rsults clarly dmonstrat th improvd bhavior of th modifid mthod whn ε is clos to zro. In particular, whn ε is larg th convrgnc appars to b linar with rspct to h, whil w obsrv narly quadratic convrgnc whn ε is small. his is in fact consistnt with horm 4.2, sinc u 3 is indpndnt of ε. Lt us compar th dimnsion of th spac W h with th dimnsion of th Morly spac M h.lt X h b th numbr of intrior vrtics in th triangulation h and lt E h b th corrsponding numbr of intrior dgs. It follows from th discussion abov that dim(w h )= X h +2 E h, whil th dimnsion of th Morly spac M h is X h + E h. Hnc, sinc E h 3 X h, th rplacmnt of M h by W h lads to an incras in th numbr of unknowns of approximatly 75 prcnt. Blow w shall brifly discuss an altrnativ nonconforming finit lmnt spac W h. his spac has a similar robustnss proprty as W h with rspct to paramtr ε, and th dimnsion of th spac is 3 X h + E h, which rprsnts an incras of approximatly 50 prcnt as compard to th Morly spac. On th othr hand, th sparsity structur of th spac W h is mor favorabl than th sparsity structur of th nw spac. If is a triangl, lt W ( )={w P 4 : w P 3 E( )}.

14 502 RYGVE K. NILSSEN, XUE-CHENG AI, AND RAGNAR WINHER It can asily b vrifid, using argumnts as abov, that W ( ) can b quivalntly dfind as all functions (4.16) w = q + pb, whr q P 3, p P 1 and b is th cubic bubbl function. Hnc, W ( ) is a linar spac of dimnsion twlv. h basis givn blow will b usd to dfin a propr finit lmnt spac. Lmma 4.3. h spac W ( ) is a linar spac of dimnsion twlv. Furthrmor, an lmnt w W ( ) is uniquly dtrmind by th following dgrs of frdom: th valus of w and Dw at th cornrs; w n at th midpoint m of E( ). Proof. Sinc dim W ( ) = 12 it is nough to show that th twlv dgrs of frdom dtrmin lmnts of W ( ) uniquly. Assum w W ( ) such that th twlv dgrs of frdom ar all zro. Sinc w P 3, with a doubl zro at ach ndpoint w must hav w = 0. From (4.16) w obtain that w is of th form w = pb, whr p P 1 and b is th cubic bubbl function. Not that (Dw) =(pd(b)) for ach dg, andthat (Db) n)(m ) 0. hrfor, th thr conditions ( w/ n)(m )=0implythatp(m ) = 0. Hnc, p 0, or quivalntly w 0. As abov th polynomial spac W ( ) can b usd to dfin a finit lmnt spac W h consisting of functions which ar locally in W ( ), with w and Dw continuous at ach vrtx, and with w/ n continuous at th midpoint of ach dg. If w W h and E h,thn[ w/ n] is a cubic function which has a root at th two ndpoints and th midpoint. hrfor, th proprty (4.5) holds for all functions in W h. Hnc, by arguing almost xactly as abov, w can also driv a rsult lik horm 4.2 in this cas. 5. h influnc of boundary layrs hroughout this sction w assum that th domain Ω is a convx, boundd polygonal domain in R 2. As in most of th prvious sction w considr th nonconforming finit lmnt mthod (4.9), drivd from th spac W h, approximating th singular prturbation problm (1.1). As w obsrvd abov, horm 4.2 nsurs linar convrgnc with rspct to h, uniformly in ε, aslongasthsminorm u 2 +ε u 3 is uniformly boundd. Howvr, by studying th ffct of boundary layrs in th on dimnsional analogs of (1.1), on will quickly b convincd that th bst on can hop for is that this quantity bhavs lik O(ε 1/2 )asεtnds to zro. In fact, ths on dimnsional analogs admit solutions of th form (5.1) u(x) =ε x/ε p(x), whr p is a cubic polynomial which is boundd indpndntly of ε, chosn such that th Dirichlt boundary conditions hold, and f = p.

15 A ROBUS NONCONFORMING H 2 -ELEMEN 503 In th lmma blow w stat som usful stimats in th two dimnsional cas. Hr, and in th rst of this sction, u = u ε H 2 0 H 3 dnots th wak solution of problm (1.1), whil u 0 H 1 0 H2 is th corrsponding solution of th rducd problm (4.15). Lmma 5.1. hr is a constant c, indpndnt of ε and f, such that u 2 + ε u 3 cε 1/2 f 0 and u u 0 1 cε 1/2 f 0 for all f L 2. Proof. W will stablish th bounds abov by nrgy argumnts. hroughout th proof c will dnot a gnric constant, indpndnt of ε and f, and not ncssarily th sam at diffrnt occurncs. Lt us first rcall th standard stimats for convx domains. It follows from [5, horm ] that, (5.2) u 0 2 c f 0 and, sinc 2 u = ε 2 (u u 0 ), it is a consqunc of [5, Corollary ] that (5.3) u 3 cε 2 (u u 0 ) 1 cε 2 (u u 0 ) 1. Furthrmor, from th wak formulations of th problms (1.1) and (4.15), and th fact that u H0 2 H 3,wdrivthat ε 2 ( u, v)+(d(u u 0 ),Dv)=ε 2 ( u) v Ω n ds for all v H0 1 H 2.Inparticular,bychoosingv = u u 0 w obtain (5.4) ε 2 u u u0 2 1 ε2 ( u) u0 Ω n ds ε2 ( u, f). Howvr, (5.5) ε 2 ( u, f) ε2 2 ( u f 2 0). Furthrmor, standard trac inqualitis and (5.2) imply u0 n 2 ds c f 2 0 and Ω Ω u 2 ds c u 0 u 1. Hnc, from th arithmtic gomtric man inquality w obtain that for any δ>0 thr is a constant c δ such that (5.6) ε 2 ( u) u0 Ω n ds c δε f δε 3 u 0 u 3. Howvr, from (5.3) w driv ε 3 u 0 u ( ε2 u ε 4 u 2 3) (5.7) 1 2 ε2 u c u u h inqualitis (5.4) (5.7) lad to th bound ε 2 u u u0 2 1 cε f 2 0, and togthr with (5.3) this implis th dsird stimats.

16 504 RYGVE K. NILSSEN, XUE-CHENG AI, AND RAGNAR WINHER h rgularity rsult givn in th prvious lmma lads to th following uniform convrgnc proprty for th nonconforming finit lmnt mthod (4.9). horm 5.2. Assum that f L 2 and u H 2 0 H 3 is th corrsponding wak solution of (1.1). Furthrmor, lt u h W h b th solution of (4.9). hn thr is a constant c, indpndnt of ε, h and f, such that u u h ε,h ch 1/2 f 0. Proof. As in th proof of horm 4.2 w start with th basic stmat (4.10), i.., th scond Strang lmma. Hr, and blow, c will dnot a constant indpndnt of ε, h and f. W first show that (5.8) inf u v ε,h u I h u ε,h ch 1/2 f 0. v W h From (4.3) and Lmma 5.1 w obtain ε u I h u 2 cε u 1/2 2 u I h u 1/2 2 cεh 1/2 u 1/2 2 u 1/2 3 ch 1/2 f 0. In ordr to stimat th H 1 -part of th nrgy norm w us th triangl inquality to obtain u I h u 1 u u 0 I h (u u 0 ) 1 + u 0 I h u 0 1. From (4.4), (5.2) and Lmma 5.1 it follows that whil (4.3) and (5.2) givs u u 0 I h (u u 0 ) 1 ch 1/2 u u 0 1/2 1 u u 0 1/2 2 ch 1/2 f 0, u 0 I h u 0 1 ch u 0 2 ch f 0. Hnc, th intrpolation stimat (5.8) is stablishd. Furthrmor, it follows from (4.8) and (4.14) that th consistncy rror, E ε,h (u, w), is boundd by E ε,h (u, w) cεh 1/2 u 1/2 2 u 1/2 3 w ε,h for any w W h. Hnc, Lmma 5.1 implis that E ε,h (u, w) ch 1/2 f 0 w ε,h, and togthr with (4.10) and (5.8) this complts th proof. Rmark. It follows from (5.1) that th Sobolv norm u s will blow up as ε tnds to zro for any s>3/2. hrfor, sinc th nrgy norm ε,h bounds th H 1 -norm, th uniform stimat givn in horm 5.2 sms to b th bst w can possibly obtain for any finit lmnt mthod, vn if w us a mor complx conforming mthod. Acknowldgmnt. h authors ar gratful for th hlp of an anonymous rfr. h proof of Lmma 5.1 is ssntially du to th rfr.

17 A ROBUS NONCONFORMING H 2 -ELEMEN 505 Rfrncs [1] D.N. Arnold and F. Brzzi, Mixd and nonconforming finit lmnt mthods: implmntation, postprocssing and rror stimats, RAIRO Mod. Math. Anal. Num. 19 (1985), pp MR 87g:65126 [2] S.C. Brnnr and L.R. Scott, h mathmatical thory of finit lmnt mthods, Springr Vrlag MR 95f:65001 [3] P.G. Ciarlt, h finit lmnt mthod for lliptic problms, North-Holland Publishing Company, MR 58:25001 [4] R.S. Falk and M.E. Morly, Equivalnc of finit lmnt mthods for problms in lasticity, SIAM J. Num. Anal. 27 (1990), pp MR 91i:65177 [5] P. Grisvard, Elliptic problms on nonsmooth domains, Monographs and studis in mathmatics vol. 24, Pitman Publishing Inc., MR 86m:35044 [6] P. Lascaux and P. Lsaint, Som nonconforming finit lmnt for th plat bnding problm, RAIRO Sér. Roug Anal. Numér. R-1 (1975), pp MR 54:11941 [7] M. Wang, h ncssity and sufficincy of th patch tst for th convrgnc of nonconforming finit lmnts, Prprint, Dpt. of Math., Pnn. Stat Univrsity (1999). [8] A.H. Schatz and L.B. Wahlbin. On th finit lmnt for singularly prturbd ractiondiffusion problms in two and on spac dimnsions, Math.Comp. 40 (1983), pp MR 84c:65137 [9] Z. Shi, Error stimats of Morly lmnt, Math. Numr. Sinica 12 (1990), pp MR 91i:65182 Dpartmnt of Mathmatics, Univrsity of Brgn, Johanns Brunsgt. 12, 5007 Brgn, Norway addrss: rygv.nilssn@mi.uib.no Dpartmnt of Mathmatics, Univrsity of Brgn, Johanns Brunsgt. 12, 5007 Brgn, Norway addrss: Xu-Chng.ai@mi.uib.no Dpartmnt of Informatics and Dpartmnt of Mathmatics, Univrsity of Oslo, P.O. Box 1080 Blindrn, 0316 Oslo, Norway addrss: ragnar@ifi.uio.no