MAXIMUM NORM STABILITY OF DIFFERENCE SCHEMES FOR PARABOLIC EQUATIONS ON OVERSET NONMATCHING SPACE-TIME GRIDS

MATHEMATICS OF COMPUTATION Voume 72 Number 242 Pages 619 656 S 0025-57180201462-X Artce eectroncay pubshed on November 4 2002 MAXIMUM NORM STABILITY OF DIFFERENCE SCHEMES FOR PARABOLIC EQUATIONS ON OVERSET NONMATCHING SPACE-TIME GRIDS T P MATHEW AND G RUSSO Abstract In ths paper theoretca resuts are descrbed on the maxmum norm stabty and accuracy of fnte dfference dscretzatons of paraboc equatons on overset nonmatchng space-tme grds We consder paraboc equatons contanng a near reacton term on a space-tme doman Ω [0T] whch s decomposed nto an overappng coecton of cyndrca subregons of the form Ω [0T] for =1p Each of the space-tme domans Ω [0T] are assumed to be ndependenty grded n parae accordng to the oca geometry and space-tme reguarty of the souton yedng space-tme grds wth mesh parameters and τ In partcuar the dfferent space-tme grds need not match on the regons of overap and the tme steps τ can dffer from one grd to the next We dscretze the paraboc equaton on eacoca grd by empoyng an expct or mpctθ-scheme n tme and a fnte dfference scheme n space satsfyng a dscrete maxmum prncpe The oca dscretzatons are couped together wthout the use of Lagrange mutpers by requrng the boundary vaues on each space-tme grd to match a sutabe nterpoaton of the souton on adjacent grds The resutng goba dscretzaton yeds a arge system of couped equatons whch can be soved by a parae Schwarz teratve procedure requrng some communcaton between adjacent subregons Our anayss empoys a contracton mappng argument Appcatons of the resuts are brefy ndcated for reacton-dffuson equatons wth contractve terms and heterogeneous hyperboc-paraboc approxmatons of paraboc equatons 1 Introducton In ths paper theoretca bounds are descrbed extendng resuts n [32 10] for the maxmum norm stabty and convergence of dscretzatons of paraboc equatons on nonmatchng overset space-tme grds Nonmatchng overset spata grds are popuar n severa fud dynamcs computatons nvovng compex geometres [33 13] They permt ndependent parae generaton of oca grds adapted to the oca geometry wthout the restrcton of matchng the grds on the regons of overap at the cost of ncreased computatons n coupng the varous oca dscretzatons For evouton probems addtona fexbty can be obtaned Receved by the edtor Juy 25 2000 2000 Mathematcs Subject Cassfcaton Prmary 65N20 65F10 Key words and phrases Nonmatchng overset space-tme grds maxmum norm stabty composte grds parae Schwarz aternatng method paraboc equatons dscrete maxmum prncpe dscrete barrer functons 619 c 2002 Amercan Mathematca Socety Lcense or copyrght restrctons may appy to redstrbuton; see http://wwwamsorg/journa-terms-of-use

620 T P MATHEW AND G RUSSO by permttng dfferent tme steps and choce of expct or mpct schemes on each of the dfferent space-tme subregons [33 13 17 18 7 23 5 7 20 22] In the computatona terature severa approaches have been proposed for coupng dscretzatons on nonmatchng grds These ncude Lagrange mutpers ncudng mortar methods and east squares based technques see [33 13 32 5 4 9 23 20 1 2 10 19] The method consdered n ths paper does not use ether Lagrange mutpers or east squares to coupe the varous oca probems It s smper to mpement see [32 10]; however t appes ony to a certan cass of paraboc equatons exhbtng a contracton property and t requres overap amongst adjacent grds Our study w be restrcted to a sma cass of paraboc equatons of the form 11 u t a u + bx u + cxu = fx t n Ω [0T] ux t = 0 on Ω [0T] ux 0 = u 0 x on Ω where fx t bx cxandu 0 x are suffcenty smooth functons and a>0 Here Ω R d for d =1 2 In order to have a contracton property for homogeneous soutons we w requre that cx c 0 > 0 for some postve constant c 0 Gven the cyndrca space-tme doman Ω [0T] we decompose t nto an overappng coecton of cynders of the form {Ω [0T]}p =1 that form a coverng of Ω [0T] Each cynder Ω [0T] w be assumed to be tranguated by a space-tme grd wth mesh and tme parameters and τ see Fgure 11 We empoy fnte dfference methods n space and mpct or expct θ-schemes n tme ndependenty on each space-tme grd On each subdoman boundary Ω [0T] we requre the oca souton to match some sutaby chosen nterpoant of the souton from adjacent grds see [33 32 13 10] Our man resut n the paper stated n Theorem 44 concerns the accuracy of the goba dscretzaton Let u hτ denote the restrcton of the exact souton u of the paraboc equaton to a the space-tme grdponts and et U hτ denote the computed souton of the goba dscretzaton Suppose the truncaton and ntergrd nterpoaton errors for the dscretzaton and boundary condtons respectvey on the th space-tme grd Ω [0T]satsfy Loca truncaton error on Ω q [0T]= u q;1 +2q ;2 +2 Ω [0T ] h ;1 + τ q ;2 Loca nterpoaton error on Ω [0T]= u r r ;1 r ;2 B h ;1 2 [0T ] + τ r ;2 where q ;1q ;2 Ω [0T ] and r ;1r ;2 B 2 [0T ] denote Soboev norms Here B 2 s a sma spata regon coverng the the boundary segment B2 = Ω Ω of the th spata subdoman Theorem 44 states that under sutabe assumptons the maxmum norm of the goba error u hτ U hτ satsfes the bound u hτ U hτ C max{ u q ;1+2q ;2+2 Ω [0T ]h q ;1 + τ q ;2 + u r;1 r ;2 B ;1 2 [0T ]hr + τ r ;2 } where C>0sndependent of the mesh szes From ths we deduce that deay the oca grd szes and τ shoud be chosen so that a the oca error terms are baanced Ths woud mean smaer and τ on regons where the souton s Lcense or copyrght restrctons may appy to redstrbuton; see http://wwwamsorg/journa-terms-of-use

DIFFERENCE SCHEMES FOR PARABOLIC EQUATIONS 621 Overappng space-tme subregons Tme T τ 1 Nonmatchng overset grds τ 2 Ω 1 Ω 2 Space h 1 h 2 Fgure 11 Sampe nonmatchng overset space-tme grd ess reguar Addtonay the ntergrd nterpoaton maps for determnng oca boundary data shoud be chosen so that the nterpoaton errors are baanced wth the oca truncaton errors Then the resutng accuracy of the goba dscretzaton w be of optma order The rest of the paper s outned as foows In Secton 2 we ntroduce notaton for the overappng space-tme subregons dscuss expct and mpct oca dscretzatons on each space-tme grd ntergrd nterpoaton maps the goba dscretzaton and a parae Schwarz teratve procedure for sovng the resutng system of equatons In Secton 3 we dscuss theoretca propertes of the oca schemes such as aprorestmates maxmum prncpes comparson theorems barrer functons and contracton propertes of homogeneous soutons In Secton 4 we anayze the stabty and accuracy of our goba space-tme dscretzaton by empoyng Pcard s contracton mappng theorem 2 Goba dscretzaton on nonmatchng overappng space-tme grds In ths secton we descrbe the constructon of a goba dscretzaton of 11 and a parae Schwarz teratve method for sovng the resutng arge system of equatons 21 Space-tme subdomans Let Ω [0T] denote the space-tme regon on whch the paraboc equaton 11 s posed We w descrbe here the constructon of an overappng coecton of space-tme subregons that covers the above regon Let {Ω } p =1 denote a partton or a coverng of the spata doman Ω: p Ω Ω =1 Lcense or copyrght restrctons may appy to redstrbuton; see http://wwwamsorg/journa-terms-of-use

622 T P MATHEW AND G RUSSO In practce the subregons Ω may be chosen accordng to the geometry of Ω or the reguarty of the souton f known or ese by estmatng the reguarty from pror numerca approxmatons For each subregon Ω choose a parameter β > 0and enarge Ω to Ω as Ω {x Ω:dstx Ω <β } The coecton of subregons {Ω }p =1 w form an overappng coverng of Ω wth overap parameters β An overappng coverng of the space-tme regon Ω [0T] can be mmedatey constructed: p Ω [0T]= Ω [0T] =1 where each space-tme subregon Ω [0T] s cyndrca We w denote the boundary of each spata subregon Ω by B Ω It w be convenent to further partton each boundary B = Ω nto two segments B1 and B2 we w omt the superscrpt when the subregon s cear from the context: B1 Ω Ω and B2 Ω Ω Correspondng to ths the space-tme boundary B [0T] of eacoca subregon can be decomposed nto B1 [0T]andB2 [0T] 22 Loca space-tme grds On each of the oca space-tme cynders Ω [0T] we assume that a space-tme grd Ω {0τ 2τ T τ T} s constructed takng nto account the geometry of Ω and the reguarty of the souton on ths space-tme regon see Fgure 11 Here denotes the mesh sze on Ω and τ denotes the tme step wth τ = T N for some nteger N 1 Throughout the paper x w denote the th grdpont n Ω We w use I to denote the nteror nodes n the grd Ω and B to denote ts boundary nodes Snce B s decomposed nto B1 and B2wedenotebyB 1 and B 2 the grdponts on B 1 and B 2 respectvey We w denote a grd functon on Ω by w h Correspondng to the partton of grdponts n Ω nto I and B we obtan the bock vector w h =w h Iw h B T When t s necessary to dstngush the bock components of w h correspondng to the boundary subgrds B 1 and B 2 we w use the notaton w B 1 and w h B 2 respectvey: w h B =w h B 1 w h B 2 T We w denote a grd functon on Ω at tme kτ by wh k A space-tme grd functon on Ω {0τ T} w be denoted by w h τ wth w h τ = { w k } N k=0 Lcense or copyrght restrctons may appy to redstrbuton; see http://wwwamsorg/journa-terms-of-use

DIFFERENCE SCHEMES FOR PARABOLIC EQUATIONS 623 A grd functon on the entre famy of space-tme grds w be denoted by w hτ where w hτ w h τ p =1 {w } k N p = h k=0 =1 Gven a contnuous spata functon wx we w use π h w to denote ts spata nterpoaton onto the grdponts n Ω π h w wx x Ω Smary π h Iw w denote nterpoaton of wx onto the nteror grdponts n Ω Ifwx t s a contnuous space-tme functon we w use π h τ to denote the nterpoaton of w ontotheth space-tme grd π h τ w {π h w kτ } N k=0 If wx t s a contnuous space-tme functon we w use π hτ to denote the nterpoaton of w onto a the space-tme grds π hτ w π h τ w p =1 23 Loca subprobems and oca dscretzatons On each of the space-tme subdomans Ω [0T] the orgna paraboc equaton 11 w be repaced by the foowng oca paraboc nta boundary vaue probem wth sutaby chosen boundary condtons g x t that coupe the adjacent probems: u t + Lu = fx t x t Ω [0T] u = 0 x t B1 [0T] u = g x t x t B2 [0T] ux 0 = u 0 x t =0 where L denotes the eptc operator Lu a u + b u + cxu Here the choce of oca nta data s u 0 x restrcted to Ω snce the exact souton restrcted to Ω [0T] woud satsfy ths nta condton and snce u 0x s assumed to be known The boundary data on B1 [0T] s zero snce the exact souton satsfes ths boundary condton The boundary data g x t w pay a cruca roe as t s not known We w requre see subsecton 25 that g x t equasa sutabe nterpoaton of the soutons from adjacent regons Ths w coupe the varous oca probems and requre an teratve process to compute g x t Eacoca paraboc equaton w be dscretzed on the space-tme grd Ω {0τ T} by a fnte dfference scheme n space and an mpct or expct θ- scheme n tme The eptc operator L w be dscretzed on each spata grd Ω by a fnte dfference scheme wth coeffcent matrx A If x s the th nteror grdpont n Ω then the dscretzaton of L at ths grdpont w be denoted by Lwx = j A j wx j +C w x where Cw x s the oca truncaton error at x for an arbtrary smooth functon wx The matrx A w be rectanguar; the frst ndex n the precedng Lcense or copyrght restrctons may appy to redstrbuton; see http://wwwamsorg/journa-terms-of-use

624 T P MATHEW AND G RUSSO equaton corresponds to nteror grdponts whe the second ndex j above corresponds to nteror and/or boundary grdponts Correspondng to the partton w h =w h Iw h B T the rectanguar matrx A can be bock parttoned A = [ A A IB ] wth A w h = A w I + A IB w B Assumpton A1 We w assume the foowng about the entres of A 1 A > 0 for a I 2 A j 0when j wth I and j I B 3 j A j = c c 0 > 0 Remark 1 Fnte dfference dscretzatons satsfyng Assumpton A1 canbecon- structed n many ways If the grd Ω s unform then the standard second-order fve-pont stenc may be apped to approxmate x a w x If a s not sma n reaton to b Ω then centered fnte dfferences may be apped to obtan a second order approxmaton to bx w provded a oca ce Pecet restrcton of the form b Ω h<2a s satsfed If a s sma n reaton to b Ω or zero then a frst order upwnd dscretzaton can be apped to approxmate bx w The term cxwx can be approxmated by a one-pont stenc at each grdpont If the grd s nonunform fnte voume based fnte dfferences may be apped to construct the desred approxmatons For nstance f Ω R 2 then a Deaunay tranguaton need frst to be constructed for the grd and fnte voume based fnte dfferences can be apped see [6] Remark 2 If matrx A satsfes Assumpton A1 then A w be strcty dagonay domnant and A 1 j 0 for a j In partcuar A w be an M-matrx see for nstance [31] Sem-dscretzaton of the oca paraboc nta boundary vaue probem on the spata grd Ω yeds 21 du h I dt + A U I + A IB U B = f h It U h B 1 t = 0 U h B 2 t = g h B 2 t U h 0 = π h u 0 x where f h It π h If t The boundary condtons g h B 2 t w be specfed n subsectons 24 and 25 Lcense or copyrght restrctons may appy to redstrbuton; see http://wwwamsorg/journa-terms-of-use

DIFFERENCE SCHEMES FOR PARABOLIC EQUATIONS 625 To dscretze 21 n tme we choose 0 θ 1 for =1p ndependenty and appy a θ-scheme to obtan I U h k I + θ A τ I + θ A IB B + θ A U h k I + θ A IB U h k B = θ f k+1 I + θ fh k I 22 B 1 = 0 B 2 = g k+1 B 2 U 0 0 = π h Iu 0 for k =0N 1 where θ 1 θ The dscrete boundary condtons {g k B 2 } for k =1N and =1pare cruca for coupng the varous oca paraboc dscretzatons and w be descrbed n the next secton 24 Intergrd nterpoaton The oca space-tme dscretzatons 22 w be couped together by requrng that the oca boundary data gh k B 2 match a sutabe nterpoaton Ih k U hτ of the dscrete souton U hτ from adjacent grds The near map Ih k s descrbed beow Let x from Ω be a grdpont on B2 At tme kτ the boundary data gh k B 2 =Uh k ux kτ w be approxmated by near combnatons of noda vaues of U h k j ux h j kτ from adjacent space-tme grds Ω [0T] wth U k h = Ih k U hτ where the nterpoaton map Ih k s defned n terms of a tensor α k kĩ : 23 p N I k h U hτ =1 k=0 ĩ α k U k kĩ h ĩ Beow we st assumptons about the weghts α k kĩ used to defne Ih k U hτ 1 Assumpton A2 The ntergrd nterpoaton map Ih k must use ony vaues from adjacent grds In terms of the coeffcents ths paces the foowng requrement on the weghts α k kĩ : α k kĩ =0 when = 2 Assumpton A3 Gven the ntergrd nterpoaton map Ih k shoud nvove ony noda vaues from grdponts n the unextended subregons Ω [0T]for e t shoud not nvove noda vaues from grdponts n the extended regons Ω \ Ω [0T] In terms of the coeffcents α k ths requrement s kĩ α k kĩ =0 when xh j Ω \ Ω or = Lcense or copyrght restrctons may appy to redstrbuton; see http://wwwamsorg/journa-terms-of-use

626 T P MATHEW AND G RUSSO Gven B2 etb 2 Ω denote the smaest regon such that B 2 [0T]contans a the ces and grdponts from adjacent subregons used n defnng the nterpoaton map Ih k for k =0N 3 Assumpton A4 Let wx t be a smooth space-tme functon whch s zero on Ω [0T] For each space-tme grdpont x j kτ on the boundary segment B2 [0T] we assume that the nterpoaton s chosen so that the error satsfes wx j kτ Ih k w hτ j = Dk w x j where the oca nterpoaton error can be estmated by Tayor seres expanson Dh k w x j C w r r 1 r 2 B h 1 2 [0T ] + τ r 2 for some ntegers r 1 1andr 2 1 where C s ndependent of τ and w r1 r 2 B 2 [0T ] denotes a Soboev norm of the space-tme functon w on the regon B 2 [0T] Defnton Throughout the paper we w use σ hτ to denote the maxmum norm of the ntergrd nterpoaton map σ hτ max k p N α k kĩ =1 k=0 Exampe We ncude a smpe exampe to ustrate the ntergrd nterpoaton map for a one-dmensona regon Ω = 0 4 wth T = 1 and a two-subdoman decomposton wth Ω 1 =0 2 and Ω 2 =2 4 Let the overap parameters be β =1for =1 2 Then Ω 1 =0 3 and Ω 2 =1 4 Let the space-tme grds be chosen wth h 1 =3/10 τ 1 =1/10 h 2 =3/4 andτ 2 =1/3 Let the grdponts be x h1 = h 1 for =010 n Ω h 1 and x h2 =1+h 2 for =04 We w consder a second-order accurate nterpoaton scheme In ths exampe B h1 1 =0 B h1 2 =3B h2 1 =4andB h2 2 = 1 The space-tme grdponts on B2 1 [0 1] are {3k/10 : k =010} We w descrbe how the entres of the map Ih k can be constructed for defnng the nterpoated vaues at x kτ fork =1 =1 and = 10 The other grd vaues can be constructed smary The boundary grdpont 3 1/10 on boundary B2 1 [0 1] s encosed n the ce wth vertces 25 0 325 0 25 1/3 and 325 1/3 whose vertces are a grdponts n the space-tme subdoman Ω 2 [0 1] Note that these four nodes are contaned n Ω 2 [0 1] We w defne our approxmaton to Uh 1 1 by usng second-order 10 bnear nterpoaton: U 1 h1 10 = 23 90 U h 2 25 0 + 46 90 U h 2 325 0 + 7 90 U h 2 25 1/3 + 14 90 U h 2 325 1/3 The nterpoaton map can be defned smary for the other grdponts on B h1 2 [0 1] so that Assumptons A2 anda3 are satsfed If a the stencs nvove convex combnatons as n the above stenc then the nterpoaton map w have maxmum norm σ hτ = 1 In ths exampe the nterpoaton error s second order n the mesh parameters h 2 and τ 2 of the space-tme grd on Ω 2 [0T] wth coeffcents ĩ Lcense or copyrght restrctons may appy to redstrbuton; see http://wwwamsorg/journa-terms-of-use

DIFFERENCE SCHEMES FOR PARABOLIC EQUATIONS 627 that depend on second dervatves of the functon beng nterpoated n the convex hu of the four nodes nvoved n the stenc: 23 wxh1 10 τ 1 90 W h 2 25 0 + 46 90 W h 2 325 0 + 7 90 W h 2 25 1/3 + 14 90 W h 2 325 1/3 C w 22 [25325] [01/3] h 2 2 + τ2 2 If the exact souton s ess smooth near B h1 2 [0T] then hgher accuracy stencs shoud be apped However determnng the regon where the souton s ess smooth woud requre some estmates see [24] for the smoothness of the souton 25 Goba dscretzaton A goba dscretzaton of 11 can be obtaned from the oca dscretzatons 22 once the boundary data gh k B 2 are specfed usng the ntergrd nterpoaton map Ih k For eac =1pet0 θ 1 be the choce of the θ-scheme on Ω [0T] Mutpyng each equaton n 22 by τ and rearrangng terms the goba dscretzaton of 11 becomes I + τ θ A I = I τ θ A Uh k I 24 + τ θ A IB I + θ A + τ θ f k+1 I + θ f k I IB U k I B 1 = 0 B 2 = I k+1 T h Uh1τ 1 U hpτ p Uh 0 I = π h Iu 0 for =1p and k =0N 1 The above system coupes the p oca paraboc dscretzatons through the B 2 = I k+1 U hτ terms It yeds a very arge system of near equatons whose parae teratve souton w be descrbed next Technques w aso be descrbed for reducng the sze of the system and the oca memory requrements 26 A parae Schwarz teratve method The near system 24 can be soved by a parae verson of the Schwarz teratve method [26 15 16 35 28 12] whch w under assumptons stated n Secton 4 converge geometrcay On a parae archtecture each processor can n prncpe be assgned to a dfferent spacetme grd Some communcaton between processors w however be necessary to compute the boundary condtons gh k B 2 nvovng the ntergrd nterpoaton maps Ih k The oads may be we baanced f the subprobems are of comparabe sze System 24 w be very arge n genera nvovng a the unknowns on a the space-tme grds However wth some care the number of unknowns and the memory requrements can be reduced 1 Frst suppose m s a common factor of the number of tme steps N on each grd for =1p e m =gcdn 1 N p Then T can be reduced by a factor m by defnng T T/m and repeatedy appyng the goba scheme on the tme ntervas [0 T ] [ T2 T ][m 1 Tm T ] Lcense or copyrght restrctons may appy to redstrbuton; see http://wwwamsorg/journa-terms-of-use

628 T P MATHEW AND G RUSSO For exampe suppose there are two subregons e p = 2 andthetme steps are τ 1 = T/100 and τ 2 = T/200 Then choose T = τ 1 =2τ 2 and repeatedy appy the goba dscretzaton scheme m = 100 tmes on [0τ 1 ] [τ 1 2τ 1 ][99τ 1 100τ 1 ] to obtan a souton on the tme nterva [0T] Ths procedure w reduce the sze of the near system by a factor 100 2 Wthn each space-tme grd Ω {0τ 1 N τ } the dscrete souton Uh k need not be stored for k = 0N Store the nta data Uh 0 I and the boundary data gh k B 2 for k =0N Usng these the oca dscrete souton can be generated by sovng the oca equatons For the parae Schwarz agorthm descrbed next t woud aso be necessary to store the noda vaues of Uh k that w be used to compute the ntergrd nterpoaton maps Once the sze of the goba system has been reduced by reducng N so that gcdn 1 N p = 1 then system 24 can be soved by a parae Schwarz teratve agorthm To dstngush the dfferent terates n the Schwarz procedure we ntroduce the foowng notaton: U k;n w denote the nth Schwarz terate at tme kτ on the grd Ω Parae Schwarz teraton Let {{U k;0 } N k=0 }p =1 be a gven startng guess 1 For n =0 1 unt convergence do 2 For =1p n parae do 3 Compute the oca boundary condtons for k =1N : T g k;n+1 B 2 = Ih k U n h 1τ 1 U n h pτ p 25 4 EndFor 5 For =1p n parae do 6 Let U 0;n+1 I = π h Iu 0 7 For k =0N 1 sove I + τ θ A ;n+1 I = ;n+1 B 1 = 0 ;n+1 B 2 = g k+1;n+1 B 2 I τ θ A U k;n+1 I + τ θ A IB ;n I + θ A IB U k;n+1 I + τ θ f k+1 I + θ fh k I 8 EndFor 9 Endfor 10 Endfor Under sutabe assumptons the terates {U k;n } above can be shown to converge geometrcay to the exact souton of 24 as n see Secton 4 Lcense or copyrght restrctons may appy to redstrbuton; see http://wwwamsorg/journa-terms-of-use

DIFFERENCE SCHEMES FOR PARABOLIC EQUATIONS 629 3 Maxmum norm propertes of the oca dscretzatons In ths secton we descrbe severa background resuts For oca dscretzatons of paraboc equatons we descrbe maxmum norm stabty maxmum prncpes comparson theorems barrer functons and contracton propertes These resuts w be used n Secton 4 to study the maxmum norm stabty of the goba dscretzaton 24 The proofs are presented here for the convenence of the reader though most of the resuts are scattered n the terature [34 14 25 21 28 10] 31 Maxmum norm stabty and a pror estmates for oca dscretzatons The foowng premnary resut provdes the bass for maxmum norm estmates of soutons to θ-schemes Lemma 31 Suppose the foowng hod 1 Let A = [ A A h ] IB satsfy Assumpton A1 2 Let 0 θ 1 and et 0 <τ 3 Let w h =[w h Iw h B] T satsfy I + τ θ A w h I + τ θ A IB w B = f h I w h B = g h B Then the foowng hods: { w h max 1 1+τ θ c 0 f h I g h B Proof Wthout oss of generaty f needed mutpy w h by 1 et w h = w h e w h w h j for a j If B thensncew h B = g h B the desred resut hods Therefore n the foowng suppose I Then the foowng hods: fh I = 1+τ θ A w + τ θ A j w j j = 1+τ θ A w + τ θ A j w j j 1 + τ θ A w + τ θ A j w j } snce A j 0for j = = 1+τ θ A + j 1+τ θ c w h τ θ A w h j 1 + τ θ c 0 w h Thus w h fh I /1 + τ θ c 0 f h I /1 + τ θ c 0 The precedng resut can be apped to the near system occurrng at each tme step n the θ-scheme provded a stabty constrant s satsfed by τ Lemma 32 Suppose the foowng hod 1 Let matrx A satsfy Assumpton A1 2 Let 0 θ 1 and defne θ 1 θ Lcense or copyrght restrctons may appy to redstrbuton; see http://wwwamsorg/journa-terms-of-use

630 T P MATHEW AND G RUSSO 3 Let 0 <τ satsfy the stabty constrant τ mn I 1 θ A h f θ 0 4 Let satsfy I + τ θ A I + τ θ A IB B = I τ θ A Uh k I τ θ A + τ θ f k+1 I + θ fh k I B = gk+1 B where Uh k I U h k B gk+1 B f k+1 I and f h k I are gven Then the foowng hods: max { g k+1 B Proof We appy the precedng emma wth f h I I τ θ A + τ θ 1+τ θ c 0 f k+1 I τ θ 1+τ θ c 0 f k I + 1 τ θ c 0 1+τ θ c 0 U k } U k I τ θ A IB U k B + τ θf k+1 I + θ f k I IB U k B g h B g k+1 B We need to estmate f h I Snce1 τ θ A 0 for a by the stabty crteron for τ and τ θ A j 0for j weobtan fh I 1 τ θ A U k h τ θ A j Uh k j + τ θ f k+1 I j + θ f k h 1 τ θ A j U h k j + τ θ f k+1 I + θ fh k I 1 τ θ c 0 Uh k + τ θ f k+1 I + θ fh k I whereweusedthat τ θ j A j τ θ c 0 The desred resut now foows by an appcaton of the precedng emma By recursvey appyng the precedng resut for k =0 1N 1 one can obtan an aprorestmate for the souton to the dscretzed oca paraboc equatons Lemma 33 Suppose the foowng hod 1 Let matrx A satsfy Assumpton A1 2 Let 0 θ 1 and defne θ 1 θ Lcense or copyrght restrctons may appy to redstrbuton; see http://wwwamsorg/journa-terms-of-use

DIFFERENCE SCHEMES FOR PARABOLIC EQUATIONS 631 3 Let 0 <τ satsfy the stabty constrant τ mn I 1 θ A h f θ 0 4 Let I satsfy I + τ θ A I + τ θ A IB B = I τ θ A Uh k I τ θ A + τ θ f k+1 I + θ fh k I B = gk+1 B for k =0 1N 1 where{gh k B }N k=0 {f h k I }N k=0 and U h 0 I Then the foowng hods: U N max gn B 1 τ θ c 0 g N 1 1+τ θ c 0 N 1 + τ + k=0 1 τ θ c 0 θ 1+τ θ c 0 + θ B N k 1 τ θ c 0 1+τ θ c 0 N 1 τ θ c 0 Uh 0 1+τ θ c I 0 f k+1 I N k 1 τ θ c 0 1+τ θ c 0 fh k I N g 0 IB U k B are gven B Proof The proof foows by a recursve appcaton of the precedng emma Remark 3 By settng c 0 = 0 n the precedng resut we may obtan a ess sharp resut: { } U N max g N B g N 1 B gh 0 B N + τθ f N I 1 + τ fh k I + τ θ f h 0 I + U 0 I Remark 4 If fh k I 0fork =0N and Uh 0 I = 0 then the above resut yeds { } U N max g N B g N 1 B gh 0 B whch s a form of the dscrete maxmum prncpe for homogeneous soutons of the oca dscretzed paraboc equaton 32 Maxmum prncpes and comparson theorems The foowng s a maxmum prncpe for the ocay dscretzed paraboc equatons k=1 Lemma 34 Suppose the foowng hod 1 Let matrx A satsfy Assumpton A1 2 Let 0 θ 1 and defne θ 1 θ Lcense or copyrght restrctons may appy to redstrbuton; see http://wwwamsorg/journa-terms-of-use

632 T P MATHEW AND G RUSSO 3 Let 0 <τ satsfy the stabty constrant 1 τ mn I θ A h f θ 0 4 Let I satsfy I + τ θ A I + τ θ A IB B = I τ θ A Uh k I τ θ A + τ θ f k+1 I + θ fh k I IB U k B B = gk+1 B for k =0 1N 1 where{gh k B }N k=0 {f h k I }N k=0 and U I 0 are gven 5 Let the nta and boundary data satsfy U 0 h 0 I fh k I 0 I k =0N gh k B 0 j B k =0N j Then the foowng hods: U k h I 0 I and k =1N Proof Let k 0 1 denote the smaest nteger such that there s an 0 wth U k0 I 0 < 0 for f there does not exst such a k 0 and 0 thenuh k I 0 for a k and and the desred concuson hods Wthout oss of generaty suppose that U k0 I =mn U k0 0 I h I < 0 Consder the oca dscretzed equaton at tme k 0 τ I + τ θ A U k0 I + τ θ A IB U k0 B = I τ θ A U k0 1 I τ θ A IB U k0 1 B + τ θ f k0 I + θ f k0 1 I U k0 B = gk0 B At the 0 -th grdpont ths becomes 1+τ θ A 0 0 U k0 + τ θ 0 = A h 0j j 0 U k0 j I τ θ A 0 0 U k0 1 τ θ A h 0j 0 j 0 + τ θ f k0 I + 0 θ f k0 1 I 0 U k0 1 j where U k0 I j =gh k B j for j B Usng the nonnegatvty of f k0 I weobtan 1+τ θ A 0 0 U k0 + τ θ A h 0j U k0 I τ θ A 0 j 0 0 U k0 1 0 j 0 τ θ 0j U k0 1 j j 0 A h Lcense or copyrght restrctons may appy to redstrbuton; see http://wwwamsorg/journa-terms-of-use

Snce U k0 1 DIFFERENCE SCHEMES FOR PARABOLIC EQUATIONS 633 j 0 for a j 1 τ θ A 0 0 0and τ θ j 0 A 0j 1+τ θ A 0 0 U k0 + τ θ A h 0j 0 j 0 U k0 j 0 0 we obtan 1 τ θ A 0 0 U k0 1 0 τ θ A h 0j j 0 U k0 1 j By assumpton U k0 0 U k0 j and A 0forj 0j 0 Rearrangng terms n theefthandsdeaboveweobtan 0 1+τ θ A 0 0 U k0 + τ θ A h 0j U k0 0 j j 0 1+τ θ A 0 0 U k0 + τ θ A h 0j U k0 0 0 j 0 = 1+τ θ A 0j U k0 0 j = 1+τ θ c 0 U k0 < 0 0 snce 1 + τ θ c 0 1andU k0 0 < 0 by assumpton We have thus arrved at a contradcton Therefore our assumpton was ncorrect and we must have U k 0 for k =0N h and I whch s the desred concuson Remark 5 A smar resut can be shown to hod when c 0 =0provdedA s an M-matrx As an mmedate appcaton of the precedng emma we obtan the foowng comparson prncpe Lemma 35 Suppose the foowng hod 1 Let matrx A satsfy Assumpton A1 2 Let 0 θ 1 and defne θ 1 θ 3 Let 0 <τ satsfy the stabty constrant 4 Let I I + τ θ A τ mn I 1 θ A h f θ 0 satsfy I + τ θ A IB B = B = gk+1 B I τ θ A Uh k I τ θ A IB U h k B + τ θ f k+1 I + θ fh k I for k =0 1N 1 where{f k I }N k=0 {gk B }N k=0 and U 0 I are gven Lcense or copyrght restrctons may appy to redstrbuton; see http://wwwamsorg/journa-terms-of-use

634 T P MATHEW AND G RUSSO 5 Let Ũ k+1 I I + τ θ A satsfy Ũ k+1 I + τθa IBŨ k+1 B = Ũ k+1 B = gk+1 B I τ θ A Ũ k τ θ hi A k+1 +τ θ f I + θ k f h I IBŨ k B for k =0 1N 1 where{ f h k I }N k=0 { gk B }N k=0 and Ũ h 0 I are gven 6 Let the forcng terms the nta and boundary data satsfy: Uh 0 0 I Ũ hi I fh k I f k hi I k =0N gh k B g h k j B j j B k =0N Then the foowng w hod: U k h I Ũ k h I I and k =1N Proof Subtract the two sets of dscretzed paraboc equatons to obtan I + τ θ A I Ũ k+1 I + τ θa IB B Ũ k+1 B = I τ θ A Uh k I Ũ h k I τ θ A IB Uh k B Ũ h k B + τ θ f k+1 k+1 I f I + τ θ fh k I f h k I B Ũ k+1 B = gk+1 B gk+1 B for k =0 1N 1 for the dfferences {Uh k I Ũ h k I } Snce U h 0 I Ũ h 0 I gh k B gk B andf h k I f h k I a have nonnegatve entres by assumpton we can appy the precedng maxmum prncpe to obtan Uh k I Ũ h k I 0 I for k =1N whch s the desred resut Our prmary appcaton of the comparson prncpe for dscretzed paraboc equatons w be to estmate on eacoca grd the moduus of dscrete homogeneous soutons Ũ h τ wth trva nta data usng a sutabe comparson or barrer grd functon w h τ Accordngy gven a dscrete homogeneous souton Ũ h τ satsfyng trva nta condtons Ũ h 0 = 0 and nontrva boundary condtons Ũ h k B 2 = g h k B 2 for k =0N we woud need to construct or guarantee the exstence of a sutabe grd functon w h τ satsfyng the requrements of the precedng emma For convenence we w seek a comparson or barrer grd functon wh k τ to be a statonary grd functon w h e w k = w h k =0N Lcense or copyrght restrctons may appy to redstrbuton; see http://wwwamsorg/journa-terms-of-use

DIFFERENCE SCHEMES FOR PARABOLIC EQUATIONS 635 To ensure appcabty of the precedng comparson prncpe the requrements on w h of the precedng emma w be reduced to A w I + A IB w B 0 I w h B max k=0n g h k B B w h 0 I snce f k =0andŨ 0 = 0 We have the foowng resut Lemma 36 Suppose the foowng hod 1 Let matrx A satsfy Assumpton A1 2 Let 0 θ 1 and defne θ 1 θ 3 Let 0 <τ satsfy the stabty constrant 4 Let Ũ k+1 I I + τ θ A τ mn I 1 θ A h f θ 0 sove Ũ k+1 I + τ θ A IBŨ k+1 B = Ũ k+1 B = gk+1 B I τ θ A Ũ k hi τ θ A IBŨ k B for k =0 1N 1 wth Ũ h 0 I =0where{ gk B }N k=0 are gven 5 Let w h =w h Iw h B T satsfy A w I + A IB w B 0 I w h B j max k=0n gh B j j B Then the foowng hods: k w h I Ũh I I for k =0N Proof We w appy the comparson prncpe Lemma 35 usng Ũ τ as defned above and Uh k w h for k =0N Due to the statonarty of Uh k the fnte dfference approxmaton of ts tme dervatve s zero and I + τ θ A B IB I + τ θ A B I τ θ A Uh k I τ θ A IB U h k B + τ f h I g k+1 B j B j hods for k =0N 1 In order to appy Lemma 35 we aso need to verfy that U 0 h I Ũ 0 =0 I j Lcense or copyrght restrctons may appy to redstrbuton; see http://wwwamsorg/journa-terms-of-use

636 T P MATHEW AND G RUSSO whch n our appcaton corresponds to w h I 0 for a From the defnton of w h weobtan 1 w h I = A fh I A IB w B Snce A satsfes Assumpton A1 t s a strcty dagonay domnant rectanguar M-matrx and satsfes 1 A 0 j I j Snce A j 0for j k f h =0andw h B 0 n the above we concude that w h I 0for I By the comparson prncpe U k h =w k Ũh k To compete the proof we observe that a of the above nequates reman vad f Ũh k I s repaced by Ũ h k I Appyng dentca arguments yeds k w h Ũh k Combnng these two resuts yeds whch s the desred resut w h Ũ k k In the next secton we descrbe a technque for constructng grd functons w h for suffcenty sma mesh sze wth h whenc 0 > 0 Such grd functons actuay standardzed versons of them where the boundary vaues are sutaby scaed w be referred to as dscrete barrer or comparson functons Once barrer functons are constructed or ther propertes are known they can be apped to derve a contracton property for homogeneous soutons 33 Exstence of contnuous and dscrete barrer functons In ths secton we prove the exstence of a dscrete barrer grd functon w h when c 0 > 0oneach oca grd and descrbe some of ts propertes We state beow the precse requrements that a grd functon must satsfy n order to be caed a barrer functon Defnton A grd functon w h =w h Iw h B T that satsfes A w I + A IB w B 0 I 31 w h B 1 0 B 1 w h B 1 1 B 2 w be referred to as a dscrete barrer or comparson grd functon The exstence of dscrete barrer grd functons w be proved n two stages Frst resuts on the exstence of a contnuous barrer functon w x assocated wth the contnuous anaog of 31 on Ω w be descrbed Second a grd functon w h = π h w w be defned by noda nterpoaton of w x ontothegrdω For suffcenty sma e for h for some h > 0 t w be shown that the resutng grd functon w h w satsfy the requrements 31 We have the foowng resut for contnuous barrer functons Lcense or copyrght restrctons may appy to redstrbuton; see http://wwwamsorg/journa-terms-of-use

DIFFERENCE SCHEMES FOR PARABOLIC EQUATIONS 637 Lemma 37 Suppose the foowng hod 1 Let 0 d x denote the dstance from x to the boundary segment B2 = Ω Ω: d x dst x B2 2 Let w x be defned as w x e αdx x Ω for α>0 Then there exsts a choce α = α > 0 so that w x defned above satsfes Lw c 0 2 x Ω w x 0 x B 1 w x = 1 x B 2 Proof We foow the constructon n Lons [27] see aso [28] Drect computaton of Le αdx yeds Le αd x = e αd x aα d x aα 2 d x 2 α bx d x+cx e αd x aα d x aα 2 d x 2 α bx d x+c 0 If we choose α = α > 0 α mn 1 c 0 2 b d Ω + a d Ω 2 Ω then + a d Ω c 0 2 + aα d x aα 2 d x 2 α bx d x 0 and we obtan Le α d x c 0 /2 > 0 Snce d x = 0 on B2 t foows that e αdx = 1 on B 2 Snce the exponenta e αdx s aways nonnegatve t foows that 0 e αdx on B1 Addtonay snce 0 d x for a x t foows that 0 e α d x 1 Remark 6 In the above constructon we tacty assumed smoothness of the dstance functon d x Unfortunatey ths may not be the case n genera even f B2 = Ω Ω s smooth However gven any 0 <ɛ β for our appcatons we may repace d x byanysmooth functon 0 d ;ɛ x satsfyng d ;ɛ x 0 x d ;ɛ x = 0 x B 2 d ;ɛ x d x+ɛ x d ;ɛ x d x ɛ x Such a pseudo-dstance functon d ;ɛ x can be constructed as foows For any γ>0 et Ω γ denote Ω γ {x :dstx Ω <γ} Then Ω =Ωβ Lcense or copyrght restrctons may appy to redstrbuton; see http://wwwamsorg/journa-terms-of-use

638 T P MATHEW AND G RUSSO 1 Let S ;ɛ denote a regon wth smooth boundares satsfyng Ω β + ɛ 2 \ Ω β ɛ 2 S ɛ Ω β +ɛ \ Ω β ɛ Then B2 S ɛ 2 Gven S ɛ etd ;ɛ x denote d ;ɛ x dst x S ɛ Then d ;ɛ x w have the foowng propertes: d ;ɛ x 0 x d ;ɛ x = 0 x S ;ɛ d ;ɛ x d x+ ɛ 2 x d ;ɛ x d x ɛ 2 x Unfortunatey d ;ɛ x w not be smooth n the regons where the eve sets of S ɛ ntersect 3 Let 0 ψ ɛ x denote a smooth probabty densty functon havng compact support of dameter ɛ /4 centered at the orgn Defne d x asthe convouton mofcaton of d ;ɛ x wthψ ɛ x as d x ψ ɛ yd ;ɛ x ydy y By constructon d x w be smooth Due to the nonnegatvty and compact support of ψ ɛ x ofdameterɛ /4 t w further satsfy d x 0 x d x = 0 x B 2 d x d x+ɛ x d x d x ɛ x Thus gven a sutabe sma but fxed choce of ɛ e α d x e αd x+ɛ w satsfy the requrements of a barrer functon for the vaue of α gven n the precedng emma wth d x repacng d x For convenence however we w henceforth assume that d x ssmooth Gven the contnuous barrer functon w x we w nterpoate t onto the grd Ω to construct a dscrete barrer functon To ensure that the resutng grd functon satsfes 31 we w requre that the dscretzaton A be at east frst order accurate and that be suffcenty sma Assumpton A5 Let x denote an nteror grdpont n Ω Then for any suffcenty smooth test functon vx we assume that Lvx = A π h v + C v x hs hods where the coeffcent C h v x nvoves hgher order dervatves of vx n the convex hu of the grdponts of the oca stenc 32 C h v x C v s +2 Ω where C s a postve constant ndependent of and v and1 s s an nteger Lcense or copyrght restrctons may appy to redstrbuton; see http://wwwamsorg/journa-terms-of-use

DIFFERENCE SCHEMES FOR PARABOLIC EQUATIONS 639 Lemma 38 Suppose the foowng hod 1 Let matrx A satsfy Assumptons A1 and A5 2 Let w x =e α d x be a smooth contnuous barrer functon satsfyng 3 Defne w h π h w where Lw c 0 2 x Ω w x 0 x B1 w x = 1 x B 2 w h = w x x Ω 4 Let satsfy 1/r h c 0 2C w s +2 Ω where C s defned n 32 Then the foowng w hod: A w I + A w h B 0 componentwse w h B 2 1 componentwse w h B 1 0 componentwse e w h s a dscrete barrer functon Proof By Assumpton A5 on the oca consstency of the fnte dfference dscretzaton matrx A weobtan A w h = Lw x +C w x hs c 0 2 + C w x hs snce Lw c 0 2 c 0 2 C w s +2 Ω hs usng A5 0 f h where h = c 0 2C w s +2 Ω 1/s The desred resut now hods due to the propertes of w x and snce w h = π h w In the next secton we use barrer grd functons to prove a contracton property of homogeneous soutons to dscretzed paraboc equatons Lcense or copyrght restrctons may appy to redstrbuton; see http://wwwamsorg/journa-terms-of-use

640 T P MATHEW AND G RUSSO 34 Contracton property of homogeneous soutons Suppose {Wh k } N k=0 denotes a homogeneous souton of the dscretzed paraboc equaton on the toca grd wth trva nta condtons Wh 0 I = 0 and nontrva boundary data Wh k B 2 = gh k B 2 on B 2 By the dscrete maxmum prncpe for paraboc equatons we obtan the bound Wh k j max max k=0n { B 2 } gh k B 2 at any nteror grdpont x j at tme kτ for k =0N However when c 0 > 0 as we have assumed a stronger property w hod n the nteror regon Ω [0T] max W k h j ρ h max max g k h j Ω h k=0n B 2 B 2 for some ρ h < 1 Ths w be referred to as the oca contracton property and w be essenta n estabshng the stabty of the goba dscretzaton We defne now the normazed contracton factor 0 ρ h 1fromadomanΩ to a subregon Ω for dscrete homogeneous soutons of the dscretzed paraboc equaton on the th grd Defnton Let {Wh k } N k=0 denote a homogeneous souton of the foowng dscretzed paraboc equaton wth trva nta condtons: I + τ θ A W k+1 I + τ θ A IB W k+1 B = I τ θ A Wh k I τ θ A IB W h k B W k+1 B 1 = 0 W k+1 B 2 = g k B 2 W 0 I = 0 where gh k B 2 =1for B 2 We defne the normazed contracton factor ρ wth 0 ρ h 1 as 33 ρ h max max W k h I k=0n Ω h Our frst resut n ths secton provdes an upper bound for ρ h ntermsofthe contracton factor for the contnuous barrer functon e α d x from the precedng secton Lemma 39 Suppose the foowng hod 1 Let 0 <β denote the overap parameter from subdoman Ω to Ω 2 Let matrx A satsfy Assumptons A1 and A5 andetτ satsfy the stabty crteron 3 Let c 0 > 0 4 Let 0 < α be chosen so that the grd functon w h = π h e α d x s a dscrete barrer grd functon for h 5 Let ρ h denote the normazed contracton factor on the th grd wth W h as empoyed n 33 Lcense or copyrght restrctons may appy to redstrbuton; see http://wwwamsorg/journa-terms-of-use

DIFFERENCE SCHEMES FOR PARABOLIC EQUATIONS 641 Then the foowng hods: ρ h max Ωh w h max Ωh e α d x Consequenty ρ h e α β f d x =d x e α β ɛ f d x ɛ d x d x for some ɛ > 0 In ether case ρ h < 1 f β ɛ Proof By assumpton on the mesh sze the precedng emma yeds that w h π h e α d x s a dscrete barrer grd functon By appyng comparson prncpe 35 from subsecton 32 wth Uh k = w h and Ũ h k = Wh k above we obtan Wh k w h k e α d x k e α β ɛ k whch s the desred resut The above estmates for the contracton factor ρ h are quatatve and nvove an unknown constant α We ndcate beow how more quanttatve theoretca bounds can be obtaned for ρ h on unform grds see [21] Exampe Consder a paraboc equaton u t + Lu = f n one space dmenson e Ω R wherelu = u + bu + cu and where 0 <band 0 <care constants For convenence consder a subdoman Ω =0 1 wth Ω =ã b where0< ã< b < 1 Suppose that a unform grd s constructed on Ω wth mesh sze =1/M and grdponts x = for 0M Dscretze u by three pont fnte dfferences bu by upwnd fnte dfferences and cu by a one pont approxmaton on the above unform grd and suppose 0 and 1 are nteror ponts n Ω To estmate the dscrete contracton factor ρ h we sove the dfference equatons bh +1u 1 + 2+bh + ch 2 u 1u +1 = 0 for =1M 1 u 0 = 1 u M = 1 whose genera souton has the form u = c 1 σ 1 + c 2 σ 2 where σ 1 and σ 2 are roots of the quadratc σ 2 2+bh + ch 2 σ +bh +1=0 Lcense or copyrght restrctons may appy to redstrbuton; see http://wwwamsorg/journa-terms-of-use

642 T P MATHEW AND G RUSSO In terms of b c and the roots are σ 1 = 1+ bh 2 + ch2 4c + b 2 2 + h +2bhc + c 2 h 2 1/2 4 σ 2 = 1+ bh 2 + ch2 4c + b 2 2 h +2bhc + c 2 h 2 1/2 4 The constants c 1 and c 2 can be computed by enforcng the boundary condtons c 1 + c 2 = 1 whch yeds σ M 1 c 1 + σ M 2 c 2 = 1 c 1 = σ M 2 1 σ M 2 σ M 1 1 σ M 1 c 2 = σ M 2 σ M 1 The contracton factor ρ h can be estmated as ρ h max {:x ã b} σ M 2 1σ 1 +1 σm σ M 2 σ M 1 1 σ 2 Gven b c ã and b these can be determned quanttatvey Our man resut n ths secton reates the dscrete contracton factor ρ h whch was defned for the normazed Drchet boundary condtons to the case of genera Drchet boundary condtons Lemma 310 Suppose the foowng hod 1 Let matrx A satsfy Assumptons A1 and A5 2 Let 0 θ 1 and defne θ 1 θ 3 Let 0 <τ satsfy the stabty condton τ mn I 1 θ A h f θ 0 4 Let Ṽ h k satsfy I + τ θ A Ṽ k+1 I + τ θ A IBṼ k+1 B = Ṽ k+1 B 1 = 0 I τ θ A Ṽ k+1 B 2 = g k+1 B 2 Ṽ 0 I = 0 Ṽ k hi τ θ A IBṼ k B for k =0N 1 5 Let ρ h denote the normazed contracton factor defned by 33 n subsecton 34 wth assocated grd functon {W k } N k=0 Lcense or copyrght restrctons may appy to redstrbuton; see http://wwwamsorg/journa-terms-of-use

DIFFERENCE SCHEMES FOR PARABOLIC EQUATIONS 643 Then the foowng hods: max {:x Ω k=0n } k Ṽh Proof Let ṼB denote the number 2 ṼB 2 ρ h max B 2 max Ω k max Ṽ h k=0n k max Ṽ h k=0n Defne a grd functon W h k by scang Ṽ h k so that ts boundary vaues have maxmum moduus one: k Ṽ W k h h k ṼB 2 Appy comparson prncpe 35 from subecton 32 empoyng Uh k = Wh k where Wh k s the grd functon assocated wth the normazed contracton factor ρ h defned n 33 of subsecton 34 and empoyng Ũ h k = ± W h k where W h k s defned above The desred resut foows mmedatey from the normazed contracton property Remark 7 Due to the statonarty of the dscrete barrer functon w h the upper bound for the contracton factor ρ h w be ndependent of τ 4 Maxmum norm stabty and accuracy of the goba dscretzaton In ths secton we prove that the goba dscretzaton 24 s stabe n the maxmum norm and anayze ts accuracy We aso show that the parae Schwarz agorthm s geometrcay convergent The proofs are motvated by [32 10] and empoy Pcard s contracton mappng theorem In the frst secton we descrbe the contracton mappng theorem and exstence and unqueness resuts for 24 We aso dscuss the geometrc convergence of the parae Schwarz terates In the second secton we descrbe the stabty of the goba dscretzaton In the thrd secton we appy the stabty theorem to estmate the accuracy of the goba nonmatchng grd dscretzaton n terms of the oca dscretzaton and nterpoaton errors 41 Contracton mappng theorem The exstence and unqueness of soutons to the goba dscretzed system 24 w be proved by appyng Pcard s contracton mappng theorem [3] whch we summarze beow for convenence In addton the geometrc convergence of the parae Schwarz terates w foow from propertes of the contracton mappng Theorem 41 Suppose the foowng hod 1 Let H be a compete metrc space wth metrc d 2 Let T : H Hbe a contractve mappng e for any X Y Hwe have d T X T Y δdx Y where δ<1 Then the foowng hod Lcense or copyrght restrctons may appy to redstrbuton; see http://wwwamsorg/journa-terms-of-use

644 T P MATHEW AND G RUSSO 1 There exsts a unque fxed pont U Hof T satsfyng U = T U 2 Let U 0 be any eement of H Then U n T n U 0 U geometrcay wth d U n+1 U δd U n U 3 For any U 0 Hwe have d U 0 U δ n d U 0 U 1 1 δ d T U 0 U 0 Proof See [3] In our appcatons we w choose the metrc space H and mappng T so that goba dscretzaton 24 s a fxed pont equaton for T Furthermore the parae Schwarz terates {U n } gven a startng guess U 0 w correspond to U n+1 = T U n For convenence we w consder a system of equatons more genera than 24 Let f h k I and gk B 2 be gven forcng terms for k =0N and =1p Let ũ 0 I be gven nta data for =1p We consder the foowng genera system of equatons for unknowns U hτ = {{Uh k } N k=0 }p =1 : 41 I + τ θ A I + τ θ A IB B = B 1 = 0 I τ θ A U k I τ θ A IB U k B k+1 + τ θ f I + θ k f h I B 2 = I k+1 U hτ + g k+1 B 2 Uh 0 I = ũ 0 I for k =0N 1and =1p Our choce for H w be based on the near system 41 Gven f h k I gk B 2 for k =0N and =1p and ũ 0 I we defne H { X hτ : Xh k satsfy foowng constrants } I + τ θ A X k+1 I + τ θ A IB Xk+1 B = I τ θ A Xh k I τ θ A IB Xk B k+1 + τ θ f I + θ k f h I X k+1 B 1 =0 Xh 0 I =ũ 0 I for k =0N 1and =1p H s not a vector space due to the near nhomogeneous constrants However H s cosed n the vector space of a grd functons endowed wth the maxmum norm and consequenty H w be a compete metrc space f the metrc d defned on H s nherted from the standard maxmum norm Gven X hτ Y hτ H we defne d X hτ Y hτ X hτ Y hτ Lcense or copyrght restrctons may appy to redstrbuton; see http://wwwamsorg/journa-terms-of-use

DIFFERENCE SCHEMES FOR PARABOLIC EQUATIONS 645 where denotes the maxmum norm on the vector space of a space-tme grd functons X hτ Y hτ max max max X k =1p k=0n Ω h Y k For the above H we defne a mappng T : H Hso that near system 41 s a fxed pont equaton of T on H GvenX hτ Hdefne X hτ = T X hτ as 42 I + τ θ A Xk+1 I + τ θ A k+1 IB X B = X k+1 B 1 = 0 I τ θ A Xk hi τ θ A k+1 + τ θ f I + θ k f h I X k+1 B 2 = I k+1 X hτ + g k+1 B 2 X 0 I = ũ 0 I IB X k B for k =0N 1and =1p It mmedatey foows that system 41 s a fxed pont equaton of T Addtonay the parae Schwarz terates from Secton 2 can be descrbed n terms of the mappng T : Gven a startng guess U 0 hτ to 41 the subsequent parae Schwarz terates {U n hτ } are U n hτ TU n 1 hτ n =1 2 The exstence and unqueness of soutons to system 41 and 24 w be guaranteedbypcard scontractonmappng theorem provdedt s a contracton mappng n the metrc space H Ths contracton property of T s pvota to the stabty anayss n ths paper and s proved next Theorem 42 Suppose the foowng hod 1 Let matrx A satsfy Assumptons A1 and A5 for =1p 2 Let 0 <τ satsfy the oca stabty crteron τ mn I 1 θ A h f θ 0 for =1p 3 Let the ntergrd nterpoaton maps {Ih k } k satsfy Assumptons A2 and A3 4 Let the oca overap parameters β be chosen arge enough and the oca mesh sze h sma enough so that the contracton factor ρ h satsfes δ h σ hτ max ρ h δ<1 where σ hτ denotes the maxmum norm of the map {Ih k } k Then the foowng hod 1 The mappng T w be a contracton mappng on H satsfyng d T X hτ T Y hτ δdx hτ Y hτ X hτ Y hτ H 2 System 41 w be unquey sovabe wth a souton U hτ Lcense or copyrght restrctons may appy to redstrbuton; see http://wwwamsorg/journa-terms-of-use

646 T P MATHEW AND G RUSSO 3 Gven any startng guess U 0 n hτ H the terates U hτ = T n U 0 hτ whch correspond to the parae Schwarz terates converge geometrcay to the unque fxed pont U hτ : T n U 0 hτ U hτ δ n U 0 hτ U hτ Proof The proof s smar to [10] and rees on the maxmum prncpe and the contracton property Gven X hτ Y hτ H we need to estmate d T X hτ T Y hτ n terms of d X hτ Y hτ For convenence denote X hτ = T X hτ and Ỹhτ = T Y hτ By appyng the defnton of T wenotethat X hτ Ỹhτ w satsfy the homogeneous system 43 I + τ θ A h Xk+1 I Ỹ I + τ θ A Xk+1 IB B Ỹ k+1 I τ θ A Xk h I Ỹ k I = X k+1 B 1 Ỹ k+1 B 1 =0 X k+1 B 2 Ỹ k+1 B 2 = I k+1 X hτ Y hτ X 0 Ỹ 0 hi hi =0 B + τ θ A IB Xk h B Ỹ k B for k =0N 1and =1p By the maxmum prncpe from Secton 3 we obtan X hτ Ỹhτ max max I k h X hτ Y hτ k=0n =1p σ hτ max k=0n max max Xh k =1p Y k Ω Here σ hτ s the maxmum norm of the ntergrd nterpoaton map Ih k Snce X hτ Y hτ H ther dfference X hτ Y hτ satsfes a dscretzed homogeneous paraboc equaton and by the contracton property we obtan max Ω X k h Yh k ρ h X hτ Y hτ and consequenty max max max X k h Y k h k=0n =1p max ρ X hτ Y hτ Ω =1p Combnng the two bounds we obtan X hτ Ỹhτ max max I h k k=0n =1p X hτ Y hτ σ hτ max max Xh k k=0n ;=1p Y k h Ω σ hτ max ρ X hτ Y hτ =1p = δ X hτ Y hτ Snce d X hτ Ỹhτ = X hτ Ỹhτ and d X hτ Y hτ = X hτ Y hτ and snce by assumpton δ<1 we obtan that T s a contracton d Ṽhτ W hτ δdv hτ W hτ Lcense or copyrght restrctons may appy to redstrbuton; see http://wwwamsorg/journa-terms-of-use

DIFFERENCE SCHEMES FOR PARABOLIC EQUATIONS 647 The unque sovabty of the near system foows from the unqueness of the fxed pont of a contracton mappng The geometrc convergence of the parae Schwarz terates foows by the geometrc convergence of the Pcard terates We note that the precedng resut does not drecty provde an aprorbound for the maxmum norm of the souton Ths w be done n the next secton 42 Stabty of the goba dscretzaton In ths secton we w derve an aprorbound for the maxmum norm of the souton to 41 Theorem 43 Suppose the foowng hod 1 Let matrces A satsfy Assumptons A1 and A5 for =1p 2 Let 0 <τ satsfy the oca stabty constrant τ mn I 1 θ A h f θ 0 for =1p 3 Let I k satsfy Assumptons A2 and A3 4 Let the overap parameters β be chosen arge enough and the mesh szes sma enough wth h so that δ σ hτ max ρ h δ<1 5 Let U hτ = {U k } k denote the unque souton of system 41 Then the maxmum norm of the souton U hτ satsfes U hτ 1+ σ hτ 1 δ N 1 + τ k=1 max =1p ũ 0 I Ω h + τ θ f N I Ω N f k I Ω + τ θ f 0 I Ω + k=0 g k B 2 B 2 Proof Choose any sutabe grd functon X hτ Hand use t as an nta guess n the Pcard fxed pont teraton By Theorem 41 the contracton mappng d X hτ U hτ 1 1 δ d X hτ T X hτ Usng that the metrc n H was nherted from the maxmum norm we obtan U hτ X hτ + U hτ X hτ trange nequaty = X hτ + d U hτ X hτ = X hτ + 1 1 δ d X hτ T X hτ from above Therefore to obtan a bound for U hτ we ony need to choose X hτ Hand estmate X hτ and d X hτ T X hτ Lcense or copyrght restrctons may appy to redstrbuton; see http://wwwamsorg/journa-terms-of-use

648 T P MATHEW AND G RUSSO Accordngy choose X hτ = {Xh k } k as the soutons to the oca dscretzed paraboc equatons wth trva boundary condtons 44 I + τ θ A X k+1 I + τ θ A IB Xk+1 B = X k+1 B 1 = 0 X k+1 B 2 = 0 X 0 I = ũ 0 I I τ θ A Xh k I τ θ A + τ θ f k+1 I + θ f k h I IB Xk B for k =0N 1and =1p Snce the oca probems for {Xh k } k are decouped we can estmate the maxmum norm of eacoca component X h τ ndependenty by usng the oca aprorestmates from Secton 3 Ths yeds X hτ max =1p ũ 0 I + τ θ f N I Ω + τ N 1 k=1 f k I Ω h +τ θ f h 0 I N Ω + g k B 2 h B 2 We next estmate X hτ TX hτ For convenence et X hτ = T X hτ We note that X hτ X hτ w satsfy the dscretzed homogeneous paraboc equaton 45 I + τ θ A X k+1 B 1 X k+1 I X k+1 B 1 =0 X k+1 k+1 B 2 X B 2 X 0 I X 0 I k+1 X I + τ θ A IB = I τ θ A = I k+1 X hτ =0 X k+1 B X k I X k I k+1 X B k=0 τ θ A IB Xh k B X h k B for k =0N 1and =1p By appyng the dscrete maxmum prncpe and usng σ hτ = I k weobtan X hτ X hτ σ hτ X hτ Substtutng these n our expresson for U hτ weobtan X hτ 1+ σ hτ 1 δ N 1 + τ k=1 whch s the desred resut max =1p ũ 0 I Ω + τ θ f N I Ω N f k I Ω + τ θ f 0 I Ω + k=0 g k B 2 B 2 Lcense or copyrght restrctons may appy to redstrbuton; see http://wwwamsorg/journa-terms-of-use

DIFFERENCE SCHEMES FOR PARABOLIC EQUATIONS 649 The above resut depends crtcay on the contracton factor δ < 1 If the overap β of the oca subregons are suffcenty arge so that σ hτ max ρ h δ<1 unformy n then the goba dscretzaton w be stabe In the next secton we appy the above stabty resut to estmate the accuracy of the nonmatchng grd dscretzaton 24 43 Accuracy of the goba dscretzaton From the genera theory for dscretzaton of near evouton equatons [30] we expect a stabe and consstent scheme to be convergent The same hods for the nonmatchng overappng grd dscretzaton scheme 24 consdered here We w now consder the consstency of the goba scheme 24 whch s measured by the magntude of the resdua when π hτ u e the exact souton ux t restrcted to the coecton of space-tme grds s substtuted nto the scheme Defnton Gven the restrcton u hτ = π hτ u of the exact souton ux t tothe space-tme grds we defne the grd functon Eh k u torepresenttheoca dscretzaton error on the grd Ω at tme kτ E k+1 u I + τ θ A u k+1 I h I τ θa u k I + τ θ A IB uk+1 B + θ A IB uk B τ θ f k+1 I + θ fh k I We use Dh k u to denote the boundary grd functon representng the oca ntergrd nterpoaton error Dh k u u k+1 B 2 I k+1 u hτ At each grdpont x kτ the oca dscretzaton error Eh k u andthenterpoaton error Dh k u can be estmated by expandng the stencs usng Tayor seres expansons centered at the grdpont The resutng estmate w nvove the oca mesh parameters and τ and hgher order dervatves of u atoneormore ponts n the convex hu of the grdponts nvoved n that stenc For convenence suppose that the dscretzaton and nterpoaton errors satsfy Eh k u q C u q;1 +2q ;2 +1 Ω h ;1 46 [0T ] + τ q ;2 Dh k u r C u r;1 r ;2 B h ;1 + τ r ;2 2 [0T ] where B 2 s a neghborhood of the boundary B2 contanng the unon of a ces nvoved n the oca ntergrd nterpoaton We w now estmate the accuracy u hτ U hτ of the goba dscretzaton 24 n terms of the oca dscretzaton and nterpoaton errors Eh k u anddh k u respectvey Theorem 44 Suppose the foowng hod 1 Let matrces A satsfy Assumptons A1 and A5 for k =1p Lcense or copyrght restrctons may appy to redstrbuton; see http://wwwamsorg/journa-terms-of-use