ETNA Kent State University

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1 Elctronic Transactions on Numrical Analysis Volum 41, pp , 2014 Copyrigt 2014, Knt Stat Univrsity ISSN ETNA Knt Stat Univrsity ttp://tnamatkntdu A UNIFIED ANALYSIS OF THREE FINITE ELEMENT METHODS FOR THE MONGE-AMPÈRE EQUATION MICHAEL NEILAN Abstract It was rcntly sown in S C Brnnr t al [Mat Comp, , pp ] tat Lagrang finit lmnts can b usd to approximat classical solutions of t Mong-Ampèr quation, a fully nonlinar scond ordr PDE W xpand on ts rsults and giv a unifid analysis for many finit lmnt mtods satisfying som mild structur conditions in two and tr dimnsions Aftr proving som abstract rsults, w lay out a bluprint to construct various finit lmnt mtods tat inrit ts conditions and sow ow C 1 finit lmnt mtods, C 0 finit lmnt mtods, and discontinuous Galrkin mtods fit into t framwork Ky words fully nonlinar PDEs, Mong-Ampèr quation, finit lmnt mtods, discontinuous Galrkin mtods AMS subjct classifications 65N30, 65N12, 35J60 1 Introduction In tis papr, w considr t finit lmnt approximations of t fully nonlinar Mong-Ampèr quation wit xact solution u and Diriclt boundary conditions [14, 15, 28, 32]: 11 { f dtd 0 Fu : 2 u, inω, u g, on Ω Hr, dtd 2 u dnots t dtrminant of t Hssian matrix D 2 u, Ω R n n 2,3 is a two or tr dimnsional, smoot, strictly convx domain, and f is a strictly positiv function T goal of tis papr is to build and analyz various numrical mtods to approximat classical convx solutions of t Mong-Ampèr quation In particular, w dvlop an abstract framwork to construct discrtizations, dnotd by F, of t nonlinar oprator F in ordr to build finit lmnt approximations ofu Tis work is motivatd by t rcnt rsults of Brnnr t al [10, 11] wr t autors dvlopd and analyzd Galrkin mtods on smoot domains using t wll-known and simpl Lagrang finit lmnts In ordr to build convrgnt mtods, t autors constructd consistnt numrical scms suc tat t rsulting discrt linarization is stabl As mpasizd in [10], tis simpl ida lads to an intricat drivation of a not-so-obvious discrtization In tis papr, w xpand on ts rsults and giv a unifid analysis of many finit lmnt mtods in two and tr dimnsions wic satisfy som gnral structur conditions Furtrmor, w lay out a simpl bluprint to construct suc scms wit ts proprtis T ky ida, as in [10], is to build consistnt and stabl in trms of t linarization discrtizations Assuming a fw mor mild conditions on t numrical scm, w us a rlativly simpl fixd-point argumnt to prov t xistnc of a solution providd t discrtization paramtr is sufficintly small Furtrmor, t rror stimats rduc to rror stimats of an auxiliary linar problm T proof of ts rsults ar t main objctivs of Sctions 2 and 4 Smingly powrful, t abstract tory dos not xplicitly tll us ow to build suc stabl and consistnt scms As prviously mntiond, t construction of t Lagrang finit Rcivd Jun 18, 2012 Accptd Jun 20, 2014 Publisd onlin on Sptmbr 30, 2014 Rcommndd by A Klawonn T work of t autor was supportd by t National Scinc Foundation undr grant numbr DMS Dpartmnt of Matmatics, Univrsity of Pittsburg, Pittsburg, PA nilan@pittdu 262

2 Knt Stat Univrsity ttp://tnamatkntdu ANALYSIS OF FINITE ELEMENT METHODS FOR THE MONGE-AMPÈRE EQUATION 263 lmnt mtod in [10] involvs a convolutd drivation, and it is not immdiatly clar ow to xtnd tis mtodology to mor gnral scms Howvr, intuitivly w xpct tat t discrt linarization sould b consistnt wit t linarization of F a justification of tis assrtion is givn in Rmark 25 blow Wit tis in mind, to construct discrtization scms F tat satisfy t conditions prsntd in Sctions 2 and 4, w i considr t linarization of F, ii discrtiz t linarization, iii d-linariz to obtain discrtizations for F Using tis mtodology, in Sction 3 w ar abl to systmatically dvlop various numrical mtods including C 1 finit lmnt mtods, C 0 finit lmnt mtods, and discontinuous Galrkin DG mtods In t first cas, w rcovr som spcial cass of t rsults of Bömr [7, Torm 87] and driv H 2 rror stimats; s also [8] In addition, w driv L 2 and H 1 rror stimats wic ar nw in t litratur In t cas of C 0 finit lmnt mtods, w rcovr t rsults in [10], but again, in a mor compact and systmatic approac T dvlopmnt and analysis of DG mtods is compltly nw, and as far as t autor is awar, tis is t first tim DG mtods av bn itr dvlopd or analyzd for fully nonlinar scond ordr quations Du to tir important rol in many application aras suc as diffrntial gomtry and optimal transport [15, 44, 45], tr as bn a growing intrst in rcnt yars towards dvloping numrical scms for fully nonlinar scond ordr quations Hr, w giv a brif rviw in tis dirction T first numrical mtod for t Mong-Ampèr quation is du to Olikr and Prussnr [39] wo constructd a numrical scm for computing an Alksandrov masur inducd by D 2 u and obtaind t solution of problm 11 as a by-product Mor rcntly, Obrman [26, 38] constructd t first practical wid stncil diffrnc scms for nonlinar lliptic PDEs wic can b writtn as functions of ignvalus of t Hssian matrix, suc as t Mong-Ampèr quation It was provd tat t finit diffrnc scm satisfis t convrgnc critrion consistncy, stability, and monotonicity stablisd by Barls and Souganidis [4], altoug no rats of convrgnc wr givn T clar advantag of tis mtod is t ability to comput t convx solution vn if u is not smoot Howvr, t mtod also suffrs from low rats of convrgnc, vn if t solution is smoot Obrman t al as also constructd a simpl finit diffrnc mtod in two dimnsions in [5], altoug no convrgnc analysis was prsntd Dan and Glowinski [19] prsntd an augmntd Lagrang multiplir mtod and a last squars mtod for t Mong-Ampèr quation by trating t nonlinar quations as a constraint and using a variational principl to slct a particular solution T convrgnc analysis of t scm rmains opn Bömr [7] s also [8] introducd a projction mtod using C 1 finit lmnt functions for classical solutions of gnral fully nonlinar scond ordr lliptic PDEs including t Mong-Ampèr quation and analyzs t mtods using consistncy, stability and linarization argumnts As far as t autor is awar, tis is t first convrgnc proof of fully nonlinar scond ordr problms in a finit lmnt stting Fng and t autor considrd fourt ordr singular prturbations of 11 by adding a small multipl of t biarmonic oprator to t PDE [24, 25] Otr rlvant paprs includ [2, 20, 34, 35, 43, 46] Tr ar svral advantags of t C 0 and DG mtods compard to t mtods mntiond abov As xplaind in [10], t advantags of ts scms includ t rlativly simplicity of t mtod and t ability to asily implmnt t mtod wit svral finit lmnt softwar packags Tis tn allows on to us faturs suc as fast solvrs multigrid and domain dcomposition and adaptivity off-t-slf Furtrmor, in comparison to finit diffrnc mtods, bot typs of finit lmnt mtods can asily andl curvd boundaris wit littl modification of t finit lmnt cod Morovr, t advantags of t DG mtod for linar and mildly nonlinar problms carry ovr for t Mong-Ampèr quation as wll Ts includ t as of implmntation spcially in t contxt of

3 Knt Stat Univrsity ttp://tnamatkntdu 264 M NEILAN p-adaptivity, t ability to asily andl inomognous boundary conditions and curvd boundaris, t ability to us igly nonuniform unstructurd mss wit anging nods, and t ability to accuratly captur socks and discontinuitis of t function and gradint Admittdly tr ar not many advantags of using C 1 finit lmnts du to tir computational complxity; w includ tis spcific cas to sow tat it fits witin our framwork and to simplify som of t analysis in [7] Tr ar svral notions of solutions for t Mong-Ampèr quation including classical, viscosity, and Alkandrov solutions [18, 32, 33, 42, 45] Solutions in gnral ar not smoot Howvr, if Ω, g, and f ar sufficintly rgular, Ω is strictly convx, and f is uniformly positiv in Ω tn all of ts notions of solutions ar quivalnt, and t rgularity of u follows from t rsults of Caffarlli, Nirnbrg, and Spruck [16] T mtods dvlopd in tis papr ar usful for svral applications in diffrntial gomtry suc as t prscribd Gauss curvatur quation and t affin maximal surfac quation [2, 28, 29, 30, 31, 44] In rgard to optimal transport, it is important to b abl to comput wak solutions Sinc t analysis blow rlis on t smootnss of u, it is not immdiatly obvious ow to xtnd t rsults to t cas of nonclassical solutions On way to ovrcom tis issu is to us t mtods dscribd blow in conjunction wit t vanising momnt mtodology [24, 25] Tis simpl procdur involvs computing a fourt ordr prturbation of 11 by adding a small multipl of t biarmonic oprator to t fully nonlinar PDE Numrical xprimnts in [10, 11, 24, 25] indicat tat tis is a powrful tool not only to comput wak solutions, but it also provids an avnu to obtain good initial gusss to start Nwton s mtod Unlik t discrtizations in [24, 25], t mtods in tis papr ar dsignd to b stabl witout rgularization As suc, w xpct our mtods in conjunction wit t vanising momnt mtod will b mor robust wit rspct to t prturbation paramtr Finally, w mntion tat tr ar svral discrtization mtods for t biarmonic problm using C 1, C 0, and DG spacs; s, g, [3, 13, 17, 22, 27, 36] Trfor, t formulation of our mtods wit t vanising momnt mtod is asily obtaind by adding a small multipl of ts biarmonic discrtizations to t discrt nonlinar oprator F Howvr, t convrgnc analysis of t rgularizd discrtization is byond t scop of tis articl T rsults blow sow tat tr xists a solution to t numrical scms providd tat t discrtization paramtr is sufficintly small, namly, 0 for som 0 0,1 T xact valu of 0 is not addrssd in tis papr and in gnral is not known a priori as it dpnds on t t xact solution u Tis issu as wll as t ffct of numrical rrors of t scms will b studid in futur work T rst of t papr is organizd as follows Sctions 2 and 3 ar dvotd to t dvlopmnt and analysis of finit lmnt mtods in two dimnsions In Sction 2 w stat and prov som abstract rsults, sowing tat tr xists a solution to t discrt problm if som conditions on t scm old W nd tis sction by driving L 2 stimats using a standard duality argumnt Aftr stting som notation, in Sction 3 w apply t abstract framwork to tr xampls, namly, C 1 finit lmnt mtods, C 0 finit lmnt mtods, and DG mtods In Sction 4, w xpand on t rsults of Sction 2 by driving som abstract convrgnc rsults in t tr dimnsional cas Finally, w nd t articl wit som concluding rmarks and discuss possibl xtnsions 2 Abstract rsults in two dimnsions Trougout t papr, w us H r Ω r 0 to dnot t st of all L 2 Ω functions wos distributional drivativs up to ordr r ar in L 2 Ω, andh r 0Ω to dnot t st of functions wos tracs vanis up to ordrr 1 at Ω For a normd linar spac Y, w dnot by Y its dual and, t pairing btwn Y and Y Lt X, X b a finit dimnsional spac suc tat t inclusion X L 2 Ω

4 Knt Stat Univrsity ttp://tnamatkntdu ANALYSIS OF FINITE ELEMENT METHODS FOR THE MONGE-AMPÈRE EQUATION 265 olds W considr t following discrt vrsion of 11: find u X suc tat 21 F u,v 0, v X, wr F : X X is a smoot oprator W mak t following assumptions: a1 a Tr xists an auxiliary normd linar spac Y, Y wit X Y and u Y suc tat X is wll dfind ony b T oprator F can b xtndd to a smoot oprator F : Y X wit Fw,v F w,v, w,v X c Tr xists a constant α > 0, wic may dpnd on, suc tat t following invrs stimat olds: 22 v Y α v X, v X a2 a T nonlinar opratorf is consistnt witf in t sns tatfu 0 b F can b dcomposd as 23 F F 2 +F 1 +F 0, wr F 2 is quadratic i, F 2 tw t 2 F 2 w for all w Y and t R, F 1 is linar, and F 0 is constant in tir argumnts a3 Dfin t linar oprator 24 wr Fu+tw Fu Lw : DF[u]w : lim F 1 w+df 2 [u]w, t 0 t 25 DF 2 F 2 u+tw F 2 u [u]w : lim t 0 t Not tat, by dfinition of L and F, w av L : Y X Tn tr ar constants β,c cont > 0 suc tat β v 2 X L v,v, v X, Lw,v Ccont w X v X, w Y, v X, wr L : X X dnots t rstriction of L to X, tat is, L w,v Lw,v for allv,w X a4 Tr xists a constant γ > 0 suc tat for allv,w Y DF 2 [v]w X γ v Y w Y, wr DF 2 [v]w DF 2 [v]w,y X : sup y X y X

5 Knt Stat Univrsity ttp://tnamatkntdu 266 M NEILAN THEOREM 21 Suppos tat assumptions a1 a4 ar satisfid Ltu c L 1 Lu X, i, L u c,v Lu,v, v X 28 Assum furtr tat 29 u u c Y τ 0 β 2α γ for som τ 0 0,1 Tn tr xists a locally uniqu solution u X to 21 Morovr, tr olds 210 u u X u u c X + 1 α u u c Y, u u Y 2 u u c Y W prov Torm 21 using t Banac fixd-point torm as our main tool T ssntial ingrdints of t proof of Torm 21 is to construct a mapping suc tat i t mapping is a contraction in a subst in our cas, a ball B ρ wit radius ρ of X ; ii t mapping maps tis ball into itslf Bot of ts rsults ar drivd by t following lmma LEMMA 22 Suppos tat conditions a1 a4 old Dfin t mapping M : Y X by 211 Mw L 1 Lw Fw Tn for any v,w Y, w av 212 Mw Mv X γ 2β u w Y + u v Y w v Y Proof By t dcomposition 23, w avfw Fv F 2 w F 2 v+f 1 w v, wr w av usd t proprty tat F 1 is a linar oprator Morovr, by 24 tr olds Lw v F 1 w v+df 2 [u]w v Consquntly, w av Lw v Fw Fv DF 2 [u]w v F 2 w F 2 v DF 2 [u]w v 1 0 DF 2 [tw +1 tv]w vdt SincF 2 is smoot and quadratic, t mappingw,v DF 2 [w]v is bilinar It tn follows tat Lw v Fw Fv [ 213 DF 2 u 1 ]w 2 w +v v Trfor, by t dfinition of M 211 along wit t idntity 213, w arriv at Mw Mv L 1 Lw v Fw Fv L 1 DF [u 2 12 ] w +v w v

6 Knt Stat Univrsity ttp://tnamatkntdu ANALYSIS OF FINITE ELEMENT METHODS FOR THE MONGE-AMPÈRE EQUATION 267 By t stability stimat in a3, w asily obtain Mw Mv X β 1 [ DF 2 u 1 2 w +v] w v X, and trfor by a4, w av Mw Mv X γ 2β u w Y + u v Y w v Y Proof of Torm 21 Dfin t closd discrt ball wit cntr u c as 214 B ρ u c {v X ; u c v X ρ}, and lt M : X X b t rstriction of M to X T proof procds by sowing tat M as a fixd point in a B ρ0 u c wit ρ 0 : 1 α u u c Y By t dfinition of M, 211, w clarly s tat tis fixd point is a solution to 21 First, by 212, a1c, 214, and t dfinition ofρ 0, w av for allv,w B ρ0 u c, M w M v X γ 2β u w Y + u v Y w v Y α γ β u uc Y +α ρ 0 w v X 2α γ β u u c Y w v X Hnc, by 29 w obtain 215 M w M v X τ 0 w v X for som τ 0 0,1 Nxt, it is clar from 28, 211, and t consistncy off tatu c Mu Trfor by 212, a1c, t dfinition of ρ 0, and 29, w av for any w B ρ0 u c, 216 uc M w X Mu Mw X γ u w 2 Y 2β γ β u uc 2 Y + u c w 2 Y γ β u uc 2 Y +α 2 ρ 2 0 2α γ β u u c Y ρ 0 ρ 0 From 215 and 216 it tn follows tat M as a uniqu fixd point u in t ball B ρ0 u c wic is a solution to 21 Also, by t triangl inquality w av u u X u u c X +ρ 0 u u c X + 1 α u u c Y Finally, to prov 210, w us t triangl inquality onc again, t invrs stimat 22, and t dfinition of ρ 0 to gt u u Y u u c Y +α ρ 0 2 u u c Y COROLLARY 23 Suppos tat assumptions a1 a4 old and dfin C : 1 + C cont /β Tn, tr olds providd u u X inf v X 2C u v X + 1 α u v Y, u u Y 2 inf v X u v Y +α C u v X, inf u v Y +α C u v X τ 0 β /2α γ for somτ 0 0,1 v X

7 Knt Stat Univrsity ttp://tnamatkntdu 268 M NEILAN 217 Proof In ligt of 26, 28, and 27, w find β u c v 2 X L u c v,u c v Lu v,u c v C cont u c v X u c v X, for any v X, and trfor u u c X C inf v X u v X Furtrmor, by 217 and 22 w av 218 u u c Y inf v X u v Y +α C u v X Hnc, by Torm 21 w obtain u u X inf v X 2C u v X + 1 α u v Y, u u Y 2 inf v X u v Y +α C u v X, providd tat t rigt-and sid of 218 is smallr tan τ 0 β /2α γ REMARK 24 In all t xampls considrd blow, X is a discrth 1 -typ norm, Y is a discrth 2 -typ norm, α O 1, β O1, and C O1 REMARK 25 Som practical considrations Torm 21 stats tat if t discrt linarization is stabl and if som otr mild conditions old, tn tr xists a solution to 21 clos to u Howvr, it dos not indicat ow to construct discrtizations wit stabl linarizations On natural way to do tis is to construct a scm suc tat t opratorl is consistnt witl, wr 219 Fu+tw Fu Lw : lim cofd 2 u : D 2 w cofd 2 u w t 0 t Hr, cofd 2 u dnots t cofactor matrix of D 2 u, cofd 2 u : D 2 w n cofd 2 u i,j D 2 w i,j, i,j1 and w av usd t divrgnc-fr row proprty of cofactor matrics cf Lmma 31 blow to obtain t last quality In otr words, it is dsirabl tat t diagram in Fig- Fu 0 linariz discrtiz Fu 0 linariz Id F u 0 linariz Lw 0 Lw 0 Id L w 0 discrtiz FIGURE 21 An abstract commuting diagram Hr, Id dnots t rstriction of an oprator to t finit lmnt spac X T pat w tak to driv convrgnt finit lmnt mtods is indicatd by t doubl-lind arrows ur 21 commuts Sinc u is a classical convx solution to t Mong-Ampèr quation, t matrix cofd 2 u is positiv dfinit, and so t oprator L is a uniformly lliptic As

8 Knt Stat Univrsity ttp://tnamatkntdu ANALYSIS OF FINITE ELEMENT METHODS FOR THE MONGE-AMPÈRE EQUATION 269 discrtization scms for lliptic scond ordr linar PDEs ar wll undrstood and dvlopd namly, many stabl discrtizations of L xist, it is bttr to build t nonlinar discrtization F basd on t discrt linar problm L In fact w us tis proprty wn driving L 2 stimats blow; cf Assumption a5c Tis is t approac w tak in Sction 3 wn constructing finit lmnt scms 21 L 2 stimats W nd tis sction by driving som L 2 rror stimats Tis is acivd by using duality argumnts in conjunction wit t following additional st of assumptions a5 a T oprator L is symmtric and can b naturally xtndd suc tat L : H 2 Ω Y b T norm X is wll-dfind on H 2 Ω c T oprator L is consistnt witl dfind by 219 in t sns tat Lv,w Lv,w, v H 2 Ω H 1 0Ω, w Y d L is boundd in t sns tat tr xists anm > 0 suc tat Lv,w M v X w X, v,w X +H 2 Ω u is strictly convx and u W 3, Ω THEOREM 26 In addition to t assumptions of Torm 21, suppos tat condition a5 is satisfid Tn tr olds 220 u u L 2 Ω sup wr C E is dfind by 222 blow inf C E ϕ H 2 Ω ϕ X ϕ H 2 Ω Proof Lt ψ H 1 0Ω b t solution to M u u X ϕ ϕ X + γ 2 u u 2 Y ϕ X, 221 Lψ u u, ψ 0, inω, on Ω Sinc u is strictly convx in Ω and u W 3, Ω, by lliptic rgularity tr olds ψ H 2 Ω and 222 ψ H 2 Ω C E u u L 2 Ω for somc E > 0 It follows from a5a and 221 tat for any ψ X u u 2 L 2 Ω 223 Lψ,u u Lψ,u u Lu u,ψ ψ + Lu u,ψ Bounding t first trm in 223, w us a5d to obtain Lu u,ψ ψ M u u X ψ ψ X

9 Knt Stat Univrsity ttp://tnamatkntdu 270 M NEILAN To bound t scond trm in 223, w first not by 23 and 24 tat Lu u,ψ Lu u Fu Fu,ψ DF 2 [u]u u F 2 u F 2 u,ψ 1 2 F 2 [u]u u 1 DF 2 [u u ]u u,ψ, 0 F 2 [tu +1 tu]u udt,ψ wr w av again usd t fact tat t mapping w,v F 2 [w]v is bilinar Combining tis last idntity wit assumption a4, w obtain t stimat γ 224 Lu u,ψ 2 u u 2 Y ψ X Finally, combining , w av u u 2 L 2 Ω M u u X ψ ψ X + γ 2 u u 2 Y ψ X Dividing bot trms by u u L 2 Ω and using t lliptic rgularity stimat 222, w obtain Som spcific xampls in two dimnsions In tis sction w apply t abstract framwork st in t prvious sction to som concrt xampls, namly,c 1 finit lmnt, C 0 finit lmnt, and discontinuous Galrkin mtods In t first cas, w rcovr som of t rror stimats obtaind by Bömr in [7], but also obtain L 2 and H 1 rror stimats In t scond cas, w rcovr t sam rsults rcntly sown in [10], but in a mor compact form T mtod and rror analysis of DG mtods for t Mong-Ampèr quation is compltly nw Bfor procding, w first giv som notation and standard lmmas tat will b usd trougout t rst of t papr 31 Notation and som prliminary lmmas Lt T b a quasi-uniform, simplicial, and conforming triangulation [6, 12, 17] of t domain Ω wr ac triangl on t boundary as at most on curvd sid W dnot bye i t st of intrior dgs,eb t st of boundary dgs, and E E i Eb t st of all dgs in T W st T diamt for all T T, diam for all E, and not tat by t assumption of t quasiuniformity of t ms, T : max T T T Dfin t brokn Sobolv spac, norm, and smi-norm associatd wit t ms as H r T : H r T, v 2 H r T : v 2 H r T, v 2 H r T : v 2 H r T T T T T T T W dfin t jump of a vctor function w on an intrior dg T + T as follows: [w] w + n + +w n R, wr w ± w T ± and n ± is t outward unit normal of T ± On a boundary dg E b, w dfin [w] w n R T jump of a scalar function w is a vctor and is dfind as [w] w + n + +w n, T + T E i, [w] wn, T Ω E b

10 Knt Stat Univrsity ttp://tnamatkntdu ANALYSIS OF FINITE ELEMENT METHODS FOR THE MONGE-AMPÈRE EQUATION 271 For a matrixw R 2 2, w dfin t avrag of w on T + T by {{ }} w 1 w + 2 +w R 2 2, and on a boundary dg E b w tak {{ w }} w R 2 2 Similarly, for a vctor w R 2, w dfin t avrag of w on by {{ }} w 1 w + 2 +w R 2, T + T E, i {{ w }} w R 2, E b W nd tis subsction wit som lmmas tat will b usd many tims trougout t papr T first stats t divrgnc-fr row proprty of cofactor matrics [23, p 440] LEMMA 31 For any smoot function v, cofd 2 v i n j1 x j cofd 2 v ij 0, fori 1,2,,n, wr cofd 2 v i and cofd 2 v ij dnot rspctivly t it row and t i,j-ntry of t cofactor matrixcofd 2 v Nxt, w stat som standard invrs inqualitis [12, 17], as wll as a discrt Sobolv inquality [9] LEMMA 32 Tr olds for all T T v Hm T q m T v Hq T, v P k T, 0 q m, wr P k T dnots t st of all polynomials up to dgr k rstrictd to T Furtrmor, for any picwis polynomial wit rspct to t partitiont, tr olds v 2 L Ω 1+ ln v 2 H 1 T + 1 [v ] 31 2 L 2 E REMARK 33 In ordr to avoid t prolifration of constants, w sall us t notation A B to rprsnt t rlation A constant B, wr t constant is indpndnt of t ms paramtr and any pnalty paramtrs 32 C 1 finit lmnt mtods As a primr for mor complicatd looking mtods to com, w considr a simpl xampl to us t abstract framwork st in Sction 2, namly C 1 finit lmnt mtods To simplify mattrs, w assum in tis subsction tat Ω is a polygonal domain and tat g 0 in 11 so tat t Diriclt boundary conditions can b imposd xactly in t finit lmnt spac T assumptions do not guarant t smootnss of u, but tis xampl as t advantag of bing simpl T issus of curvd boundaris and inomognous boundary data will b andld using pnalization tcniqus in t nxt two subsctions, and it is straigtforward to apply tis mtodology toc 1 finit lmnt mtods W tak our finit lmnt spac and auxiliary spac to b 32 X {v H 2 Ω H 1 0Ω; v T P k T T T }, Y H 2 Ω,

11 Knt Stat Univrsity ttp://tnamatkntdu 272 M NEILAN wit norms 33 v X v H 1 Ω, v Y v H 2 Ω REMARK 34 To nsur tat t inclusion X C 1 Ω olds, w rquir k > 4 in t dfinition 32; s [17] As discussd in Rmark 25, w first considr finit lmnt discrtizations of t linar oprator 219 To tis nd, w dfin 34 Lw,y cofd 2 u w y dx, w Y,y X Ω T goal now is to build F suc tat 24 olds If w intgrat by parts in 34 and us Lmma 31 and t C 1 continuity of t finit lmnt spac, w obtain t following idntity: 35 Lw,y cofd 2 u : D 2 w y dx Basd on 35, w tn dfin t nonlinar oprator F as Fw,v f dtd 2 w v dx Ω Ω REMARK 35 Tis is t sam discrtization on gts witout considring t linar problm, but tis will not b t cas for otr discrtization scms drivd blow THEOREM 36 Suppos tat u H s Ω wit s > 3 Tn tr xists an 0 > 0 dpnding on u suc tat for 0 tr xists a solution u X to 36 F u,v 0, v X, wr F is t rstriction of F tox Morovr, tr olds u u Y + u u X l 1 u Hl Ω, wr t norms X and Y ar dfind by 33 and l min{k +1,s} In addition, ifu W 3, Ω tn u u L2 Ω l u Hl Ω + 2l 4 1+ ln 1 2 u 2 H l Ω Proof T proof is acivd by vrifying tat conditions a1 a5 old and mploying Corollary 23 and Torm 26 First, by t dfinitions ofy, X, andf, and t assumptions onu, conditions a1a a1b ar satisfid Furtrmor, by t invrs inquality w av v X v H 1 Ω 1 v H 2 Ω v Y, and so a1c olds witα O 1 Nxt w obsrv tat F as t dcomposition 23 wit F 0,v fv dx, F 1 u,v 0, Ω 37 F 2 u,v dtd 2 u v dx, Ω

12 Knt Stat Univrsity ttp://tnamatkntdu ANALYSIS OF FINITE ELEMENT METHODS FOR THE MONGE-AMPÈRE EQUATION 273 and sinc Fu 0, assumption a2 olds Continuing, w us 37 and 25 to conclud tat for any v,w Y and y X, DF 2 38 [v]w,y cofd 2 v : D 2 w y dx Trfor by 35, w av DF[u]w,y F 1 w+df 2 [u]w,y 39 Ω cofd 2 u : D 2 w y dx Lw,y Ω Sinc u is convx, t matrix cofd 2 u is positiv dfinit Trfor by 34 and an application of t Poincaré inquality, w av v 2 X v 2 L 2 Ω cofd 2 u v v dx L v,v Ω Morovr sinc u H s Ω, wits > 3, implis u W 2, Ω, w av Lv,w u W 2, Ω v H 1 Ω w H 1 Ω, v,w H 2 Ω+X Hnc bot assumptions a3 and a5d old Furtrmor, it is asy to s tat assumptions a5a a5c ar tru by t dfinitions of X, Y, and L T last assumption to vrify is a4; tat is, to bound t oprator DF 2 To tis nd, w us 38, t discrt Sobolv inquality 31, and t dfinition of t norms X and Y to bound DF 2 as follows: DF 2 [v]w,y v H 2 Ω w H 2 Ω y L Ω 1+ ln 1 2 v Y w Y y X Trfor, condition a4 olds witγ O1+ ln 1 2 It rmains to sow tat inf u v Y +α C u u X τ0 β /2αγ for somτ 0 0,1, v X in ordr to apply Corollary 21 and Torm 26 Hr, C 1+C cont /β O1 Tus, by 33 and sincα O 1 and γ O1+ ln 1/2, tis last xprssion rducs to inf u v H2 Ω v X u u H1 Ω O1+ ln 1/2 By standard approximation proprtis of X [12, 17], w av inf u v H2 Ω + 1 v X u u H1 Ω l 2 u H Ω l Trfor sinc s > 3 and k > 4 cf Rmark 34, condition 310 olds providd is sufficintly small Finally, applying Corollary 23 w obtain u u X inf v X u v X + u v Y l 1 u Hl Ω, u u Y inf v X u v Y + 1 u v X l 2 u Hl Ω,

13 Knt Stat Univrsity ttp://tnamatkntdu 274 M NEILAN and by applying Torm 26 w obtain u u L 2 Ω sup ϕ H 2 Ω inf ϕ 1 ϕ X H 2 Ω u u X ϕ ϕ X l u H l Ω +1+ ln 1 2 2l 4 u 2 H l Ω +1+ ln 1 2 u u 2 Y ϕ X 33 C 0 finit lmnt mtods T us of Lagrang finit lmnts and Nitsc s mtod [37] to comput t solution of t Mong-Ampèr quation was rcntly introducd and analyzd in [10] In tis sction, w sow ow tis mtod can fit into t abstract framwork st in Sction 2 To tis nd, w dfin t finit lmnt spac X H 1 Ω as follows: if T T dos not av a curvd dg, tn v T is a polynomial of total dgr k in t rctilinar coordinats fort ; if T T as on curvd dg, tn v T is a polynomial of dgr k in t curvilinar coordinats of T tat corrspond to t rctilinar coordinats on t rfrnc triangl s [6, Exampl 2, p 1216] W sty H 3 T, and dfin t norms v 2 X v 2 H 1 Ω + 1 v 2 L 2 + v 2, L 2 E b v 2 Y v 2 H 2 T + E b 1 3 v 2 L [ v] 2 + {{ L 2 D 2 v }} 2 L 2 E Applying Nitsc s mtod to t linar oprator L, w dfin L as 313 Lw,y cofd 2 u w y dx Ω + η wy [[ cofd 2 u w ]] y [[ cofd 2 ]] u y w ds, E b wr η is a positiv pnalization paramtr Hr, t tird trm in t rigt-and sid of 313 nsurs consistncy of t oprator, wil t fourt trm imposs symmtry W now driv t discrtization F basd on 313 Intgrating by parts and using t divrgnc-fr row proprty of cofactor matrics, w obtain Lw,y cofd 2 u : D 2 w y dx 314 T T T + [[{{ cofd 2 u }} w ]] y ds E i + η wy [[ cofd 2 ]] u y w ds E b

14 Knt Stat Univrsity ttp://tnamatkntdu ANALYSIS OF FINITE ELEMENT METHODS FOR THE MONGE-AMPÈRE EQUATION 275 Basd on t idntity 314, w dfin t discrt nonlinar oprator as Fw,v f dtd 2 w v dx+ [[{{ 315 cofd 2 w }} w ]] v ds T T E i + η w gv [[ cofd 2 ]] w v w g ds E b REMARK 37 T oprator F can b dcomposd as in 23 wit F 0,v fv dx η gv ds, T T T E b F 1 η w,v wv + [[ cofd 2 ]] 316 w v g ds, 317 E b F 2 w,v T T E b dtd 2 wv dx+ T E i [[ cofd 2 ]] w v wds [[{{ cofd 2 w }} w ]] v ds THEOREM 38 Suppos tatu H s Ω for soms > 3 and tatk 3 in t dfinition of X Tn tr xists an η 0 > 0 and 0 0 η suc tat for η η 0 and 0 η tr xists a solution to 318 F u,v 0, v X, wr F is t rstriction to X of F dfind by 315 Morovr, tr olds t following rror stimats u u Y + u u X 1+η l 1 u Hl Ω, wr l min{k +1,s} Ifu W 3, Ω, tn u u L 2 Ω 1+η 2 l u Hl Ω +1+ ln 1 2 2l 4 u 2 H l Ω Proof T proof is vry similar to tat of t proof of Torm 36; tat is, w vrify tat conditions a1 a5 old and apply t abstract rsults st in Sction 2 First, w obsrv tat a1a a1b and a2 old by t dfinitions off, X, Y, Rmark 37, and t assumptions ofu Morovr, by scaling argumnts, v Y 1 v X, and so assumption a1c olds witα O 1 Nxt, w us t dfinition of F and DF 2 25, to conclud tat for any v,w Y and y X, tr olds 319 DF 2 [v]w,y cofd 2 v : D 2 w y dx T T T [[{{cofd 2 v }} w ]] + [[{{ cofd 2 w }} v ]] y ds + E i E b [[cofd 2 ]] [[ w y v + cofd 2 ]] v y w ds

15 Knt Stat Univrsity ttp://tnamatkntdu 276 M NEILAN In particular, by sttingv u, noting t boundary condition 11 and [[{{ cofd 2 w }} u ]] y ds 0, E i w av DF 2 [u]w,y cofd 2 u : D 2 w y dx 320 T T Trfor by 24, 316, 320, and 314, w obtain DF[u]w,y DF 2 [u]w+f 1 w,y T + [[{{ cofd 2 u }} w ]] y ds E i [[cofd 2 ]] [[ w y g + cofd 2 ]] u y w ds E b T T T cofd 2 u : D 2 w y dx + [[{{ cofd 2 u }} w ]] y ds E i + η wy [[ cofd 2 ]] u y w ds Lw,y E b By using standard finit lmnt tcniqus cf [10, Lmma 31] and [37], t rstriction L of L to X is corciv on X providd η 0 is sufficintly larg and is boundd in t sns of assumption a5d wit M C cont 1 + η Tus, assumptions a3 and a5 old Nxt by 319, 311, t invrs inquality, and t discrt Sobolv inquality 31, w av for v,w Y and y X, DF 2 {{ [v]w,y D 2 v }} 2 L [ w] L 2 E i + {{ D 2 w }} L2 [ v] L2 y L Ω + D 2 w v L 2 L 2 + D 2 v L 2 w L 2 y L Ω E b + v H 2 T w H 2 T y L Ω 1+ ln 1 2 {{ D 2 v }} L2 [ w] L2 E i + {{ D 2 w }} L 2 [ v] L 2 + E b 1 D 2 w v L 2 L 2 + D 2 v L2 w L2 + v H2 T w H2 T y X

16 Knt Stat Univrsity ttp://tnamatkntdu ANALYSIS OF FINITE ELEMENT METHODS FOR THE MONGE-AMPÈRE EQUATION 277 Tus by 312 and many applications of t Caucy-Scwarz inquality, w conclud DF 2 [v ]w,y 1+ ln 1 2 v Y w Y y X From tis calculation, condition a4 olds witγ 1+ ln 1 2 As all of conditions a1 a5 av bn vrifid, it rmains to sow tat inf u v Y + 1 β η u v X O v X α γ O 1+ ln 1 2 to apply Corollary 23 and Torm 26 By approximation proprtis of t finit lmnt spac X [6] and scaling, w av inf v X u v Y η u v X 1+η l 2 u Hl Ω Tus, by t dfinition of l, w s tat 321 olds providd s > 3, k 3, and is sufficintly small Finally, applying Corollary 23 and Torm 26, w obtain u u X inf v X 1+η u v X + u v Y 1+η l 1 u Hl Ω, u u Y inf u v Y η u v X 1+η l 2 u Hl Ω, v X u u L 2 Ω sup ϕ H 2 Ω inf ϕ 1 ϕ X H 2 Ω 1+η u u X ϕ ϕ X +1+ ln 1 2 u u 2 Y ϕ X 1+η 2 l u H l Ω +1+ ln 1 2 2l 4 u 2 H l Ω 34 Discontinuous Galrkin mtods As our last xampl, w construct and analyz discontinuous Galrkin mtods for t Mong-Ampèr quation W tak our finit lmnt spac to consist of totally discontinuous picwis polynomial functions In particular, w dfin X L 2 Ω to consist of functions v suc tat if T T dos not av a curvd dg, tn v T is a polynomial of total dgr k in t rctilinar coordinats fort ; if T T as on curvd dg, tn v T is a polynomial of dgr k in t curvilinar coordinats of T tat corrspond to t rctilinar coordinats on t rfrnc triangl W sty H 3 T and dfin t norms 322 v 2 X v 2 H 1 T + 1 [v] 2 L 2 + {{ v }} 2, L 2 E v 2 Y v 2 H 2 T + E 1 3 [v] 2 L [ v] 2 L 2 {{ + D 2 v }} 2 L 2

17 Knt Stat Univrsity ttp://tnamatkntdu 278 M NEILAN REMARK 39 By t discrt Sobolv inquality 31 and t dfinition of X, w av v L Ω 1+ ln 1 2 v X, v X Similarly to t prvious two subsctions, w bas t nonlinar mtod F on t corrsponding discrt linar problm In tis cas, w dfin t discrt linar problm corrsponding to t linar oprator 219 as 323 Lw,y cofd 2 u w y dx T T T {{cofd 2 u w }} [y ] E +γ {{ cofd 2 u y }} [w] η [w] [y ] ds, wr γ is a paramtr tat can tak t valus {1, 1,0}, wic corrspond to t SIPG mtod γ 1, NIPG mtod γ 1, and IIPG mtod γ 0 [1, 8, 21, 40, 41] T constant η > 0 is again a pnalty paramtr Intgrating by parts of t first trm in 323 givs us 324 cofd Lw,y 2 u : D 2 w y dx T T [[{{ cofd 2 u }} w ]]{{ }} y ds + E i E γ {{ cofd 2 u y }} [w] η [w] [y ] ds Basd on t idntity 324, w dfin F suc tat Fw,v f dtd 2 w v dx+ η [w] [v ] T T T E i + [[{{ cofd 2 w }} w ]]{{ }} {{ v γ cofd 2 w v }} [w] ds + η w gv γ [[ cofd 2 ]] w v w g ds E b

18 Knt Stat Univrsity ttp://tnamatkntdu 325 ANALYSIS OF FINITE ELEMENT METHODS FOR THE MONGE-AMPÈRE EQUATION 279 REMARK 310 T dcomposition 23 olds wit F 0,v T T F 1 w,v E fv dx gv ds, E b η [w] [v ] ds [[ cofd 2 ]] w v gds, T 326 F 2 w,v T T T +γ E b dtd 2 wv dx [[{{cofd 2 w }} w ]]{{ }} v + E i γ E {{ cofd 2 w v }} [w] ds THEOREM 311 Suppos tat u H s Ω for som s > 3 and tat k 3 in t dfinition of X Tn tr xists an η 1 η 1 γ > 0 and 1 1 η > 0 suc tat for η η 1 and 1 η, tr xists a solution u X satisfying 327 F u,v 0, v X Morovr, tr olds u u Y + u u X 1+η l 1 u Hl Ω, wr l min{k +1,s} Ifu W 3, Ω and γ 1, tn u u L2 Ω 1+η 2 l u H l Ω +1+ ln 1 2 2l 4 u 2 H l Ω REMARK 312 For t NIPG cas γ 1, t pnalization paramtr can b takn to b any positiv numbr Proof T proof follows t sam argumnts as t proof of Torms 36 and 38 First, by 326 and 25 for any v,w Y and y X, w av 328 DF 2 [v]w,y cofd 2 v : D 2 w y dx T T T [[{{cofd 2 v }} w ]] + [[{{ cofd 2 w }} v ]] {{ }} y + E i γ E [[{{cofd 2 v }} y ]] [w]+ [[{{ cofd 2 w }} y ]] [v] ds

19 Knt Stat Univrsity ttp://tnamatkntdu 280 M NEILAN Trfor, sttingv u in 328 w av by 24, , and 324, DF[u]w,y DF 2 [u]w+f 1 w,y T T T cofd 2 u : D 2 w y dx γ {{ cofd 2 u y }} [w] E η [w] [y ] ds+ [[{{cofd 2 u }} w ]]{{ }} y ds E i Lw,y Furtrmor, using standard DG tcniqus g [40], conditions a3 and a5d corcivity and continuity olds wit C cont M 1+η and β indpndnt of providd tat η 1 is sufficintly larg for t cas γ 1, η 1 can b takn to b any positiv numbr Nxt, it is asy to s tat assumptions a1a a1b, a2, and a5b a5c old by t dfinition of Y, X, L, Rmark 310, and t assumptions on u Furtrmor by t dfinitions of t norms 322 and t invrs inquality, assumption a1c olds as wll wit α O 1 Lastly, by t dfinition of L, assumption a5a symmtry olds providd γ 1 It rmains to sow tat assumption a4 olds Tis is acivd by using 328, t invrs inquality, 31, and t Caucy-Scwarz inquality as follows: DF 2 [v]w,y v H 2 T w H 2 T y L Ω + {{ D 2 v }} 2 L [ w] + {{ L D 2 w }} 2 L 2 [ v] L y 2 L Ω E i + {{ D 2 v }} L2 [w] L2 + {{ D 2 w }} L2 [v] L2 y L Ω E 1+ ln E i [ v H 2 T w H 2 T {{ D 2 v }} L2 [ w] + {{ L2 D 2 w }} L2 [ v] L2 + E 1 {{ D 2 v }} 2 L [w] + {{ L D 2 w }} ] 2 L 2 [v] L y 2 X 1+ ln 1 2 v Y w Y y X Trfor assumption a4 olds witγ 1+ ln 1 2 To apply Corollary 23 w must vrify tat inf u v Y η u v X Oβ /α γ v X O 1+ ln 1 2 By standard approximation proprtis ofx, tis rquirmnt rducs to 1+η l 2 u Hl Ω O 1+ ln 1 2,

20 Knt Stat Univrsity ttp://tnamatkntdu ANALYSIS OF FINITE ELEMENT METHODS FOR THE MONGE-AMPÈRE EQUATION 281 wic olds providd s > 3, k 3, and is sufficintly small Trfor by Corollary 23, w av u u X inf v X 1+η u uc X + u u c Y 1+η l 1 u Hl Ω, u u Y inf v X u v Y η u v X 1+η l 2 u Hl Ω Ifγ 1, tn by Torm 26 tr olds u u L2 Ω sup ϕ H 2 Ω inf ϕ 1 ϕ X H 2 Ω 1+η u u X ϕ ϕ X +1+ln 1 2 u u Y ϕ X 1+η 2 l u Hl Ω +1+ ln 1 2 2l 4 u 2 H l Ω 4 Abstract rsults in tr dimnsions In tis sction, w xtnd t abstract rsults of Sction 2 to t tr dimnsional cas As bfor, w lt X, X b a finit dimnsional spac and w considr t problm of findingu X suc tat 21 olds W mak similar assumptions as for t two dimnsional countrpart, but tr ar som subtl diffrncs First, sinc t PDE 11 is cubic in 3D, t dcomposition 23 nds to cang to rflct tis fatur Anotr diffrnc is tat w must introduc two auxiliary normd linar spacs to ffctivly analyz t mtod Tis in turn will allow us to ffctivly stimat t scond Gâtaux drivativ of t nonlinar componnt off, wic is a ky lmnt in t proof of Torm 41 blow Spcifically, w mak t following assumptions: A1 a Tr xists two auxiliary normd linar spacs Y 1, Y 1, Y 2, Y 2 wit X Y 2 Y 1 and u Y 2 suc tat X is wll-dfind on Y 1 b T opratorf can b xtndd to a smoot opratorf : Y 1 X wit Fw,v F w,v, w,v X c Tr xist constants a 1,a2 > 0 suc tat for all v X, 41 v Y 1 a 1 v X, v 2 Y a 2 v X A2 a T nonlinar opratorf is consistnt witf in t sns tatfu 0 b F can b dcomposd as 42 F F 3 +F 1 +F 0, wrf 3 is cubic,f 1 is linar, andf 0 is constant in tir argumnts A3 Dfin t linar oprator L : Y 1 X as 43 Fu+tw Fu Lw : DF[u]w lim F 1 w+df 3 [u]w, t 0 t and dnot by L t rstriction of L to X Tn tr xists a constant b > 0 suc tat t following corcivity condition olds: 44 b v 2 X L v,v, v X

21 Knt Stat Univrsity ttp://tnamatkntdu 282 M NEILAN A4 Dfin D 2 F 3 DF 3 [v +tz]w DF 3 [v]w [v]w,z : lim t 0 t Tn tr xists a constant c suc tat for all v,w Y 1, 45 D 2 F 3 [u]w,v X c v Y 1 w Y 1, and for allv,w,z Y 2, D 2 F 3 [v]w,z X c v Y 2 w Y 2 z Y 2 A5 a T oprator L is symmtric and can b naturally xtndd suc tat L : H 2 Ω Y 1 b T norm X is wll-dfind on H 2 Ω c T oprator L is consistnt witl dfind by 219 in t sns tat Lv,w Lv,w, v H 2 Ω H 1 0Ω, w Y 1 d L is boundd in t sns tat tr xists anm > 0 suc tat Lv,w M v X w X, v,w X +H 2 Ω u is strictly convx inωand u W 3, Ω THEOREM 41 Suppos tat assumptions A1 A4 old, and lt u c X b t uniqu solution to L u c,v Lu,v, v X Suppos tat 46 u u c 2 Y 1 + u u c 3 Y 2 τ 1 b 4c min { 1 a 1 u u c Y 1, 1 a 2 u u c Y 2 for somτ 1 0,1 Tn tr xists a solutionu X to 21 satisfying { } u u X u u c X +min u u c Y 1, u u c Y 2, u u Y 1 2 u u c Y 1, u u Y 2 2 u u c Y 2 If in addition assumption A5 olds, tn u u c L 2 Ω sup inf a 1 C E ϕ H 2 Ω ϕ X ϕ H 2 Ω a 2 M u u X ϕ ϕ X +c 1 2 u u 2 Y u u 3 Y 2 ϕ X, }

22 Knt Stat Univrsity ttp://tnamatkntdu ANALYSIS OF FINITE ELEMENT METHODS FOR THE MONGE-AMPÈRE EQUATION 283 witc E dfind by 222 REMARK 42 As in Corollary 23, it is asy to sow tat t quantity u u c X can b boundd by Cinf v X u v X for som constant C > 0 Proof T proof procds by using similar argumnts to tos of Torm 21 Namly, w considr t mapping M : Y 1 X dfind in Lmma 22, tat is, Mw L Lw Fw T goal is to sow tat M, wn rstrictd tox, as a fixd point First, w not tat, by 42, 43, and t consistncy of F, Fw F 0 +F 1 w+f 3 w F 1 u F 3 u+lw DF 3 [u]w+f 3 w 1 Lw u+ DF 3 [tw +1 tu]w u DF 3 [u]w u dt Lw u D 2 F 3[ stw u+u ] w u,tw u dtds Sinc F 3 is cubic, t mapping w,v,z D 2 F 3 [w]v,z is trilinar It tn follows tat D 2 F 3[ stw u+u ] w u,tw u dtds D 2 F 3 [w u]w u,w u +D 2 F 3 [u]w u,w u st 2 dtds tdtds 1 2 D2 F 3 [u]w u,w u+ 1 6 D2 F 3 [w u]w u,w u Substituting tis idntity into 411 w obtain 412 Fw Lw u 1 2 D2 F 3 [u]w u,u D 2 F 3 [tw+1 tu]w u,tw +1 tudt Lw u+ 1 2 D2 F 3 [u]w u,w u D2 F 3 [w u]w u,w u, and using tis last idntity in 411, w arriv at Mw L 1 Lu 1 2 D2 F 3 [u]w u,w u 1 6 D2 F 3 [w u]w u,w u

23 Knt Stat Univrsity ttp://tnamatkntdu 284 M NEILAN Hnc, for any v,w Y 1 w av Mw Mv L 1 L D 2 F 3 [u]v u,v u D 2 F 3 [u]w u,w u D 2 F 3 [v u]v u,v u D 2 F 3 [w u]w u,w u D 2 F 3 [u]v w,v u+d 2 F 3 [u]w u,v w D 2 F 3 [v w]v u,v u+d 2 F 3 [w u]v w,v u +D 2 F 3 [w u]w u,v w Nxt w apply 44 and 45 to obtain 413 Mw Mv X 1 1 D 2 F 3 [u]v w,v u b 2 X + 1 D 2 F 3 [u]w u,v w 2 X + 1 D 2 F 3 [v w]v u,v u 6 X + 1 D 2 F 3 [w u]v w,v u 6 X + 1 D 2 F 3 [w u]w u,v w 6 X c u v 2b Y 1 + u w Y 1 w v Y 1 + c u v 2 Y + u v 6b 2 Y 2 u w Y 2 c 2b + u w 2 Y w v 2 Y 2 [ u v Y 1 + u w Y w v Y 1 ] u v 2Y + u w 2Y w v 2 2 Y 2 In particular, sincu c Mu, w av uc Mw X c 2b u w 2 Y u w 3 Y 2 Lt M b t rstriction of M tox, ltb ρ u c b dfind by 214, and st ρ 1 : min { 1 a 1 u u c Y 1, 1 a 2 u u c Y 2 }

24 Knt Stat Univrsity ttp://tnamatkntdu ANALYSIS OF FINITE ELEMENT METHODS FOR THE MONGE-AMPÈRE EQUATION 285 Tn by 413, 41, and 415 forv,w B ρ1 u c, [ M w M v X c a 1 u u c b Y 1 +a 1 ρ 1 2c b +a 2 u u c 2 Y + a 2 2 ρ 2 1 ] w v X [ ] a 1 u u c Y 1 +a 2 u u c 2 Y 2 T condition 46 tn implis 416 M w M v X τ 1 w v X, τ 1 0,1 Nxt, by 414, 41, 415, and 46, for w B ρ1 u c, uc M w X c u u c 2 Y + a 1 b 1 ρ u u c 3 Y + a 2 2 ρ 3 1 2c u u c 2 Y + u u b 1 c 3 Y 2 { 1 1 min a 1 u u c Y 1, a 2 u u c Y 2 w v X } ρ 1 It tn follows from tat M as a fixd point u in B ρ1 u c wic is a solution to 21 To driv t rror stimats 47 49, w us t triangl inquality, 415, and a1b to gt u u X u u c X +ρ 1 u u c X { } 1 1 +min u u c Y 1, u u c Y 2, a 1 a 2 u u Y 1 u u Y 1 +a 1 ρ 1 2 u u c Y 1, u u Y 2 u u Y 2 +a 2 ρ 1 2 u u c Y 2 To driv t L 2 stimat 410, w lt ψ solv t auxiliary problm 221 Using similar argumnts to tat of t proof of Torm 26, w av for any ψ X, u u 2 L 2 Ω 418 Lu u,ψ ψ + Lu u,ψ Lu u,ψ ψ + Lu u +Fu,ψ M u u X ψ ψ X + Lu u +Fu,ψ Using t idntity 412 witw u, w obtain 1 Lu u +Fu,ψ D 2 F 3 [u]u u,u u,ψ D 2 F 3 [u u]u u,u u,ψ, 6

25 Knt Stat Univrsity ttp://tnamatkntdu 286 M NEILAN Trfor by applying t stimats statd in assumption A4, w av Lu u +Fu,ψ c 1 2 u u 2 Y u u 3 Y 2 ψ X Using tis last stimat in 418, w av u u 2 L 2 Ω M u u X ψ ψ X +c 1 2 u u 2 Y u u 3 Y 2 ψ X Dividing by u u L2 Ω and using t lliptic rgularity stimat 222, w obtain 410 T proof is complt REMARK 43 It was rcntly sown in [11] tat t C 0 finit lmnt mtod 318 satisfis assumptions A1 A5 wit a 1 O 1, a 2 O b O1, c O As a rsult, t autors sowd tr xists a solution u to t mtod 318 in tr dimnsions and drivd quasi-optimal rror stimats providd tat u H s Ω, s > 7/2, and cubic polynomials or igr ar usd W xpct tat similar rsults will old for t C 1 finit lmnt mtod 36 and t discontinuous Galrkin mtod 327 as wll 5 Som concluding rmarks In tis papr, w av dvlopd and analyzd various numrical mtods for t two and tr dimnsional Mong-Ampèr quation undr a gnral framwork T ky ida to build convrgnt numrical scms is to construct discrtizations suc tat t rsulting discrt linarization is stabl and consistnt wit t continuous linarization Wit tis in and, and wit a fw mor mild conditions, w provd xistnc of t numrical solution as wll as som abstract rror stimats using a simpl fixd-point tcniqu W xpct tat t analysis prsntd r can b xtndd to gnral Mong-Ampèr quations, in wic t function f dpnds on u and u, as wll as parabolic Mong-Ampèr quations Furtrmor, w conjctur tat t abstract framwork can b xpandd so tat otr numrical mtods including mixd finit lmnt mtods, local discontinuous Galrkin mtods, and Ptrov-Galrkin mtods can naturally fit into t stting, REFERENCES [1] D N ARNOLD, F BREZZI, B COCKBURN, AND L D MARINI, Unifid analysis of discontinuous Galrkin mtods for lliptic problms, SIAM J Numr Anal, , pp [2] F E BAGINSKI AND N WHITAKER, Numrical solutions of boundary valu problms fork-surfacs inr 3, Numr Mtods Partial Diffrntial Equations, , pp [3] G BAKER, Finit lmnt mtods for lliptic quations using nonconforming lmnts, Mat Comp, , pp [4] G BARLES AND P E SOUGANIDIS, Convrgnc of approximation scms for fully nonlinar scond ordr quations, Asymptotic Anal, , pp [5] J D BENAMOU, B D FROESE, AND A M OBERMAN, Two numrical mtods for t lliptic Mong- Ampèr quation, M2AN Mat Modl Numr Anal, , pp [6] C BERNARDI, Optimal finit lmnt intrpolation on curvd domains, SIAM J Numr Anal, , pp [7] K BÖHMER, On finit lmnt mtods for fully nonlinar lliptic quations of scond ordr, SIAM J Numr Anal, , pp

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27 Knt Stat Univrsity ttp://tnamatkntdu 288 M NEILAN [38] A M OBERMAN, Wid stncil finit diffrnc scms for t lliptic Mong-Ampèr quation and functions of t ignvalus of t Hssian, Discrt Contin Dyn Syst Sr B, , pp [39] V I OLIKER AND L D PRUSSNER, On t numrical solution of t quation 2 z 2 z x 2 y 2 2 z x y 2 f and its discrtizations, Numr Mat, , pp [40] B RIVIÉRE, Discontinuous Galrkin Mtods for Solving Elliptic and Parabolic Equations: Tory and Implmntation, SIAM, Piladlpia, 2008 [41] B RIVIÉRE, M F WHEELER, AND V GIRAULT, Improvd nrgy stimats for intrior pnalty, constraind and discontinuous Galrkin mtods for lliptic problms I, Comput Gosci, , pp [42] M V SAFONOV, Classical solution of scond-ordr nonlinar lliptic quations, Izv Akad Nauk SSSR Sr Mat, , pp [43] D C SORENSEN AND R GLOWINSKI, A quadratically constraind minimization problm arising from PDE of Mong-Ampèr typ, Numr Algoritms, , pp [44] N S TRUDINGER AND X-J WANG, T Mong-Ampèr quation and its gomtric applications, in Handbook of Gomtric Analysis Vol I, L Ji, P Li, R Scon, and L Simon, ds, Advancd Lcturs in Matmatics, 7, Intrnational Prss, Somrvill, pp [45] C VILLANI, Topics in Optimal Transportation, AMS, Providnc, 2003 [46] V ZHELIGOVSKY, O PODVIGINA, AND U FRISCH, T Mong-Ampèr quation: various forms and numrical solution, J Comput Pys, , pp