TR/87 April 1979 INTERPOLATION TO BOUNDARY ON SIMPLICES J.A. GREGORY

Transcription

1 TR/87 Apr 979 ITERPOLATIO TO BOUDARY O SIMPLICES by JA GREGORY

2 w60369

3 Itroducto The fte dmesoa probem of costructg Lagrage ad Hermte terpoats whch atch fucto ad derate aues at a fte umber of pots o a smpex R has bee co- sdered by a umber of authors [490] More recety Masfed [8] has cosdered the fte dmesoa probem of costructg bedg fucto terpoats whch match fucto ad derate aues o the etre boudary of a tetrahedro ad more geeray a -smpex Masfed's wor geerazes a scheme for terpoatg o trages frst descrbed Barh Brhoff ad Gordo [] I the preset paper we deeop a ew scheme for bedg fucto terpoato o smpces whch s a geerazato of a terpoat for trages descrbed [7] The esseta feature of the scheme s that t s a mog aerage or 'bed' of terpoats each of whch matches fucto ad derate aues o a but oe of the faces of the smpex The paper begs Secto 3 wth the deeopmet of a expct represetato of a fte dmesoa Hermte terpoato poyoma for the smpex Ths terpoat s a atura geerazato of Hermte two pot Tayor terpoato oe arabe ad cudes the barate trcubc poyoma terpoat of Brhoff [3] as a speca case The exstece of the terpoat the geera case s suggested by Masfed [8] It shoud be stressed that the pecewse appcato of the terpoat oer a uo of o-oerappg smpces R > ges a C 0 goba fucto ee though derates across the ertces of the smpca compex are cotuous The mportace of the Hermte terpoat to ths paper s

4 that ts bass fuctos are used to costruct the weght fuctos of the bedg fucto scheme Ths scheme s descrbed geera terms Secto 4 ad poyoma ad ratoa exampes of the scheme are deeoped Sectos 5 ad 6 These bedg fucto schemes ue the Hermte terpoat ge C goba fuctos whe they are pecewse apped oer a smpca compex A the terpoato schemes descrbed ths paper defe bouded dempotet ear operators e proectors o some approprate fucto space Thus the schemes are abe to reproduce a fuctos the rage of the terpoato proector The rage s thus caed the precso set of the terpoat The subset of a poyomas of a certa degree whch ca be cotaed the precso set s mportat determg the accuracy of each terpoat Such poyoma sets are cosdered deta for each terpoato scheme of the paper The paper begs wth a summary of otato Secto I partcuar the barycetrc coordate system for a smpex s troduced sce each terpoat w be descrbed terms of ths arat system Premary otato Let s {x λ / 0 λ λ } where () x (x x ) R defe a smpex R wth ertces ()

5 3 ( ) R (3) We assume that the smpex s o-degeerate so that the ertces do ot e a m-dmesoa hyperpae of the represetato x λ where λ R m< I ths case uquey defes the barycetrc coordate system λ λ (x) Let E deote a pot o the face λ 0 e the face opposte the ertex V The E ca be represeted as (4) E λ where λ 0 λ (5) The barycetrc coordate system ca be terpreted as defg a affe trasformato betwee (x x ) R ad (λ λ ) R Ths trasformato taes S wth ertces V oto a stadard smpex S ad 0 caoca bass of th ertces V e respectey where e deotes the R We defe a derate operator aog the edge og the ertces V ad V by D ( ) / x ad a product of such operators aog a edges whch meet at V by D D Π (6) (7) Furthermore f deote o-egate tegers ad

6 4 ( ) (8) Deotes a mut-dex of of these tegers the we defe D D Π Fay wth (9) ( ) (0) we defe D D π ( / x ) Π ( / x ) () We cocude ths secto wth a emma whch w be usefu () subsequet wor: Lemma Let f : R R be defed by f(x) g( λ λ λ ) () Where g s a rea df feretabe fucto of arabes ad λ are defed by (4) The D f ( / λ - / λ ) (3) Proof Let h (x x x ) g(λ λ λ ) where x x x are defed by λ x ad Ths trasformato s o-sguar sce the smpex s o-degeerate ad (4) s the partcuar case x ow

7 5 / λ / λ )g ( ) h/ x (( ) h/ x Thus from (6) the emma s estabshed where f(x) h(x x ) 3 A Hermte Iterpoat for the Smpex Theorem 3 Ge the o-egate teger et f be a rea aued fucto defed o S whch s such that D f (V ) are we defed aues where (3) {0 } (3) The there exsts a terpoato poyoma p expcty defed by p(x) λ P ( ) D (f/λ ) λ (V ) (33) where p ( λ ) λ /! λ (λ λ λ ) (34) whch s such that D p (V ) Proof Sce D f (35) D λ (V ) 0

8 6 t foows from (33) that for a { λ p D (f/λ ) (V ) } ' ' D p(v ) D Appcato of Lebtz's rue the ges that (V ) )} ( ' ) (f/λ β β D p (V ) {D λ } (D p '}{ D (V ) β β ' ow usg Lemma β D / λ ) β }{ λ ' / ' p (V ( / λ!} (V ) ' ) { Π Π ' f β 0 otherwse Hece substtutg ad recosttutg the summato usg Lebtz's rue ges D p (V ) β D D (λ f β (V ) β {D f/λ ) whch cpmpetes the proof of the theorem λ (V ) β }{D (f/λ )} (V ) Exampe the terpoat defed by (33) s a atura geerazato of Hermte two pot Tayor terpoato oe arabe where wth ad x λ V λ λ p(x) λ 0 λ 0 R { λ /!} { λ /!} λ λ we hae ( D (f/ λ )(V ) ( )(V ) (36) D (f/ λ

9 see Das [5 p37] Equato (36) ca be expressed the carda bass form where p(x) 0 h (λ ) D f (V ) 0 h (λ ) D f (V ) (37) ( h λ ) ( λ) λ ( )!/!! ( )! (38) Whe ad (33) the trcubc terpoat o a trage of G Brhoff [3] s obtaed The pecewse appcato of the terpoat (33) oer a bouded doma whch cossts of the uo of o-oerappg smpces R ges a C goba fucto except the case whe t s C The bedg fucto terpoats of the foowg sectos w howeer ge C goba fuctos for a It foows from the theory of fte dmesoa terpoato ad the expct represetato ge by (33) ad (34) that the ear fuctoas defed by (3) are eary depedet oer the ()() dmesoa poyoma space defed by T { λ λ / 0 } Π (39) Aso the ear operator P defed by P[f](x) p(x) f C (S ) (30) where p s ge by (33) s a proector o C (S ) wth rage T Thus P [f] - P[f] (3)

10 8 ad P[f] f for a f e T (3) The foowg theorem ges more sght to the ature of T Theorem 3 T s the space of poyomas whose restrcto aog a edge og ay two ertces V ad V s a poyoma of degree e T { λ / 0 } Π (33) Proof Ceary from (39) T { λ / 0 } Π Thus we requre to proe that f f { λ where 0 } Π (34) the f T as defed by (39) The proof of ths comprses two parts: () Suppose for The sce t foows that for ow (34) ca be wrtte as f λ λ Π λ λ ( Π λ ) Π λ ad sce -- t foows that f T () Suppose for a The K ()

11 9 Assume further the ducte hypothess that f γ for a M K () where M () ow f K M- the (3 4) ca be wrtte as f ( λ )( Π Π λ λ ) where M ad Thus f γ usg ether the ducte hypothess f or part () f - The ducte hypothess s true for M () sce ths case for a ad the (34) ca be wrtte as f ( λ )( λ Π Π λ λ ) so that f T Hece by ducto the hypothess s true for a 0 M () Ths competes the proof of the theorem The foowg coroary foows mmedatey from (33) Coroary 3 The foowg cusos hod: where P T P ( ) P () ' (35) P { x / x R 0 K } Π (36) s the set of poyomas of degree K

12 0 It shoud be oted that P T T P ad ( ) A Agebrac Idetty We are ow a posto to dere a detty whch w be esseta for the bedg fucto schemes whch foow Let f(x) The f T ad usg (3) the foowg detty ca be dered: a (x) for a x R (37) where a є T are poyoma fuctos defed by (x) a λ P (λλ( )!/! ad the P are ge by (34) (38) 4 A Geera Scheme for Bedg Fucto Iterpoato I ths secto we defe a geera scheme for terpoatg fucto ad derate aues ge a faces of the smpex S Two partcuar mpemetatos of the scheme w be ge subsequet sectos The terpoato scheme s defed the foowg theorem Theorem 4 Let f ad P [f] where the P are ear operators be rea aued fuctos defed o S whch are such that ( D P [f](e ) ) ( D ) f (E ) for a Thus P [ f ] terpoates f ad ts derates of order ad (4)

13 ess o a faces of the smpex excudg the face λ 0' The P[f](x) - a(x) P [f](x) x S (4) where the a are ge by (38) defes a ear operator P whch s such that ( D ) ( P[f] (E ) D f )(E ) for a (43) Proof The proof s amost sef edet reyg o (37) ad (38) More formay appyg Lebtz's rue ges ( D ) P[f](E ) D a P [f] (E ) β β β β D a (E ) D P[f] (E ) β β β D a where sce (38) cotas the factor That (E ) D β f (E ) λ we hae used the fact D β a (E ) 0 for a β Furthermore from (37) β f β 0 D a (E ) 0 f 0 < β ad hece (43) foows Remars Suppose H s a subspace of bouded rea aued fuctos defed o S whch s such that P :H H ad whch s such that the derates defed by (4) exst ad are bouded o H Suppose further that

14 P [g ] (x) 0 (44) for a g H such that ) ( D g (E ) 0 (45) The t foows that P (I-P )[f](x) 0 for a f H (46) where I s the detty operator ad moreoer that P(I-P) [f] (x) 0 for a f H (47) Thus P ad P defe bouded dempotet ear operators e proectors o H Aso f P [f] f for a f H (48) e f H s the precso set of the operator P the usg (37)t foows that P[f] f for a f I H (49) e the precso set of P cotas the tersecto of the H 5 Poyoma Bedg Fucto Iterpoato Scheme We ow cosder a exampe of the geera bedg fucto scheme defed Theorem 4 where proectors P are defed by Booea sums of poyoma Tayor terpoato proectors Let E λ V (λ λ ) V λ (5)

15 3 be the pot of tersecto of the face λ 0 wth the e through x ad V Aso et C (S ) whch s parae to the edge og V be the fucto spaces 0 C (S ) { f / D f C (S ) for a } (5) The Tayor terpoato proectors T ca be defed o C (S ) by ( f) ) T [f](x) (λ /!) D (E 0 (53) Some propertes of these proectors are ge the foowg emma: Lemma 5 The Tayor proectors defed by (53) hae the terp- oato propertes that ( D [f])(e ( f) (E 0 f C ) ) (S ) T ad the precso set property that D (54) T [f] f for a f H (55) where H { λ g (λ )/ g C (S ) 0 } (56) Proof Sce λ 0 at E ad sce ( / λ - / λ ) g (E ) 0 for ay dfferetabe fucto gt foows by use of Lemma that ( D ) ( ' T [f] (E ) ( / λ / λ ) (λ '/'!) D f) ' 0 ( ( f) ) ) (E ) 0 D (E (E ) (E ) ow whe x E we hae (54) foow Aso f E E ad hece the terpoato propertes

16 4 f(x) ' λ g ( λ ) 0 ' the t foows from (5) ad Lemma that D f (E ) ( / λ - / λ ) f) (E ) '! 0 f s g ( λ ) f otherwse ' Thus substtutg to (53) ges the desred precso set resut (55) whch competes the proof of the emma The pro e etors T are commutate oer C (S ) The proof of the foowg theorem s the easy supped by ducto usg the Booea sum theory of Gordo [6] where Theorem 5 The fod Booea sum P T T (57) T T T T T (58) defes a proector o C (S ) whch s such that D p [f] (E ) D f (E ) for ad a 0 (59) Furthermore p [f] for a f H H (50)

17 5 Equatos (5-9) mpy that ( P ) ( [f] (E ) D f )(E ) D for a Thus the foowg coroary foows mmedatey from Theorems 5 ad 4 : Coroary 5 P[F](X) a (x) P [f](x) x S f C (S ) (5) defes a bedg fucto terpoat o S Moreoer sce P H ( ) the (49) mpes that P[f] f for a f P ()- (5) We refer to (5) as the poyoma bedg fucto terpoat sce the P ad P oe poyoma weghts Remar The derates D f are 'compatbe' o C By ths we mea that the derates do ot deped o (S ) the order whch the dfferetato s performed Ths codto aows the commutatty of the proectors T The proectors defed the foowg secto do ot requre such strget compatbty codtos Howeer ths oes the troducto of ratoa terms ad the bedg fucto terpoat whch resuts has geera ess poyoma precso 6 Ratoa Bedg Fucto Iterpoato Scheme I ths secto we cosder a ratoa defto of the proectors P for the geera bedg fucto scheme of Theorem 4 The defto s ducte where Theorem 4 s used to defe a proector o a -smpex as a weghted aerage of proectors o (-) dmesoa

18 6 smpces I order to defe the scheme the foowg otato s troduced: Let I {ν{ν ν ν } / ν { } ν ν for a } be the ( ) (6) dmesoa set of combatos of{ } tae at a tme wthout repettos Aso et S ν νєi be the dmesoa smpex ge by the tersecto of S wth the hyperpaes through x parae to the faces λ 0 ν' where ν f {} - ν (6) The smpex S has ertces V λ υ { λ } V υ V (63) ad t s easy show that x λ V (64) where λ λ /{ λ } λ /{ λ } (65) defe the barycetrc coordates of S wth respect to the ertces V Cosder ow the partcuar case { } I where I s a ()/ dmesoa set The the oe dmesoa smpex S s the e segmet og the two ertces V V λ V λ V {λ {λ λ } λ } V V { V V } (66)

19 7 I ths case x λ V λ V { } (67) where λ λ /{λ λ } ad λ λ /{λ λ } (68) The e segmet S ν ν{ν ν } ca be terpreted as the tersecto of the smpex S wth the e through x whch s parae to the edge og V ad V Thus wth the otato of Secto 5 we hae V V E E ad Let C (S ) be the fucto space defed by C (S ) { f / f C (S ) ad D f C (E ) I C (E ) 0 < < } (69) The a Hermte two pot Tayor operator P ν{ν ν } ca be defed aog the e segmet S ν by P [f](x) 0 h (λ /{λ λ }) (λ λ ) D f (V ) 0 h (λ /{λ λ }) (λ λ ) D f (V ) ( ) f C (S ) (60) whe re the h are defed by (38) If we excude for the momet the sguarty λ λ 0 the P ν ν {ν ν } defes a terpoato proector C (S ) o ad we hae the foowg emma:

20 8 Lemma 6 The Hermte proectors defed by (60) hae the terpoato propertes that D D P [f] P [f] (E ) (E ) D D f f (E ) (E ) 0 f C (S ) (6) ad the precso set property that P [f ] f for a f є H (6) where H { λ g (λ ) / or 0 g ' C (S ) } (63) Proof The terpoato propertes foow drecty from the theory of Secto 3 the speca case We aso ow from Secto 3 that P [(λ /{λ λ } ) ] (λ /{λ λ }) 0 The precso set resut s the proed by wrtg f λ Π g (λ (λ /{λ ) λ }) {I λ } Π g (λ ) ad otg that λ ν s a scaar wth respect to the ear operator P ν I Lemma 6 we hae gored the probem of the sguarty λ λ 0 We ow show Lemma 63 that ths sguarty s remoabe by cosderg the behaour of the Hermte proector P { } a eghbourhood of the pot V S where

21 9 V V 0 λ λ λ (64) I order to proe Lemma 63 we cosder frst a premary emma whch essetay rees o the operator defed by the eft had sde of (66) ahatg poyomas of a certa degree Stadard resuts for such operators proe dffcut to appy because of the ature of partcuar operator ad fucto space oed We thus ge a drect proof of ths emma: Lemma 6 Let f Cν ν (S ) ad et p(x) (λ ν /!) D f (V ) 0 ν ν (65) ν be the Tayor terpoat to f about V ν the γ γ D { )} (λ λ ) D (P f) (V χ (γ) (66) for a γ where γ C(S m ν χ (x) 0 x S (67) x Proof For smpcty ad wthout oss of geeraty we cosder the case {} wth the derate operator D γ D γ γ (γ γ γ γ ) (68) (see (9)) where γ γ γ (D ca be expressed as a ear combato of such derates) ow f h C (S ) ( D ) h ( h) (V )f D {h(v )} (69) D (V )f

22 0 Thus sce p f C (S ) γ ad substtutg D D D t foows that γ D {( D (p f)(v ) )} D D γ γ D γ (p γ γ γ γ D D γ γ D 0 f) (V ) D γ (p f) (V ) γ γ γ γ D 0 (q g) (V ) (60) where g D γ γ D γ f g - γ C (S ) ad from (65) ad the dua of (69) q D γ - γ 0 γ ( λ /!) D γ p D g (V ) λ γ R R γ C (S ) Fay sce - γ a Tayor expaso D g about V ges D (q g) λ γ μ whe re m C(S) ad m x V μ (x) 0 Substtuto of ths resut (60) competes the proof of the emma Lemma 6 3 Let f C (S ) ad et P V be defed by the Hermte proector (60) The

23 ( ) ( ) D P [f](x) D f (V) for a m x V x S (6) Proof Let p be defed by the Tayor terpoat (65)The p H ad thus p(x) - P V [ f ](x) P [p - f](x) 0 h (λ /{λ λ })(λ λ ) D (p f) (V ) Thus by appcato of Lebtz's rue ( D (P [f]))(x) 0 β D β β [ h (λ /{λ β ) ]D λ })(λ λ [ D (p f) (V )] (λ β 0 β where t ca be show that that -β λ ) K β m 0 x V λ /{ λ λ } for a x It thus foows that m x V D P (P [f] (X) where f β (x) D K [ β D (p f) (V )](6) s bouded o S by use of the fact S ( ) 0 for x S β (6) the β β so that by Lemma 6 we may substtute β D ( ) β (p ) λ ) (x) D f)(v (λ β The proof of the emma s the competed by otg that

24 m x V D P (X) D f (V) Remar Lemma 63 has bee proed uder sghty weaer codtos tha those used Theorem of [] for the speca case of Hermte proectors o the trage Hag defed a terpoato proector P { } I wth terpoato propertes o E ad E we ow cosder the the geera case { } I I ths case a proector P s ductey defed by the foowg theorem: Theorem 6 Let { } P [f](x) (x)p [f](x)f C a I (S) (63) defe a proector P a (x) I where (λ ) (λ { ) /!}( )!/! (64) (cf (38)) ad P V D p [f] ( - ) I s defed by (60) The (E ) D f (E ) for a (65) ad PPV {} [f] f for a f f I H (66) ν where H {} s defed by (63) Proof Assume the ducte hypothess that (65) s true for a I - The (63) s smpy a appcato of Theorem 4

25 3 o the dmesoa smpex S ν Thus (65) hods for a є I where the sguartes of the ν a o I E are easy show to be remoabe sce P -{} hae commo terpoato propertes o ths set ad a ow by Lemma 6 the ducte hypothess s true for a ν є I ad hece the frst part of the theorem s proed The precso set property foows drecty from (6) ad (49) Ths competes the proof A bedg fucto scheme for the smpex S s defed the foowg coroary as a speca case of Theorem 6 We refer to ths scheme as the ratoa bedg fucto terpoat Coroary 6 ν P[f](x) P [f](x) ν { } x s f I C (S ) (67) ν defes a bedg fucto terpoat o S S Moreoer sce P C H {} for a the P[f] f for a f ν P (68) Remar The poyoma precso set (68) s cotaed the rage of each Hermte two pot Tayor proector because of the smpe addte form of the ducte scheme Masfed [8] cosders a addte ad product composto of the Hermte proectors whch ges a hgher precso ratoa scheme o the tetrahedro Ths scheme athough of a more compex form ca be geerazed to the -smpex Fay Masfed has proed that the product composto of a the Hermte proectors (60) maps a fucto f oto the fte dmesoa terpoat defed by Theorem 3 of ths paper

26 4 Refereces RE Barh G Brhoff ad WJ Gordo Smooth terpoato trages J Approxmato Theory 8 (973) 4-8 RE Barh ad JA Gregory Compatbe smooth terpoato trages J Approxmato Theory 5 (975) GBrhoff Trcubc poyoma terpoato Proc at Acad Sc USA 68 (97) PG Caret ad C Wagscha Mutpot Tayor formuas ad appcatos to the fte eemet method umer Math 7 (974) PJ Das Iterpoato ad Approxmato Basde ew Yor WJ Gordo Bedg-fucto methods of barate ad mut-arate terpoato ad approxmato SIAM J umeraa 8 (97) JA Gregory A bedg fucto terpoat for trages 977 Durham Symposum o Mutarate Approxmato Academc Press L Masfed Iterpoato to boudary data tetrahedra wth appcatos to compatbe fte eemets J Math Aa App 9 RA coades O a cass of fte eemets geerated by Lagrage terpoato SIAM J umer Aa 9 (97) RA coades O a cass of fte eemets geerated by Lagrage terpoato II SIAM J umer Aa 0 (973) 8-89 A Zese Hermte terpoato o smpces the fte eemet method Proceedgs of the Czechosoa Coferece o Dffereta Equatos ad ther Appcatos Bro A Zese Poyoma approxmato o tetrahedros the fte eemet method J Approxmato Theory 7 (973)