A C 0 INTERIOR PENALTY METHOD FOR A FOURTH ORDER ELLIPTIC SINGULAR PERTURBATION PROBLEM

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1 A C 0 INERIOR PENALY MEHOD FOR A FOURH ORDER ELLIPIC SINGULAR PERURBAION PROBLEM SUSANNE C. BRENNER AND MICHAEL NEILAN Abstract. In tis papr, w dvlop a C 0 intrior pnalty mtod for a fourt ordr singular prturbation lliptic problm in two dimnsions on polygonal domains. Using som a postriori rror analysis tcniqus, w ar abl to sow tat t mtod convrgs in t nrgy norm uniformly wit rspct to t prturbation paramtr undr minimal rgularity assumptions. In addition, w analyz t convrgnc of t numrical solution to t unprturbd scond ordr problm. Finally, w prform som numrical xprimnts tat back up t tortical rsults. Ky words. Intrior pnalty mtod, singular prturbation problms, fourt ordr AMS subjct classifications. 65N30, 65N1, 35B5 1. Introduction. In tis papr, w construct and analyz a C 0 intrior pnalty IP mtod for t following fourt ordr problm: ε v ε A v ε = ϕ in Ω, 1.1a v ε = v ε = 0 on Ω. 1.1b Hr, Ω R is a convx polygonal domain, ε is a small, positiv paramtr, ϕ L Ω, and A = Ax [W 1, Ω]. Furtrmor, A is symmtric and positiv dfinit in t sns tat tr xist constants Λ, λ > 0 suc tat Λ w H 1 Ω A w w dx λ w H 1 Ω w H0 1 Ω. 1. Ω Problm 1.1 modls a simply supportd plat problm wit v ε rprsnting t displacmnt of t plat and ε bing t ratio of bnding rigidity to tnsil stiffnss in t plat [17]. Additionally, suc problms aris wn studying t linarization of a fourt ordr prturbation of t fully nonlinar Mong-Ampèr quation [8]. W not tat t corrsponding problm for t clampd plat was invstigatd in [17, 18, 16, 1,, ]. In ordr to avoid ig ordr conforming finit lmnt mtods [17, 9], nonconforming finit lmnt mtods ar an attractiv option. Sinc t lading trm in 1.1a is t biarmonic oprator, t Morly lmnt is appaling sinc it as t last numbr of dgrs of frdom on ac lmnt for fourt ordr problms, as its basis functions consist of only quadratic polynomials [15, 19]. Unfortunatly, t Morly lmnt is not convrgnt for gnral scond ordr lliptic problms [3], and trfor, t rror dtriorats as ε 0 + as was sown in [16]. As a rsult, svral variants of t Morly lmnt mtod av bn proposd for 1.1a wit A = I, wic itr nric t Morly finit lmnt spac [16] or modify t finit lmnt formulation [1, ]. Dpartmnt of Matmatics and Cntr for Computation and cnology, Louisiana Stat Univrsity, Baton Roug, LA work of tis autor was supportd by t National Scinc Foundation undr Grant No. DMS brnnr@mat.lsu.du work of tis autor was supportd by t National Scinc Foundation undr Grant No. DMS nilan@mat.lsu.du 1

2 S. C. BRENNER AND M. NEILAN In tis papr, w tak a diffrnt approac, and formulat and study a symmtric C 0 IP mtod [10, 6] for 1.1. Similar to t Morly lmnt, w us picwis quadratic polynomials to construct our finit lmnt spac. Howvr, unlik any of t aformntiond mtods, w do not nd to modify t spac or modify t finit lmnt formulation. Furtrmor, t finit lmnts w us com in a natural irarcy, ar simpl to implmnt, and radily availabl on commrcial softwar. Finally, as ε 0 +, t mtod convrgs to t standard finit lmnt mtod for t unprturbd problm 1.3 blow. Du to t inqualitis 1., problm 1.1 is boundd and corciv on H Ω H0 1 Ω, and trfor, tr xists a uniqu wak solution v ε H Ω H0 1 Ω. Howvr unlik t cas of t clampd plat, vn on a convx domain, t bst tat can b said about t rgularity of v ε is tat v ε H +γ Ω for som γ 0, 1 wic dpnds on t angls of t cornrs of Ω [11, 4]. at is, v ε dos not ncssarily blong to H 3 Ω. W sall rfr to γ as t indx of lliptic rgularity. W discuss tis issu in furtr dtail as wll as dvlop a priori bounds in t subsqunt sctions cf. orms On of t main goals of tis papr is to rigorously prov tat t C 0 IP mtod is robust wit rspct to ε, tat is, to driv rror stimats tat ar indpndnt of ε. Anotr focal point of tis papr is t convrgnc of v ε v and v v ε, wr vε is t solution to t C 0 IP mtod, and v is t solution to t scond ordr problm A v = ϕ in Ω, 1.3a v = 0 on Ω. 1.3b Du to 1., v is uniquly dfind. Furtrmor, sinc ϕ L Ω, A [W 1, Ω], and Ω is convx, tr olds v H Ω wit [11] v H Ω C ϕ L Ω. 1.4 papr is organizd as follows. In Sction, w provid gnral notation tat will b usd in t subsqunt sctions and dfin t finit lmnt mtod for 1.1. In Sction 3, w us a postriori tcniqus to stablis som tcnical lmmas, wic w latr us to driv abstract a priori rror stimats in t nrgy norm. In Sction 4, w driv a priori stimats of t solution v ε and dtrmin t rats of convrgnc of v ε v in various norms. Sction 5 contains t main rsults of t papr. Hr, w combin t rsults of Sctions 3 and 4 to prov stimats of v ε v ε in t nrgy norm tat ar indpndnt of t paramtr ε, as wll as driv stimats of t rror v v ε wit rspct to bot ε and. Finally in Sction 6, w provid numrical xprimnts wic confirm t tortical rsults. A tcnical proof concrning t construction of a macro bubbl function is found in t Appndix.. Notation and t C 0 Intrior Pnalty Mtod. rougout t papr, w us H k Ω k 0 to dnot t st of all L Ω functions wos distributional drivativs up to ordr k ar in L Ω, and H k 0 Ω to dnot t st of H k Ω functions wos tracs vanis up to ordr k 1 on Ω. In ordr to dfin t wak solution to 1.1, w st av, w := D v, D w, bv, w := A v, w,.1

3 A C 0 IP mtod for a fourt ordr singular prturbation problm 3 wr, dnots t L innr product ovr Ω and D v, D w = Ω D v : D w dx = i,j=1 Ω v x i x j w x i x j dx. W tn dfin a function v ε V 0 := H Ω H 1 0 Ω to b a solution to 1.1 if εav ε, w + bv ε, w = ϕ, w w V 0.. Nxt, w lt b a rgular simplical triangulation [9, 5] of t domain Ω wit = diam. Lt E i dnot t st of intrior dgs in, E b t st of boundary dgs in, and st E = E i E b. For any E i tr ar two triangls + and suc tat = +. St ω = +, }, and lt n b t unit normal of pointing from + to. For any w H + H, dfin t avrag and jump of t normal drivativ of w on as follows: n w }} = 1 n w + + n w, [[ n w ]] = n w + n w. Hr, n w := w n and w ± = w ±. Similarly, w dfin t jump of t gradint of w across by [[ w ]] = w + w. Nxt, for w H 3 + H 3, dfin t avrag and jump of t scond ordr normal drivativ of w on, and t jump of t Laplacian of w on by nn w }} = 1 nn w + + n n w [[, nn w ]] = n n w + n n w, [[ ]] w = w + w, wr n n w := D wn n. For E b, w dfin [[ nn w ]] = n n w. W dnot by V t Lagrang finit lmnt spac consisting of globally continuous picwis quadratic polynomials and lt V 0 = V H 1 0 Ω. For a positiv ral numbr σ, w dfin t following bilinar form on H 3 : a w, z : = D w, D z nn w }}, [[ n z ]].3 E i + [[ n w ]], nnz }} σ 1 [[ n w ]], [[ n z ]] }, wr = diam and, rsp., dnots t L innr product ovr rsp.. C 0 IP mtod for 1.1 is to find v ε V 0 suc tat εa v ε, w + bv ε, w = ϕ, w w V 0..4

4 4 S. C. BRENNER AND M. NEILAN W dfin t discrt norms,, a,, and ε, as follows: w, : = w H, w a, : = w, + E i 1 w ε, : = ε w a, + w H 1 Ω. [[ n w ]] L, Rmark.1. r xists σ a > 0 dpnding only on t sap rgularity of suc tat for σ σ a [6] rfor by.5 and 1., tr olds 1 w a, a w, w w V 0..5 C w ε, ε w a, + λ w H 1 Ω εa w, w + bw, w w V 0..6 It tn follows tat tr xists a uniqu solution to.4 providd σ σ a. Rmark.. It can radily b cckd tat t numrical mtod.4 is consistnt. at is, if v ε H 3 Ω H 1 0 Ω is t solution to 1.1 tn v ε satisfis.4 wit v ε rplacd by vε. Howvr, as alrady mntiond in t Introduction, v ε dos not ncssarily blong to t spac H 3 Ω. As a rsult, standard finit lmnt analysis tcniqus for driving rror stimats i.. Ca s Lmma cannot b applid dirctly. Rmark.3. Intrior pnalty mtods for t corrsponding clampd plat problm wit ε = 1 wr proposd and studid in.g. [3, 13, 14, 7]. As t solution as H 3 Ω rgularity in tis cas, t analysis is rlativly simplr tan t simply supportd plat problm 1.1, and it is straigt-foward to sow tat t rsults of orm 3.1 blow old for t clampd plat problm. 3. Convrgnc Analysis. As mntiond in t prvious sction, w cannot us Ca s Lmma to driv rror stimats of t C 0 IP mtod.4 du to t low rgularity of v ε. o ovrcom tis difficulty, w us som of t main idas of [1] to driv our main rsult of tis sction, orm 3.1, assuming only v ε H Ω H0 1 Ω. As a first stp, w us som a postriori tcniqus [1, 0] to stablis t following tcnical lmma. Lmma 3.1. Suppos tat ϕ H m Ω and A [W m+1, Ω] for som m 0. n for any w V0 tr olds ϕ + A w L C ε 4 vε w H + vε w H m ϕ Hm + w H 1, [[ nnw ]] L C ω 1 vε w H + ε v ε w H ε m+3 ϕ H m + w H 1 } E, [[ ]] A w n C ε 3 L vε w H + 1 vε w H ω

5 A C 0 IP mtod for a fourt ordr singular prturbation problm 5 + m+1 ϕ Hm + } w H 1 E. i Proof. For, coos ϕ P m 1 if m = 0, w tak ϕ = 0 and Ā [P m ] suc tat ϕ ϕ L C m ϕ H m, A Ā L + A Ā W 1, C m A W m+1,. 3.5 Lt b P 6 H0 b t bubbl function dfind on suc tat b quals on at t barycntr of, tat is b = 7λ 1, λ λ3 i wr λ }3 i=1 dnot t barycntric coordinats of. Nxt, dfin r H0 Ω suc tat r = b ϕ + Ā w, and xtnd r to b zro outsid of. First, w not tat sinc ϕ + Ā w is a polynomial, tr xists a constant C > 0 indpndnt of suc tat C ϕ + Ā w ϕ + Ā w, r L. 3.6 Nxt, using standard invrs stimats [9, 5], 3.6,., and , w av for any w V 0 C ϕ + Ā w L ϕ + Ā w, r = ϕ ε w + Ā w, r = ϕ ϕ, r + Ā A w, r + ε D v ε w, D r + A v ε w, r C ϕ ϕ L + A Ā L w H + A Ā W 1, w H1 + ε vε w H + 1 vε w H1 r L C m ϕ Hm + w H1 + ε vε w H + 1 vε w H1 ϕ + Ā w. L Hr, w av usd t fact 0 b 1 to driv t last inquality. Dividing t last xprssion by ϕ + Ā w L, and using , and t triangl and invrs inqualitis, w gt ϕ + A w L ϕ + Ā w L + A Ā w L C ϕ + Ā w L + A Ā W 1, + 1 A Ā L w H1 C m ϕ Hm + w H1 + ε vε w H + 1 vε w H 1. stimat 3.1 follows from tis last inquality.

6 6 S. C. BRENNER AND M. NEILAN Nxt lt E b an arbitray dg. If E i wit = +, w construct ζ 1 P 1 + and ζ P 8 + suc tat [1]: ζ 1 = 0, n ζ 1 = [[ nnw ]], ζ ds, ζ + \ = 0, ζ + \ = 0, + ζ dx = + +. Otrwis if E b wit = + Ω, w construct ζ 1 P 1 + and ζ P 4 + suc tat ζ 1 = 0, n ζ 1 = [[ nnw ]], ζ ds, ζ + \ = 0, ζ + \ = 0, + ζ dx = +. Intgrating by parts, w av by. Cε [[ nnw ]] ε [[ L nnw ]], n ζ 1 ζ = ε [[ nnw ]], n ζ1 ζ + A [[ w ]] n, ζ 1 ζ = ε D w, D ζ 1 ζ ω + A w, ζ 1 ζ + A w, ζ 1 ζ = ε D w v ε, D ζ 1 ζ + A w v ε, ζ 1 ζ C ω } + ϕ + A w, ζ 1 ζ ε v ε w H ζ 1 ζ H + v ε w H 1 ζ 1 ζ H 1 ω By scaling, w av [1] and by t Poincaré inquality + ϕ + A w L ζ 1ζ L ζ 1 ζ H 1 ± C 1 [[ nn w ]] L, ζ 1 ζ L ± C ζ 1 ζ H1 ±. us, by t invrs inquality and 3.1, w av ε [[ nnw ]] L C ε 1 vε w H + v ε w H 1 ω + ϕ + A w L }. } } ζ 1 ζ H 1 ±

7 A C 0 IP mtod for a fourt ordr singular prturbation problm 7 C C It tn follows tat [[ nnw ]] L C ω ε 1 vε w H + v ε w H1 + m+1 } ϕ H m + w H 1 1 [[ nn w ]] L ε 1 vε w H + 1 v ε w H1 ω + m+ 3 ω } [[ ϕ H m + w H 1 nn w ]] L. 1 vε w H + ε v ε w H 1 + ε m+3 ϕ Hm + } w H 1. W now prov 3.3. o aciv tis, for givn = + E i, w construct χ 1 P m+1 + suc tat χ 1 = Ā[[ w ]] n, and xtnd χ 1 to + by constants along t normal dirctions of. Nxt, w lt χ b t macro bubbl function wit t following proprtis cf. Appndix A: χ = 0, χ + \ = 0, + \ n χ = 0, χ ds, χ L + 1. W tn av by. and by intgration by parts, C [[ Ā w ]] n Ā [[ w ]] n L, χ 1 χ = ε [[ nnw ]], n χ 1 χ + Ā [[ w ]] n, χ 1 χ = ε D w, D χ 1 χ + Ā w, χ 1 χ + Ā w, χ 1χ ω = ω ε D w v ε, D χ 1 χ + A w v ε, χ 1 χ + Ā A w, χ 1χ + ϕ + Ā w, χ 1χ us, by t invrs and Poincaré inqualitis, 3.1, and 3.5, [[ Ā w ]] n L C ε vε w H + 1 vε w H 1 By scaling, w av ω + 1 A Ā L w H 1 + ϕ + Ā w } L χ 1 χ L C ε vε w H + 1 vε w H 1 ω + m } ϕ Hm + w H 1 χ 1 χ L. χ 1 χ L χ 1 L χ L C 1 Ā [[ w ]] n L. }. }

8 8 S. C. BRENNER AND M. NEILAN It tn follows tat Ā [[ w ]] n L C and trfor, ω + m+1 ε 3 vε w H + 1 vε w H 1 ϕ H m + } w H 1, [[ ]] A w n L [[ Ā w ]] n L + A Ā [[ w ]] n L C ε 3 vε w H + 1 vε w H 1 ω + m+1 ϕ H m + w H 1 C ω + m+1 ε 3 vε w H + 1 vε w H 1 ϕ H m + } w H 1. } + A Ā L ω [[ w ]] L Bfor continuing, w discuss som proprtis of t nricing oprator, E : V V c, wr V c H Ω H 1 0 Ω is t Hsi-Cloug-ocr finit lmnt spac [9, 5] associatd wit. W sktc som of t main proprtis of tis linar oprator and rfr t radr to t paprs [6, 7] for a mor dtaild xposition. construction of E is don by avraging. o dscrib tis procss, w lt N b a global dgr of frdom of V c, tat is, N is itr t valuation of t sap function at a vrtx in, t valuation of t first ordr drivativs of a sap function at an intrior vrtx in, t valuation of t normal drivativ of a sap function at t midpoint of an dg in E, or t valuation of t normal drivativ of a sap function at any vrtx on t boundary Ω. n for any w V, w construct E w V c so tat NE w = 1 N Nw, N wr N dnots t st of triangls tat sar t dgr of frdom, and N is t cardinality of tis st. For any, t map E satisfis t stimat [7] w E w L C4 = C 4 1 p V E i p 1 E i [[ n w ]] L 3.7 [[ n w ]] L, wr V dnots t st of vrtics of, E i p is t st of all intrior dgs in E saring t ndpoint p, and cf. Figur 3.1 E i = p V E i p.

9 A C 0 IP mtod for a fourt ordr singular prturbation problm 9 Fig triangls in t st and t dgs in t st E i dasd lins. Using a trac and invrs inquality, it follows from 3.7 tat w E w L C4 [[ n w ]] 3.8 L C 4 E i 1 E i ω w H 1 C w H 1, wr cf. Figur 3.1 = E i ω. Nxt, using t trac and invrs inqualitis again, w conclud tat ε 1 n w E w }} C ε 4 L w E w L 3.9 E E ω C ε 1 [[ n w ]] L E ω E i C w ε,, and by 3.8 and 3.7 n w E w }} C L E E C E ω C w ε,, ε nnw E w }} C L E E C E E i C w ε,. w E w L 3.10 ω ω w H 1 ε 4 w E w L 3.11 ε 1 ω E i [[ n w ]] L Using similar argumnts, w conclud tat by 3.8 w av 1 w E w L C w E w L 3.1 E i C E i and by 3.7, w av ε 3 w E w L E i E i ω ω ω w H 1 w ε,, ε 4 w E w L 3.13

10 10 S. C. BRENNER AND M. NEILAN E i ω w ε,. ε 1 [[ n w ]] L E i Finally, from 3.7 and t trac and invrs inqualitis, w av E w ε, C w ε, w V Lmma 3.. For all q H Ω, z V c, and w V, tr olds εaq, z εa w, z + bq w, z C q w ε, z ε, Proof. For any z V c and q H Ω, tr olds [[ n z ]] = [[ n q ]] = 0 E i. rfor by.1,.3, t Caucy-Scwarz inquality and trac/invrs inqualitis, w av εaq, z εa w, z + bq w, z = ε D q w, D z ε [[ n q w ]], nnz }} + bq w, z E i ε q w, z, + A L Ω q w H 1 Ω z H 1 Ω + ε 1 [[ n q w ]] 1/ L nn z }} L E i E i C q w ε, z ε,. 1/ Wit ts rsults stablisd, and wit t lp of Lmma 3.1, w can sow t following rsult. tru powr of Lmma 3.3 will b xposd in orm 3.1. Lmma 3.3. Lt v ε dnot t solution to 1.1, and lt v ε dnot t solution to.4, and suppos tat ϕ H m Ω, A [W m+1, Ω] for som m 0. W tn av [ v ε v ε ε, C1 + σ inf ε v ε w H 3.16 w V 0 + ε v ε w H 1 + m+1 w H1 + ϕ H m + E i ε 1 [[ n v ε w ]] L [ v ε v ε ε, C1 + σ inf w V 0 + v ε w H 1 + m+1 + E i ε 1 [[ n v ε w ]] L ] 1, }, ε + ε v ε w H 3.17 w H 1 + ϕ H m } ] 1.

11 A C 0 IP mtod for a fourt ordr singular prturbation problm 11 Proof. Lt w V0 suc tat w v ε, and to as notation st z = vε w. n using similar argumnts as in [1, Lmma.1], w av by.6,.4,., 3.15, and 3.14 C v ε w ε, εa v ε w, z + bv ε w, z = ϕ, z εa w, z bw, z = εav ε, E z εa w, E z + bv ε w, E z + ϕ, z E z εa w, z E z bw, z E z C v ε w ε, E z ε, + ϕ, z E z εa w, z E z bw, z E z C v ε w ε, v ε w ε, + ϕ, z E z εa w, z E z bw, z E z. It tn follows from tis xprssion and t triangl inquality tat [ ] v ε v ε ε, C inf v ε G w, z w ε, + sup, 3.18 w V0 z V0 \0} z ε, wr G w, z := ϕ, z E z εa w, z E z bw, z E z. Nxt, to as notation, st s = z E z, so tat Intgrating by parts, w av D w, D s = G w, z = ϕ, s εa w, s bw, s = D w, s + n w, s } nn w, n s + nτ w, τ s }, wr τ dnots t unit tangnt vctor of. Sinc E zp = zp for any vrtx p in, it follows tat D w, D s = nn w }}, [[ n s ]] + [[ nn w ]], n s }}. 3.0 E i By t dfinitions of t bilinar forms a, and b, and by 3.0, w av εa w, s + bw, s = A w, s + ε [[ n D w, D s ε w ]], nns }} E i + nnw }}, [[ n s ]] [[ σ 1 n w ]], [[ n s ]] } = A w, s ε [[ n w ]], nns }} E i εσ 1 [[ n w ]], [[ n s ]] A [[ w ]] n, s } E

12 1 S. C. BRENNER AND M. NEILAN + E ε [[ nnw ]], n s }}, and trfor by 3.19 and 3.11, G w, z = ϕ + A w, s + ε [[ n w v ε ]], nns }} 3.1 εσ 1 E i [[ n w v ε ]], [[ n s ]] A [[ w ]] n, s } E ε [[ nnw ]], n s }} 1 + σ v ε w ε, z ε, + + ε [[ nnw ]] L n s }} L E ϕ + A w L s L + E i A [[ w ]] n L s L. Hr, w av usd t fact v ε H Ω so tat [[ n v ε]] = 0, E i. W now bound t last tr xprssions in t rigt-and sid of 3.1. Combining 3.1 and , w av ϕ + A w L s L 3. C m+1 C m+1 ϕ Hm + w H1 + v ε w H1 z H ε v ε w H 1 [[ n z ]] 1 } L E i ϕ Hm + 1 w H 1 + v ε w ε, z ε,. Nxt, applying 3. and 3.9, w gt [[ nn w ]], n s }} 3.3 E ε ε [[ nnw ]] L E C ε [[ nnw ]] L E [ C 1 ε 1 E 1 z ε, ε v ε w H + ε 1 v ε w H 1 + ε 1 m+ n s }} L ϕ H m + w H 1 }] 1 z ε,. 1

13 A C 0 IP mtod for a fourt ordr singular prturbation problm 13 W also av by 3.10 and 3. [[ nn w ]], n s }} 3.4 E ε 1 ε [[ nnw ]] 1 L E C [ C 1 ε [[ nnw ]] L E + m+1 E 1 z H 1 Ω ε vε w H + vε w H 1 ϕ Hm + w H 1 }] 1 z ε,. n s }} L Continuing, w us 3.3 and 3.1 to conclud [[ ]] A w n, s E i [[ ]] A w n E i L C A [[ w ]] n E i C + m+1 1 L E i 1 s L 1 z H 1 ε vε w H + vε w H 1 ϕ H m + } w H 1 z ε,, and by 3.13, w av [[ ]] A w n, s E i E i ε 1 3 A [[ w ]] n L 1 E i ε 3 C ε 1 3 [[ ]] A w n z L ε, E i C ε v ε w H + ε 1 v ε w H 1 + ε 1 m+ 1 ϕ H m + } w H 1 z ε,. s L Finally, t stimat 3.16 is obtaind by combining 3.18, 3.1, and t stimats 3., 3.3, and 3.6. Similarly, t stimat 3.17 is obtaind by combining 3.18, 3.1, and t stimats 3., 3.4, and

14 14 S. C. BRENNER AND M. NEILAN orm 3.1. Suppos tat ϕ H m Ω, A [W m+1, Ω] and t ms is quasiuniform wit. n tr olds v ε v ε ε, C1 + σ inf w V0 v ε w ε, + m+1 w H 1 Ω + ϕ H Ω m. 3.7 Proof. If ε 1, tn by 3.17, w av [ ε v ε v ε ε, C1 + σ inf 1 + ε 1 v ε w, w V0 + v ε w H 1 Ω + m+1 w H 1 Ω + ϕ H m + ε 1 [[ n v ε w ]] 1 ] L E i C1 + σ inf v ε w ε, + m+1 w H1 Ω + ϕ H m. w V0 On t otr and, if ε 1, tn by 3.16 w av [ v ε v ε ε, C1 + σ inf ε 1 v ε w, w V0 + ε v ε w H1 Ω + m+1 w H1 Ω + ϕ H m + E i ε 1 [[ n v ε w ]] L 1 ] C1 + σ inf v ε w ε, + m+1 w H 1 Ω + ϕ H m. w V0 4. A Priori Estimats and Convrgnc as ε 0 +. Sinc t rror stimat 3.1 will vntually dpnd on t norms of v ε, w must driv a priori bounds of ts quantitis in ordr to obtain uniform convrgnc of t finit lmnt mtod wit rspct to ε. orm 4.1 A Priori Estimats I. Lt v ε b t solution to 1.1 and lt v dnot t solution to 1.3. n t following old: v ε H Ω C ϕ L Ω, 4.1 v ε v L Ω Cε ϕ L Ω, 4. v ε v H 1 Ω Cε 1 ϕ L Ω. 4.3 Morovr, v ε H +γ Ω for som γ 0, 1 and v ε H +γ Ω Cε γ ϕ L Ω. 4.4 Proof. Sinc v H Ω, w av for any w V 0 ε D v ε v, D w + A v ε v, w = εd v, D w.

15 A C 0 IP mtod for a fourt ordr singular prturbation problm 15 Stting w = v ε v in t xprssion abov, w us 1. to conclud ε v ε v H Ω + λ vε v H 1 Ω ε v H Ω v ε v H Ω. Assuming λ ε, w av by 1.4 v ε v H Ω C v H Ω C ϕ L Ω, v ε v H 1 Ω Cε 1 v H Ω Cε 1 ϕ L Ω, and trfor, v ε H Ω C ϕ L Ω. 4.5 o obtain t stimat 4., w lt φ H Ω H0 1 Ω b t solution to ε φ A φ = v ε v in Ω, 4.6 φ = φ = 0 on Ω. W tn av by 4.1 φ H Ω C v ε v L Ω. 4.7 W also av by 4.6, 1.1, 1.3, and 4.7 v ε v L Ω = εd v, D φ ε v H Ω φ H Ω Cε v H Ω v ε v L Ω Cε ϕ L Ω v ε v L Ω. Dividing t last xprssion by v ε v L Ω, w obtain 4.. W now sow 4.4. o tis nd, w not tat v ε = ε 1 A v ε v. Sinc A [W 1, Ω] and v ε, v H Ω, tr olds A v ε v L Ω. us by lliptic tory for t simply supportd plat, tr xists a γ 0, 1 wic w call t indx of lliptic rgularity, suc tat v ε H +γ Ω and By , w av v ε H +γ Ω Cε 1 A v ε v H +γ Ω 4.8 Cε 1 v ε v Hγ Ω. v ε v L Ω Cε ϕ L Ω, v ε v H 1 Ω Cε 1 ϕ L Ω, and trfor, by intrpolation btwn Sobolv spacs, It tn follows from tat v ε v H γ Ω Cε γ ϕ L Ω. 4.9 v ε H +γ Ω Cε γ ϕ L Ω.

16 16 S. C. BRENNER AND M. NEILAN Nxt, undr assumption tat t solution to t unprturbd scond ordr problm as igr rgularity, w ar abl to improv som of t stimats of orm 4.1. orm 4. A Priori Estimats II.. Suppos tat v H 3 Ω. n tr olds v ε v H 1 Ω Cε 3 4 v H 3 Ω, 4.10 v ε v H Ω Cε 1 4 v H 3 Ω, 4.11 v ε H +γ Ω Cε γ 4 ϕ L Ω + v H 3 Ω. 4.1 Proof. Intgrating by parts, w av ε D v ε v, D w + A v ε v, w = εd v, D w = ε v, w ε v, n w Ω. Stting w = v ε v and using trac inqualitis givs us ε v ε v H Ω + λ vε v H 1 Ω ε v L Ω v ε v L Ω + v L Ω n v ε v L Ω Cε v ε v H1 Ω + v ε v 1 H 1 Ω vε v 1 H Ω v H 3 Ω. us by Young s inquality, w av ε v ε v H Ω + λ vε v H 1 Ω C ε + ε 3 v H3 Ω Cε 3 v H3 Ω. stimats follow from tis last inquality. Nxt, by intrpolation btwn Sobolv spacs, v ε v H γ Ω Cε 4 γ 4 ϕ L Ω + v H Ω 3 γ 0, 1. It tn follows from 4.8 tat v ε H +γ Ω Cε γ 4 ϕ L Ω + v H 3 Ω. Rmark 4.1. As Ω is a polygonal domain, w cannot xpct tat v H 3 Ω in gnral. Howvr, t rsults of orm 4. xplain som of t numrical xprimnts in Sction Convrgnc Analysis Rvisitd. In tis sction, w combin t rror analysis drivd in orm 3.1 wit t a priori bounds dvlopd in t prvious sction. First, w driv stimats of t rror v ε v ε, wic w summariz in t following torm. orm 5.1. Lt γ 0, 1 dnot t indx of lliptic rgularity. n tr olds v ε v ε ε, C1 + σ γ ϕ L Ω. 5.1

17 A C 0 IP mtod for a fourt ordr singular prturbation problm 17 Proof. Lt I v ε dnot t standard finit lmnt intrpolant of v ε. By 3.7, 4.1 and 4.4, w av v ε v ε ε, C1 + σ ε 1 v ε I v ε a, + v ε I v ε H 1 Ω + I v ε H 1 Ω + ϕ L Ω C1 + σ ε 1 γ v ε H +γ Ω + v ε H Ω + ϕ L Ω C1 + σ ε 1 γ γ + ϕ L Ω C1 + σ γ ϕ L Ω. Nxt, w turn our attntion to dvloping stimats of v v ε. Of cours, t most natural approac is to combin t stimats drivd in orm 5.1 wit t convrgnc rats drivd in orms and us t triangl inquality. Howvr, a bttr stratgy is to us t stimat 3.7 dirctly and avoid t lliptic rgularity of v ε. Furtrmor, by considring t discrt vrsion of 1.3, and wit t lp of t invrs inquality, w ar also abl to trad-in powrs of for powrs of ε. orm 5.. r olds v v ε ε, C1 + σε 1 + ϕ L Ω, 5. v v ε H 1 Ω C + ε σ ϕ L Ω. 5.3 Morovr, if v H 3 Ω, ϕ H 1 Ω, and A [W, Ω], tn v v ε ε, C1 + σ ε ϕ H1 Ω + v H3 Ω, 5.4 v v ε H 1 Ω C v H 3 Ω + ε σ ϕ L Ω, 5.5 v v ε a, C v H 3 Ω + ε 1 + σ ϕ L Ω. 5.6 Proof. Lt I v dnot t finit lmnt intrpolant of v. By 3.7, 1.4, and t triangl inquality, w av v ε v ε ε, 5.7 C1 + σ v ε I v ε, + I v H1 Ω + ϕ L Ω C1 + σ v ε I v ε, + v I v H1 Ω + v H1 Ω + ϕ L Ω C1 + σ v ε v ε, + v I v ε, + ϕ L Ω. By standard intrpolation stimats and 1.4, w av v I v ε, Cε 1 + v H Ω Cε 1 + ϕ L Ω. 5.8 Morovr, by 4.1, 1.4, and 4.3, v ε v ε, Cε 1 ϕ L Ω. 5.9

18 18 S. C. BRENNER AND M. NEILAN Applying t bounds to t inquality 5.7, and using t triangl inquality, w obtain v v ε ε, v ε v ε, + v ε v ε ε, C1 + σε 1 + ϕ L Ω. In a similar fasion, if v H 3 Ω, ϕ H 1 Ω, and A [ W, Ω ], tn by 3.7 w av v ε v ε ε, C1 + σ v ε I v ε, + I v H 1 Ω + ϕ H Ω C1 + σ v ε v ε, + v I v ε, + ϕ H 1 Ω. W also av by and standard intrpolation stimats and trfor, v I v ε, C ε 1 + v H3 Ω, v ε v ε, Cε 3 4 v H 3 Ω, v v ε ε, v ε v ε, + v ε v ε ε, C1 + σ ε ε + ϕ H1 Ω + v H3 Ω Nxt, w lt v dnot t solution to C1 + σ ε ϕ H 1 Ω + v H 3 Ω. b v, w = ϕ, w By standard finit lmnt analysis tcniqus, w av and trfor, v I v H 1 Ω C v I v H 1 Ω, w V v v H1 Ω C v H Ω C ϕ L Ω. 5.1 Also, by t invrs inquality tr olds v a, C 1 v I v H1 Ω + I v a, C v H Ω C ϕ L Ω Furtrmor, if v H 3 Ω, tn v v H 1 Ω C v H 3 Ω, 5.14 v v a, C 1 v I v H1 Ω + v I v a, 5.15 C 1 v I v H 1 Ω + v v H 1 Ω + v I v a, C v H3 Ω. From t dfinition of v ε and v and by Rmark.1, w av C v ε v ε, εa v ε v, v ε v + bv ε v, v ε v = εa v, v ε v Cε1 + σ v a, v ε v a,.

19 A C 0 IP mtod for a fourt ordr singular prturbation problm 19 us by t invrs inquality, w obtain v ε v ε, Cε1 + σ v a, v ε v a, Cε σ v a, v ε v H 1 Ω Cε σ v a, v ε v ε,. rfor, dividing by v ε v ε, and using 5.13, w av and nc, v ε v ε, Cε σ v a, Cε σ ϕ L Ω, v ε v H1 Ω Cε σ ϕ L Ω, v ε v a, Cε σ ϕ L Ω. rfor, by t triangl inquality and 5.1 v v ε H1 Ω v v H1 Ω + v ε v H1 Ω C + ε σ ϕ L Ω, and if v H 3 Ω, by 5.14 w av v v ε H1 Ω C v H3 Ω + ε σ ϕ L Ω. Also, if v H 3 Ω, tn by t invrs inquality and 5.15 v v ε a, C 1 v v ε H1 Ω + v v a, C v H 3 Ω + ε 1 + σ ϕ L Ω. Rmark 5.1. By t proof of orm 5., v ε v ε ε, C1 + σ ε 1 + ϕ L Ω, 5.16 and if ϕ H 1 Ω, v H 3 Ω, and A [ W, Ω ], tn v ε v ε ε, C1 + σ ε ϕ H1 Ω + v H3 Ω Corollary 5.1. Suppos tat = ε α for som α > 0. n tr olds v v ε ε, C1 + σ ϕ L Ω ε α α 1 ε 1 α 1, 5.18 and if v H 3 Ω, ϕ H 1 Ω, and A W, Ω, tn v v ε H1 Ω C1 + σ ϕ H1 Ω + v H3 Ω ε α α 3 8 ε 3 4 α 3 8, 5.19 v v ε a, C1 + σ ε α α 1 3 ϕ H1 Ω + v H3 Ω ε 1 α 1 3 α ε 1 4 α 3 8

20 0 S. C. BRENNER AND M. NEILAN Proof. stimat 5.18 is a dirct consqunc of t rror stimat 5.. If v H 3 Ω, ϕ H 1 Ω, and A [ W, Ω ] tn by v v ε H 1 Ω C1 + σ min ε ε α, ε α + ε 1 α} ϕ H 1 Ω + v H 3 Ω } C1 + σ ε α + min ε 3 4, ε 1 α ϕ H 1 Ω + v H Ω 3. If α 1 4, tn v v ε H1 Ω C1 + σ ε α + ε 1 α ϕ H1 Ω + v H3 Ω Otrwis, if 1 4 α 3 8, tn and if α 3 8, tn v v ε H1 Ω C1 + σ v v ε H1 Ω C1 + σ C1 + σε α ϕ H 1 Ω + v H 3 Ω. ε α + ε 3 ϕ H1 4 Ω + v H3 Ω C1 + σε α ϕ H 1 Ω + v H 3 Ω, ε α + ε 3 ϕ H1 4 Ω + v H3 Ω C1 + σε 3 4 ϕ H1 Ω + v H3 Ω, wic stabliss Nxt, by 5.4 and 5.6, w av v v ε a, C1 + σ min ε ε α 1, ε α + ε 1 α} ϕ H 1 Ω + v H Ω 3. If α 1 3, tn min ε ε α 1, ε α + ε 1 α} } min ε α 1, ε α = ε α. If 1 3 α 3 8, tn min ε ε α 1, ε α + ε 1 α} } min ε α 1, ε 1 α = ε 1 α, and if α 3 8, tn min ε ε α 1, ε α + ε 1 α} } min ε 1 4, ε 1 α = ε 1 4. stimat 5.0 follows from ts last tr inqualitis. 6. Numrical Exprimnts. In tis sction, w provid svral numrical xprimnts tat back up t tortical findings in t prcding sctions. All of t tsts prsntd blow ar on t domain Ω = 0, 1 and w st A = I so tat 1.1a and 1.3a rad, rspctivly ε v ε v ε = ϕ in Ω, v = ϕ in Ω. Furtrmor, w rplac t omognous Diriclt boundary conditions 1.1b, 1.3b wit v ε Ω = g, v Ω = g.

21 A C 0 IP mtod for a fourt ordr singular prturbation problm st 1. For tis tst, w calculat t rror v v ε for fixd =6E-3, wil varying ε. W coos our data so tat t xact solution is v = x + y. W solv.4 wit =6E-3, and list t rrors and rats of convrgnc in abl 6.1. Not tat by 5.4, for ε E-6, w av v v ε ε, C1 + σ ε ϕ H 1 Ω + v H 3 Ω C1 + σε 3 4 ϕ H1 Ω + v H3 Ω. is bavior is rflctd in abl 6.1, wr t scond column sows tat v v ε H 1 Ω = Oε 3 4 and t tird column sows tat v v ε, = Oε 1 4. Also, sinc v H3 Ω = 0, t stimats rad v v ε H1 Ω Cε σ ϕ L Ω, v v ε a, Cε 1 + σ ϕ L Ω. rfor, as ε 0 + and for fixd, w xpct to s linar convrgnc wit rspct to ε in all norms. Again, tis bavior is sn in t scond and tird columns of abl 6.1. abl 6.1 st 1. Error v v ε wit rspct to ε. =6E-3, σ = 0. ε v v ε L Ω rat v v ε H 1 Ω rat v v ε, rat 1.00E E E-01.85E E E E E E E E E E-04.93E E E E E E E E E E E E-06.55E E E E E E E E E E E E-08.05E E E E E E E E-09 8.E E E E E E E st. For tis tst, w fix a rlation = 4ε α wit α = 1, α = 1 3, and α = 1 4 and stimat t rat of convrgnc of t rror v vε wit rspct to ε. W st y x = 0, ϕ = 3x + y + x y + x 3 x x, g = x y = 0, 1 + y x = 1, 1 + x x y = 1, so tat t xact solution to t scond ordr problm 1.3 is v = x + y x.

22 S. C. BRENNER AND M. NEILAN By , w av v v ε H 1 Ω C1 + σ ε 1 α = 1 4 ϕ H 1 Ω + v H 3 Ω ε 3 α = 1 3 ε 3 4 α = 1, v v ε, C1 + σ ε 1 4 α = 1 4 ϕ H 1 Ω + v H 3 Ω ε 1 3 α = 1 3 ε 1 4 α = 1 All of ts xpctd rats of convrgnc ar obsrvd in abls abl 6. st a. Error v v ε wit rspct to ε. = 4ε 1, σ = 0. ε v v ε L Ω rat v v ε H 1 Ω rat v v ε, rat 1.00E E E E E+00.00E E-01.06E E E E E E E E E E E E E E E-0.11E E E abl 6.3 st b. Error v v ε wit rspct to ε. = 4ε 1 3, σ = 0. ε v v ε L Ω rat v v ε H 1 Ω rat v v ε, rat 1.00E E E E E E-04.3E E E E E E E E E E E-0.05E E E E-07.3E-0.06E E E abl 6.4 st c. Error v v ε wit rspct to ε. = 4ε 1 4, σ = 0. ε v v ε L Ω rat v v ε H 1 Ω rat v v ε, rat 1.00E-05.5E E E-0.06E E E-01.4E E E E E-0 4.5E E E E E E E E E-09.5E E E E

23 A C 0 IP mtod for a fourt ordr singular prturbation problm 3 REFERENCES [1] M. Ainswort and J.. Odn, A Postriori Error Estimation in Finit Elmnt Analysis, Wily-Intrscinc Jon Wily & Sons, 000. [] G. Awanou, Robustnss of a splin lmnt mtod wit constraints, J. Sci. Comput., , pp [3] G.A. Bakr, Finit lmnt mtods for lliptic quations using nonconforming lmnts, Mat. Comp., , pp [4] H. Blum and R. Rannacr, On t boundary valu problm of t biarmonic oprator on domains wit angular cornrs, Mat. Mtods Appl. Sci., 1980, pp [5] S.C. Brnnr and L.R. Scott, Matmatical ory of Finit Elmnt Mtods, tird dition, Springr, 008. [6] S.C. Brnnr and L.-Y. Sung, C 0 intrior pnalty mtods for fourt ordr lliptic boundary valu problms on polygonal domains, J. Sci. Comput., /3 005, pp [7] S.C. Brnnr,. Gudi, and L.-Y. Sung, An a postriori rror stimator for a quadratic C 0 intrior pnalty mtod for t biarmonic problm, IMA J. Numr. Anal., , pp [8] S.C. Brnnr,. Gudi, M. Nilan, and L.-Y. Sung, C 0 pnalty mtods for t fully nonlinar Mong-Ampèr quation, Mat. Comp., to appar. [9] P.G. Ciarlt, Finit Elmnt Mtod for Elliptic Problms. Nort-Holland, Amstrdam, [10] G. Engl, K. Garikipati,.J.R. Hugs, M.G. Larson, L. Mazzi, and R.L. aylor, Continuous/discontinuous finit lmnt approximations of fourt ordr lliptic problms in structural and continuum mcanics wit applications to tin bams and plats, and strain gradint lasticity, Comput. Mt. Appl. Mc. Eng., , pp [11] P. Grisvard, Elliptic Problms on Nonsmoot Domains, Pitman Publising Inc., [1]. Gudi, A nw rror analysis for discontinuous finit lmnt mtods for linar lliptic problms, Mat. Comp., , pp [13] I. Mozolvski and E. Süli, A priori rror analysis for t p-vrsion of t discontinuous Galrkin finit lmnt mtod for t biarmonic quation, Comput. Mtods Appl. Mat., 3 003, pp [14] I. Mozolvski, E. Süli, and P.R. Bösing, p-vrsion a priori rror analysis of intrior pnalty discontinuous Galrkin finit lmnt approximations to t biarmonic quation, J. Sci. Comput., , pp [15] L.S.D. Morly, triangular quilibrium lmnt in t solution of plat bnding problms, Aro. Quart., , pp [16].K. Nilssn, X.C. ai, and R. Wintr, A robust nonconforming H -lmnt, Mat Comp., , pp [17] B. Smpr, Conforming finit lmnt approximations for a fourt-ordr singular prturbation problm, SIAM J. Numr. Anal., 9 199, pp [18] B. Smpr, Locking in finit-lmnt approximations to long tin xtnsibl bams, IMA J. Numr. Anal., , pp [19] Z. Si, Error stimats of Morly lmnt, Mat. Numr. Sinica, , pp [0] R. Vrfürt A Postriori Error Estimation and Adaptiv Ms-Rfinmnt cniqus, Wily-ubnr, Cicstr, [1] M. Wang, J. Xu, and Y. Hu, Modifid Morly lmnt mtod for a fourt ordr lliptic singular prturbation problm, J. Comput. Mat., 4 006, pp [] M. Wang and Y. Hu, A robust finit lmnt mtod for 3-D lliptic singular prturbation problm, J. Comput. Mat., 5 007, pp [3] M. Wang, ncssity and sufficincy of t patc tst for t convrgnc of nonconforming finit lmnts, SIAM J. Numr. Anal., , pp Appndix A. Construction of Macro Bubbl Function. Lmma A.1. For any E i wit = +, tr xists a macro bubbl function χ suc tat χ + \ = 0, χ + \ = 0, n χ = 0, χ L + 1, χ ds.

24 4 S. C. BRENNER AND M. NEILAN Proof. Witout loss of gnrality, w may assum tat t dg is orizontal. First, w considr t cas wr + is t triangl wit vrtics 0, 0, 0.5, a, 1, 0, and is t triangl wit vrtics 0, 0, 0.5, a, 1, 0, wr a is som fixd positiv numbr. Hnc, is t lin sgmnt wit vrtics 0, 0 and 1, 0. W tn dfin χ P 8 + to b χ = y + ax y ax + a y + ax a y ax. A.1 It is asy to vrify tat χ vaniss up to t first ordr on + \ and tat n χ = 0, χ 0, χ L + = a 8. Nxt, w considr t gnral cas wr + = ABC, = ACD, and is t lin sgmnt AC s Figur A.1. Lt M b t midpoint of dg, and lt E rsp. F b t point on BC rsp. AD suc tat EM rsp. F M is prpndicular to AC. Witout loss of gnrality, w may assum tat EM F M. W tn st G to b t midpoint of EM and st H to b t point on t sgmnt F M suc tat HM = GM. On ACG ACH, w tak χ to b t scald vrsion of χ dfind by A.1. W not tat by construction χ vaniss up to t first ordr on AG, CG, CH, AH, and n χ = 0. On ac of t triangls AF H, CDH, BCG, and ABG, w tak χ to b t polynomial of dgr six so tat χ vaniss up to t first ordr on AF, F H, AH, CD, DH, CH, BC, CG, and BG, and χ quals on at t cntr of ac of t triangls. It follows from tis construction tat χ C 1 +, χ vaniss up to first ordr on + \, n χ = 0, and χ 0. Furtrmor, by scaling χ L + 1 and χ ds. Fig. A.1. Partition of macro bubbl function