Anisotropic mesh renement in stabilized Galerkin methods. Thomas Apel y Gert Lube z. February 13, 1995

Transcription

1 Anisotropic ms rnmnt in stabilizd Galrkin mtods Tomas Apl y Grt Lub z Fbruary 13, 1995 Abstract. T numrical solution of t convction-diusion-raction problm is considrd in two and tr dimnsions. A stabilizd nit lmnt mtod of Galrkin/Last-squars typ accomodats diusiondominatd as wll as convction- and/or raction-dominatd situations. T rsolution of boundary layrs occuring in t singularly prturbd cas is accomplisd using anisotropic ms rnmnt in boundary layr rgions. In tis papr, t standard analysis of t stabilizd Galrkin mtod on isotropic mss is xtndd to mor gnral mss wit boundary layr rnmnt. Simplicial Lagrangian lmnts of arbitrary ordr ar usd. Ky Words. Elliptic boundary valu problm, convction-diusion-raction problm, nit lmnt mtod, Galrkin/Last-squars, ms rnmnt, anisotropic nit lmnts AMS(MOS) subjct classication. 65N30, 65N50 Contnts 1 Introduction Notation and local stimats for gnral nit lmnts 3 3 A stabilizd Galrkin mtod on gnral mss Statmnt of t problm : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 3. Existnc and stability of discrt solutions : : : : : : : : : : : : : : : : : : : Convrgnc towards t wak solution : : : : : : : : : : : : : : : : : : : : : Convrgnc towards rgular solutions : : : : : : : : : : : : : : : : : : : : : : Coic of t numrical damping paramtrs : : : : : : : : : : : : : : : : : : 10 4 Anisotropic rnmnt Ncssity of boundary and intrior layr rnmnt : : : : : : : : : : : : : : : Ms gnration wit anisotropic boundary layr rnmnt : : : : : : : : : : Modid rror stimat on anisotropic lmnts : : : : : : : : : : : : : : : : : 13 5 Application to problms of cannl typ Dnition and proprtis of t solution : : : : : : : : : : : : : : : : : : : : : Gnration of t anisotropic ms in t boundary layr : : : : : : : : : : : Error stimats : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 16 Rfrncs 17 T papr appard as Prprint SPC95 1 in t sris if t Cmnitz rsarc group \Scintic Paralll Computing". y Tcnisc Univrsitat Cmnitz-Zwickau, Fakultat fur Matmatik, D{09107 Cmnitz, Grmany; mail: na.apl@na-nt.ornl.gov z Gorg-August-Univrstat Gottingn, Facbric Matmatik, Institut fur Numrisc und Angwandt Matmatik, Lotzstr. 16{18, D{37083 Gottingn, Grmany; mail: lub@namu01.gwdg.d 1

2 1 Introduction Tis papr is concrnd wit t nit lmnt solution of t following lliptic boundary valu problm in a boundd polydral domain IR d, d = ; 3, wit Lipscitz L " u "u + b ru + cu = f in ; (1.1) u = 0 (1.) " (0; 1] is a paramtr. (1.1) (1.) is a linar(izd) diusion-convction-raction modl. In particular, arbitrary ratios P (x) " 1 kb(x); R d k (Pclt numbr) and (x) " 1 jc(x)j will b considrd. Hnc t wol rang from (locally) diusion-dominatd (P; 1) to (locally) convction- and/or raction-dominatd problms (P 1 and/or 1) is of intrst. In cas of P 1 and/or 1, (1.1) (1.) is of singularly prturbd typ and t solution u may gnrat sarp boundary or intrior layrs wr t solution of t limit problm wit " = 0 is not smoot or cannot satisfy t boundary condition (1.). T rsolution of suc layrs is oftn t main intrst in applications and will b considrd in tis papr. Standard Galrkin nit lmnt solutions may sur from numrical instabilitis wic ar gnratd by dominant convction and/or raction trms unlss t ms is sucintly rnd. As a rmdy, stabilizd Galrkin mtods av bn proposd: t stramlin diusion mtod (SD) [7, 1, 17], t Galrkin/Last-squars mtod (GLS), s for xampl [13], and sock-capturing variants of tm, s for xampl [9, 10, 14, 15]. In contrast to standard mtods of upwind typ, stabilizd Galrkin mtods av t advantag to b consistnt wit t wak formulation of (1.1) (1.). W will focus on t (GLS)-mtod. Up to now, stabilizd Galrkin mtods wr analyzd for isotropic mss, tat mans =% = O(1) for "! 0,! 0, wr and % dnot t diamtr of t nit lmnt and t diamtr of t largst inscribd ball in, rspctivly. But a rsolution of boundary and intrior layrs wit isotropic lmnts lads to an ovrrnmnt. An anisotropic ms rnmnt in t sns lim =% = 1 is muc mor cint in suc tin layrs. "!+0 W rmark tat t prmission of % = o ( ) for! 0 was alrady discussd in [6, 16, 18, 19] but ty did not driv an advantag (from t point of viw of numrical analysis) of using dirnt lmnt diamtrs in dirnt dirctions. Tis rmdy was rmovd in [3, 4, 5] by proving sarpr stimats on t rfrnc lmnt, and t improvd stimats wr applid to stablis a-priori ms rnmnt nar gomtrical singularitis (dgs) in t cas of diusion-dominatd quations (Poisson typ problms) [3, 5]. In tis cas anisotropy was usd in a sligtly dirnt sns tan w do r, namly lim =% = 1.!+0 But tis maks no dirnc for t anisotropic local stimats. W not tat anisotropic lmnts wr also considrd from otr points of viw in [0, 3, 4, 5, 6, 8, 9, 30]. In tis papr, w xtnd t numrical analysis of t Galrkin/Last-squars mtod to mss wic ar anisotropically rnd at last in boundary layrs. T aim is to driv rror stimats in t nrgy norm uniformly wit rspct to " (0; 1]. Suc an approac is tortically possibl also in intrior layrs. But, unfortunatly, it turns out tat t lmnts in t layr av to b orintd wit rspct to t manifold wr t layr is locatd; in gnral tis cannot b don a-priori. A numrical localization procdur for intrior layrs is dscribd in [30]. W rmark tat "-uniform stimats wr also drivd using xponntially ttd Galrkin mtods [1, ] or nit dirnc mtods on crtain ortogonal mss [7]. T outlin of t papr is as follows: In Sction w considr Lagrangian intrpolation on simplicial lmnts and rviw local inqualitis in t anisotropic cas. In Sction 3 w introduc t stabilizd Galrkin mtod (GLS) for problm (1.1) (1.). Undr wak conditions to t ms (maximal angl condition instad of minimal angl condition) w prov xistnc and stability of t discrt solution, as wll as convrgnc to t wak

3 ;3 ; ;1 E ; ;1 E Figur.1: Illustration of t dnition of t lmnt rlatd ms sizs. solution u W 1; () and to rgular solutions u W r+1; (), r 1. Morovr w driv t optimal coic of t numrical damping paramtrs. Sction 4 is dvotd to t analysis of anisotropic ms rnmnt in boundary layrs of problm (1.1) (1.) using t rsults of Sction. Howvr, t critical point is an assumption on t Sobolv norms of u wic ar ard to prov in gnral cass. For tis rason w apply ts quit gnral rsults to a spcial class of problms wr suc a priori knowldg on u is availabl and tus an almost optimal rnmnt stratgy can b proposd. Tis is don in Sction 5 by considring domains of cannl typ. In particular, t actual coic of t lmnt diamtrs in t rnmnt zon and t dtrmination of t numrical damping paramtrs is addrssd. T nal rror stimat is almost uniform wit rspct to t small paramtr ". Notation and local stimats for gnral nit lmnts W considr Lagrangian lmnts on simplics IR 3, d = ; 3; wit spacs P k of polynomials of maximal dgr k 1. T intrpolant of a continuous function v is uniquly dtrmind by (I (k) v)(x(i) ) = v(x (i) k+d ) (i = 1; : : :; n, n = dim(p k ) = d ), wr x (i) ar t nodal points of t lmnt. In tis sction, w summariz t local inqualitis and a dnsity rsult wic wr provd in [4]. For xploring t dirnt sizs of t lmnt in dirnt dirctions w introduc t following notation, compar Figur.1. For IR lt E b t longst dg of. Tn w dnot by 1; mas 1 (E ) its lngt and by ; mas ()= 1; t diamtr of prpndicularly to E. In t tr-dimnsional cas, w procd by analogy. Lt again E b t longst dg of, and lt F b t largr of t two facs of wit E F. Tn w dnot by 1; mas 1 (E ) t lngt of E, by ; mas (F )= 1; t diamtr of F prpndicularly to E, and by 3; 6 mas 3 ()=( 1; ; ) t diamtr of prpndicularly to F. Not tat for t lmnt sizs t rlation 1; : : : d; ; (.1) olds. Introduc furtr a Cartsian coordinat systm (x 1; ; x ; ; x 3; ) suc tat (0; 0; 0) is a vrtx of ^, E is part of t x 1; {axis, and F is part of t x 1; ; x ; {plan. T twodimnsional cas is tratd by analogy. Subsquntly, tis systm will b calld lmnt rlatd coordinat systm. By contrast w considr a discrtization indpndnt coordinat systm (x 1 ; x ) or (x 1 ; x ; x 3 ) wic may b global or rlatd to t boundary or it may b problm rlatd in any otr sns but indpndnt of t nit lmnt ms. For anisotropic intrpolation rror stimats w av to assum tat t lmnts fulll a maximal angl condition. Maximal angl condition (D): Tr is a constant < (indpndnt of and T ) suc tat t maximal intrior angl of any lmnt is boundd by : : 3

4 Maximal angl condition (3D): Tr is a constant < (indpndnt of and T ) suc tat t maximal intrior angl f; of t four facs as wll as t maximal angl E; btwn two facs of any lmnt is boundd by : f; ; E; : Morovr, w nd for all anisotropic stimats t coordinat systm condition. Coordinat systm condition (D): T lmnt rlatd coordinat systm (x 1; ; x ; ) can b transformd into t discrtization indpndnt coordinat systm (x 1 ; x ) via a translation and a rotation by an angl, wr j sin j C ; = 1; : Coordinat systm condition (3D): T transformation of t lmnt rlatd coordinat systm (x 1; ; x ; ; x 3; ) into t discrtization indpndnt systm (x 1 ; x ; x 3 ) can b dtrmind as a translation and tr rotations around t x j; -axs by angls j; (j = 1; ; 3), wr j sin 1; j C 3 = ; j sin ; j C 3 = 1 ; j sin 3; j C = 1 : (.) Not tat w us t symbol C for a gnric positiv constant, tat mans, C may b of dirnt valu at ac occurrnc. But C is always indpndnt of t function undr considration, of t nit lmnt ms, and particularly of ". On t contrary, som constants ar indxd wit a lttr for latr rfrnc to tm. Lt W m; (), m IN; b t usual Sobolv spacs wit t norm and t spcial sminorm kv; W m; ()k 8 < : Z jjm jd vj dx W us a multi-indx notation wit 9 = ; 1= ; jv; W m; ()j = ( 1 ; : : :; d ); jj = 1 + : : : + d ; D 1 t numbrs i (i = 1; : : :; d) ar non-ngativ < d d Z jj=m jd vj dx 9 = ; 1= ; = 1 1; d d; ; Lmma.1 (Invrs inquality) Assum tat for t lmnt t coordinat systm condition olds. Tn for v P k, k IN arbitrary, t stimat olds. T particular rsult kv; L ()k C d i=1 ; L i! 1= : (.3) kv; L ()k C s 1 d; jv; W 1; ()j (.4) is valid witout t coordinat systm condition. Morovr, tr is C s = 0 for k = 1. Lmma. (Anisotropic intrpolation rror stimats) Assum tat for an lmnt t maximal angl condition as wll as t coordinat systm condition old. Tn for v W k+1; () and m = 0; : : :; k t stimat jv I (k) v; W m; ()j C jj=k+1 m olds, if d = or m < k. If v W k+; (), tr olds jv I (k) for d = ; 3, m = 0; : : :; k. v; W m; ()j C k+1 mjjk+ m jd v; W m; ()j (.5) jd v; W m; ()j (.6) 4

5 Rmark.3 T siz of t constants C in t coordinat systm condition inuncs t siz of t constants in (.5) and (.6). Witout t coordinat systm condition w can only prov stimats witout driving advantag of t dirnt lmnt diamtrs, s t following lmma. Lmma.4 Assum tat t lmnt fullls t maximal angl condition. Tn for v W k+1; () and m = 0; : : :; k t stimat jv I (k) v; W m; ()j C k+1 m 1; jv; W k+1; ()j (.7) olds, if d = or m < k. If v W k+; () tr olds for d = ; 3, m = 0; : : :; k. jv I (k) v; W m; ()j C k+ `=k+1 ` m 1; jv; W `; ()j (.8) Not tat t coordinat systm condition is not ncssary for Lmma.4. Lt T = fg b an admissibl triangulation of = S, tat mans, lt proprtis (T 1) (T 5) of [8, Captr ] b fullld. Assum tat T satiss t maximal angl condition. Morovr, introduc t spacs V and V by V W 1; 0 () fv W 1; () : = 0g; (.9) V fv V : vj P k () 8 T g: (.10) T indx indicats tat w ar considring a family of spacs for! +0, itslf caractrizs t ms siz; w can for xampl tink of = max T 1;. Lmma.5 (Dnsity of V in V ) Lt u V b an arbitrary function, tn lim!+0 inf ku v ; W 1; ()k = 0: v V Rmark.6 If v as t proprty v W r+1; () wit r > k (or r > k + 1) tn t stimats (.5) and (.7) (or (.6) and (.8), rspctivly) old tru. If r < k (or r < k + 1) w sould us I (r) for intrpolation. Not tat I (r) u V, too. 3 A stabilizd Galrkin mtod on gnral mss 3.1 Statmnt of t problm W considr t scond ordr lliptic boundary valu problm wit t basic assumptions L " u "u + b ru + cu = f in IR d ; d = ; 3; (3.1) u = 0 (3.) (H.1) 0 < " 1, b [W 1;1 ()] d, c L 1 (), f L (), (H.) r b = 0, c 0 almost vrywr in. T variational formulation of (3.1) (3.) rads Find u V; suc tat B G (u; v) = L G (v) 8v V: (3.3) 5

6 wr B G (u; v) "(ru; rv) + 1 f(b ru; v) (b rv; u) g + (cu; v) ; (3.4) L G (v) (f; v) ; (3.5) and (: ; :) G dnots t innr product in L (G), G. Morovr, t standard Galrkin mtod (G) of (3.3) is introducd by Find u V ; suc tat B G (u ; v ) = L G (v ) 8v V : (G) V and V ar introducd in Sction. W rmind t wll-known fact tat t solution u of (G) on isotropic mss may sur from non-pysical oscillations unlss t lmntwis numbrs P " 1 1; kb; [L 1 ()] d k; " 1 1;kc; L 1 ()k (3.6) ar sucintly small. As a rmdy, w considr t following stabilizd mtod of Galrkin/ Last-squars typ: wit Find U V ; suc tat B SG (U ; v ) = L SG (v ) 8v V : (GLS) B SG (u; v) B G (u; v) + (L " u; L " v) ; (3.7) L SG (v) L G (v) + (f; L " v) ; (3.8) and a st f g of non-ngativ numrical diusion paramtrs to b dtrmind blow. 3. Existnc and stability of discrt solutions First of all, w stat lowr and uppr bounds of t bilinar form B SG (: ; :). Lmma 3.1 Undr t assumptions (H.1), (H.), tr olds for v V wit vj L () 8 T tat B SG (v; v) = jjj v jjj "; wit jjj v jjj "; "krv; L ()k + k p cv; L ()k + kl " v; L ()k : (3.9) Proof St u = v in (3.7). Lmma 3. For v V and u V wit uj L () 8 T tr olds jb SG (u; v )j jjj v jjj "; 8 < : jjj u jjj "; + Z ku; L ()k! 1= 9 = ; (3.10) wit C s is t constant from (.4). Z minfb " 1 ; 1 + "C s d; + C g; (3.11) B kb; [L 1 ()] d k; C kc; L 1 ()k: (3.1) Proof Intgration by parts of t non-symmtric part of B SG (: ; :) togtr wit (H.) yilds for all u; v V nc wit (3.4), (3.7), and (3.9) 1 f(b ru; v) (b rv; u) g = (b rv; u) ; 6

7 jb SG (u; v)j "kru; [L ()] d k krv; [L ()] d k + k p c u; L ()k k p c v; L ()k + + kl " u; L ()k! 1= kl " v; L ()k! 1= + j(b rv; u) j jjj u jjj "; jjj v jjj "; + j(b rv; u) j: (3.13) Considr t last trm at t rigt and sid. W gt for v V via invrs inquality (.3) j(b rv ; u) j ku; L ()k minfkb rv ; L ()k; k "v + b rv + cv ; L ()k + k"v ; L ()k + kcv ; L ()kg ku; L ()k minfb krv ; [L ()] d k; kl " v ; L ()k + "C s 1 d; krv ; L ()k + p C k p c v ; L ()kg minfb " 1= ; maxf 1= ; " 1= C s 1 d; ; C1= g g jjj v jjj ku; L ()k (3.14) wr jjj v jjj is dnd in analogy to (3.9) by jjj v jjj "krv ; L ()k + k p cv ; L ()k + kl " v ; L ()k : (3.15) Using (3.13) { (3.15) w gt t assrtion by standard inqualitis. Furtrmor, w nd t following a-priori stability stimat. Lmma 3.3 For t solution U V and t rsidual L " U f of scm (GLS) tr olds jjj U jjj "; + kl " U f; L ()k D C(minfC F " 1 ; 1 g + )kf; L ()k (3.16) wit max, inf c(x) and Fridrics' constant C F. Proof St v = U in (GLS). Lmma 3.1, togtr wit Holdr's and Fridrics' inqualitis, implis jjj U jjj "; B SG (U ; U ) = L SG (U ) kf; L ()k ku ; L ()k + kf; L ()k! 1= kl " U ; L ()k! 1= nc kf; L ()k minf 1= k p cu ; L ()k; C F kru ; [L ()] d kg + + p kf; L ()k 1 jjj U jjj ";0 + 1 minf 1 ; " 1 C F gkf; L ()k kl " U ; L ()k! 1= kl " U ; L ()k + kf; L ()k ; jjj U jjj "; (minf 1 ; " 1 C F g + )kf; L ()k A sligt modication of t proof yilds t wigtd control of t discrt rsidual. Lmma 3.3 implis uniqunss and stability of t (GLS)-solution on an gnral admissibl ms (including anisotropic ms rnmnt). Torm 3.4 Undr assumptions (H.1), (H.) tr xists on and only on solution U V of scm (GLS) wic additionally satiss (3.16). 7

8 3.3 Convrgnc towards t wak solution Lt us considr now t strong convrgnc of t family fu g of solutions of (GLS) to t wak solution u V of (3.3). Not tat w us only data undr t assumptions (H.1), (H.) and a tcnical condition (H.3) on t paramtr st f g: (H.3) lim!+0 maxf ("C s d; + B " 1 + C )g = 0 Torm 3.5 Assum tat (H.1) { (H.3) ar valid. Tn t solution U V of (GLS) convrgs strongly in V to t solution u V of (3.3) according to Proof W split t rror u U as follows: lim jjj u U jjj ";0 = 0: (3.17)!+0 u U = (u u ) + (u U ) w 1 + w (3.18) wit t Galrkin solution u V of (G), tat mans of (GLS) wit = 0 8. Lt v b t bst approximat of v in V : jjj v v jjj ";0 = min v V jjj v v jjj ";0 : Dnoting by ~ u u t approximation rror, tr olds via (3.3) { (3.5) and (3.10) wit = 0 8: jjj u u jjj ";0 B G (u u; u u) = B G (u u; u u) jjj u u jjj ";0 4 jjj ~ jjj";0 + B " 1 k~; L ()k! 1= 3 5 ; nc jjj u u jjj ";0 jjj ~ jjj ";0 + B " 1 k~; L ()k! 1= F (~): Lmma.5 yilds tat for u V lim F (~) = 0 and!+0 lim jjj w 1 jjj ";0 lim (jjj u u jjj ";0 + jjj u u jjj ";0 ) = 0: (3.19)!+0!+0 For w = u U V w av by (3.4), (3.7), (3.8), (3.9), (3.18), (G), (GLS), and (3.16) jjj w jjj ";0 = B G (w ; w ) = (L " U f; L " w )! 1= kl " U f; L ()k kl " w ; L ()k! 1= D k "w + b rw + cw ; L ()k! 1= DT: (3.0) T invrs inquality (.3) yilds T " C s d; krw ; [L ()] d k + B krw ; [L ()] d k + C k p cw ; L ()k maxf ("C s d; + B " 1 + C )g jjj w jjj ";0 : (3.1) A sucint condition for lim jjj w jjj ";0 = 0 is tn (H.3), nc via (3.18), (3.19) w arriv!+0 at (3.17). 8

9 Rmark 3.6 W conjctur tat t asymptotic rror stimat (3.17) rmains valid undr t wakr condition (H.3') j "C s d; j 1 8 T and lim!+0 maxf (B " 1 + C )g = 0. Tis assrtion is clar in t cas of picwis linar, simplicial lmnts (k = 1) bcaus tr olds C s = 0. In t gnral cas k 1 w wr not abl to prov tis, vn if w assumd t wak rgularity assumption L " u = f in L () 8 T. W found only a simplication of t proof of (3.17) but w wr not abl to avoid (H.3). 3.4 Convrgnc towards rgular solutions W considr now t cas of smoot solutions of (3.3) according to (H.4) u V \ W r+1; () for som r IN, r 1. Consquntly, w av B SG (u U ; v) = 0 8v V : (3.) Not tat (H.4) is valid wit r = 1 if is convx. In ordr to simplify t fortcoming analysis w assum tat t following modication of (H.3') olds: (H.3") 1 "C s d; + C, wic is wit 1 1; d; and " 1 1; C quivalnt to Hnc w rplac Z in (3.11) by 1; "( C s + ) : ~Z minfb " 1 ; 1 g: (3.3) Torm 3.7 Lt (H.1), (H.), (H.3"), (H.4), as wll as t maximal angl condition b satisd. Tn tr old for t rror u U jjj u U jjj "; C A E minfk;rg 1; ju; W 1+minfk;rg; ()j (3.4) if minfk; rg 3 or d =, and jjj u U jjj "; C A witout ts conditions. E is dnd by minfk;r 1g E `=minfk;r 1g 1 ` 1;ju; W `+; ()j ; (3.5) E " + C 1; + (" 1; + B + C 1;) + 1; minf" 1 B ; 1 g: (3.6) Proof Using t rror splitting u U = (u I (minfk;rg) u) + (I (minfk;rg) u U ) + ; w conclud from Lmmata 3.1, 3. and (3.), (3.3) tat jjj jjj "; B SG (; ) = B SG ( ; ) = B SG (; ) jjj jjj "; 8 < : jjj jjj "; + ~Z k; L ()k! 1= 9 = ; ; (3.7) jjj u U jjj "; jjj jjj "; + ~Z k; L ()k! 1= : (3.8) 9

10 T local intrpolation rror stimat (.7) yilds (3.4). Not tat in t two-dimnsional cas m = k is allowd. Furtrmor, for k = 1 tr is v j = 0 8v V, tat mans, (.7) is usd only for m = 0; 1. For (3.5) w us I (minfk;r 1g) instad of I (minfk;rg) and (.8) instad of (.7) in ordr to b abl to trat also linar and quadratic lmnts in t tr-dimnsional cas. 3.5 Coic of t numrical damping paramtrs A suitabl stratgy is to coos t numrical damping paramtrs in suc a way, tat t trms E in (3.4) and (3.5) ar minimizd wit rspct to. Lmma 3.8 T trm E dnd in (3.6) is minimal for 1; = p q if P " 1 + P + P ~ 1 + P + (3.9) (convction-raction dominatd cas), and ( " 1; = min ; B " 1 + P P + ) if 0 P ~ P (3.30) (diusion dominatd cas). Hnc tr olds E C"(1 + P + ) = C(" + 1; B + 1;C ); (3.31) d = ; 3, 1 r k. For t dnition of P and s (3.6). Proof Lt rst b P " 1 1 1; suc tat minf" 1 B ; 1 g = 1. Tn E is minimizd for Tn t condition P 1; = p : (3.3) " 1 + P + " 1 1 1; is quivalnt by (3.3) to P p 1 + P +, tat mans P (1 + p )=, and w av E = "(1 + + p 1 + P + ) and tus (3.31). Considr now t cas tat 1; minf" 1 B ; 1 g = "P. If w dmand tat t trm (" 1; + B + C 1; ) in (3.6) is not gratr tan t otr trm " + C 1; + 1; " 1 B tn w nd for t inquality " + C 1; + 1; B " 1 " 1; + B + C 1; = 1; " 1 + P + : (3.33) 1 + P + A simpl calculation givs via (3.6) tat E C"(1+P + ) C"(1+ p 1 + P + + ), nc (3.31). Rmark 3.9 Not tat assumption (H.3) in Subsction 3.3 is not guarantd by (3.9) (3.30) if k, s also Rmark 3.6. Rmark 3.10 T analysis of Lmma 3.8 is valid only modulo multiplicativ constants in (3.4) (3.5) wic ar indpndnt of ", 1;, and. Trfor it is possibl to improv formula (3.9) (3.30). Lt us considr picwis linar lmnts (k = 1) and t cas c = 0. In cas of d = 1 wit constant cocints " and b, w av t wll-known suprconvrgnc rsult of nodally xact solutions for = 1; cot P P : (3.34) 10

11 T proposd tuning approac rsults in = 1; B P p P < : 1; B if P 1 P 1; 1B if P 1 (3.35) wic rcts t asymptotic baviour of (3.34) for bot P! +1 and P! Anisotropic rnmnt 4.1 Ncssity of boundary and intrior layr rnmnt A critical inspction sows tat rror stimats (3.4){(3.6) may b uslss in boundary or intrior layr rgions R " unlss t ms is sucintly n: ju; W r+1; ()j = O(1) for "! +0 if 6 R " ; ju; W r+1; ()j! +1 for "! +0 if R " : Practical calculations undrlin tis and sow t occurrnc of so-calld wiggls in t cas of larg numbrs P 1 and/or 1, for tir dnition s (3.6). Typically, ty occur globally in for t standard Galrkin mtod, but ty ar rstrictd to a numrical layr rgion R of widt O( j ln j) for t (GLS)-scm. It turns out tat t layrs R ar in gnral largr tan t boundary and intrior layrs R " of widt O(" 1j ln "j). T sizs of 1 and dpnd on t problm and caractriz t layr, for 1 s t xampl blow, dpnds on t discrtization and is not known in gnral. Nvrtlss, a rsolution of sarp layr gradints is oftn t main intrst in applications, and improvd mtods ar ncssary. Usually tis is accomplisd by using xponntially ttd mtods [1] or isotropic ms rnmnt. W try to rsolv R " by mans of anisotropic ms rnmnt in ordr to dcras t complxity of t discrt problm. T anisotropic ms in t boundary layr sould giv uniform bounds for jjj u U jjj "; wit rspct to " and contain a minimal numbr of nit lmnts. In ordr to xploit t anisotropic intrpolation rsults, s Sction, w nd sarp local Sobolv norm stimats of u. Suc stimats dpnd strongly on t asymptotic structur of u for 0 < " 1, for xampl on t typ of t boundary and intrior layrs, on t xistnc of turning points wit kbk = 0, or on priodic caractristics. In t cas of sucintly smoot data w can tak advantag of asymptotic xpansions, s [7]. Unfortunatly, suc stimats ar rar in t litratur for t cas of Lipscitzian domains IR d, d, and lss rgular data, s [] for t problms apparing. Futur rsarc sould xtnd t knowldg about t solutions. T rst task is to dtct t location of t manifolds wr boundary and intrior layrs manat. Tis could b accomplisd in an adaptiv mtod, s [30]. Nvrtlss, w focus r on incomprssibl ow lds b. In contrast to comprssibl ow problms, intrior layrs (as socks) ar rar, and t location of boundary layrs is wll-known. To gt an xampl w considr a simpl but typical boundary layr problm for t diusion-convction-raction modl (3.1) (3.) in a squar or cub = (0; 1) d : L " u "u d i=1 cos( i + cu = f in ; (4.1) u = g (4.) wit i [0; ]. In cas of i (0; ) tr occur only ordinary (or outow) boundary layrs of ticknss O(" ln 1 " ) at x i = 0, i = 1; : : :; d. In t cas of i =, i = 1; : : :; d, (no convction) and c > 0 tr xists a boundary layr of ticknss O( p " ln 1 " ) along t For 1 = 0 and (= 3 ) = parabolic (or caractristic) layrs of ticknss 11

12 T ) anisotropic lmnt K K T = T ) ;K Figur 4.1: Anisotropic ms in t boundary layr rgion O( p " ln 1 " ) ar locatd wit t xcption of t inow boundary part at x 1 = 1 (no layr) and t outow boundary part at x 1 = 0 wr again ordinary boundary layrs of ticknss O(" ln 1 " ) occur [1]. In Sction 5 w considr a mor gnral typ of domain, but only in t two-dimnsional cas. 4. Ms gnration wit anisotropic boundary layr rnmnt T ida is now to construct a xd ms in t boundary layr rgion wit anisotropic rnmnt and to us an isotropic ms away from t boundary layrs, possibly constructd by an advancing front tcniqu and (isotropically) rnd via standard adaptiv mtods (including intrior layr rnmnt). Witout loss of gnrality w assum tat a boundary layr of ticknss O(" ln 1 ") is locatd at som lin or W av = 1 or = 1 in xampl abov but it can b mor gnral. W introduc local coordinats (; ) or (; ; ) wit = 0 T. As a starting point, w gnrat an ortogonal ms via lins (plans) = i, = j, ( = k ) wit ral numbrs i, j, k (i = 0; : : :; M, j = 0; : : :; j 0, k = 0; : : :; k 0 ) and particularly 0 = 0, M = d(") " ln 1 ". W assum tat for a boundary layr rctangl (rctangular cub) K = [ i ; i+1 ] [ i ; i+1 ] or K = [ i ; i+1 ] [ i ; i+1 ] [ i ; i+1 ] t following rlation olds clos to t boundary: ;K i+1 i K maxf ;K ; ;K g maxf j+1 j ; k+1 k g: (4.3) T xcptions ar gomtric singularitis (cornrs, dgs) of t wr possibly dirnt boundary layr parts intrsct. Not tat our approac guarants an strongr rnmnt tr. T lmnts K ar split into simplicial lmnts ( triangls or 6 ttradra) wic satisfy t maximal angl condition and t coordinat systm condition wit rspct to t boundary ttd coordinat systm, s Figur 4.1. T ms outsid t (xd) boundary layr rgions sould b of isotropic typ. T rsults of Sction on invrs and intrpolation rror stimats ar tn applicabl. Not tat an isotropic ms rnmnt is possibl via standard rror stimators in t rgion away from t boundary layrs. Tis is vn dsirabl in t cas of intrior layrs. Bcaus of t diculty wit t coordinat systm condition, no attmpt will b mad r to rsolv intrior layrs (wic ar in gnral locatd at caractristic lins or surfacs) wit anisotropic lmnts. Howvr, tis problm was attackd xprimntally in [30]. W rfr also to t tst in [4] wr a numrical xampl is givn for t snsibility of t solution wit rspct to t coordinat systm condition. 1

13 4.3 Modid rror stimat on anisotropic lmnts W try to rn t rror analysis of Torm 3.7 on anisotropic boundary lmnts R. In ordr to apply t anisotropic intrpolation rsults of Sction, it is ssntial tat ac lmnt R satiss t maximum angl condition and t coordinat systm condition wit rspct to t boundary ttd systm, s gur 4.1. Starting again from (3.8) and using Lmma., w nd jjj u U jjj "; I (u) wit I (u) "kr; L ()k + C k; L ()k + + " k; L ()k + B kr; L ()k + C k; L ()k + + minfb " 1 ; 1 gk; L ()k (4.4) C " jd u; W ; ()j + (" + B ) jd u; W 1; ()j + jj=r 1 + C + C + min C jj=r 1 jj=1 jj=1 n B " 1 ; 1 oi jj=r+1 jj=r jd u; L ()j ; E an ;; (+) kd ++ u; L ()k (4.5) E;; an " + C + (" + B + C ) + minf" 1 B ; 1 g (4.6) providd tat u W r+1; () and d =, 1 r k, or d = 3, 3 r k. In t otr cas d = 3, k = 1;, w conclud from (4.4) and (.6) I (u) C E;; an (+) kd ++ u; L ()k + jj=r 1 jj=1 jj=1 + s kd +++s u; L ()k (4.7) s=1 A suitabl stratgy is now to gnrat t anisotropic ms (via coic of ; = d; ) and to coos t numrical damping paramtrs in suc a way, tat t rror trm I (u) is minimizd. Tat mans, tat t task is to minimiz t dirnt trms E;; an, but t problm is tat tr is only on fr paramtr. On account of t prsumably largst u u, w propos as a rst attmpt to minimiz r+1 ;; an in t cas d = = (0; 1) for d = and = = (0; 0; 1) for d = 3, rspctivly. Considring E an " + C d; + (" d; + B + C d;) + d; min n B " 1 ; 1 o ; (4.8) w procd as in t proof of Lmma 3.8 and coos according to (3.9), (3.30) wit, P and rplacd by d;, P an and an, P an d;kb; [L 1 ()] d k ; an " d; kc; L1 ()k ; (4.9) " rspctivly. But for a conclusiv rror stimat w must also considr t trms (+) kd ++ u; L ()k : Tat is wy w giv a mor rnd analysis for a spcial cas in t nxt sction. 13

14 Figur 5.1: Problms of cannl typ. 5 Application to problms of cannl typ 5.1 Dnition and proprtis of t solution In viw of t dicultis to gt a priori information on t solution u w rstrict our considration in tis sction to a crtain class of problms wic sould b introducd in t following. T main point is a corrspondnc of t domain and t ow ld b considrd. Givn a subdomain G and a ow ld b w dnot by (@G), (@G) +, and (@G) 0 t inow, outow and caractristic parts t indx dnots t sign of (b G )(x) wr G is t outward unit normal Lt x () b t solution of _() = b(()); (0) = x ; t stramlin of b passing troug x. Dnoting for any point x G [ (@G) by G + (x) inff > 0 : x () 6 Gg t rst xit tim of x () from G, w dn t domain of inunc of any by E( 0 ) f x () G : x 0 ; 0 G + (x)g: W say now tat a domain G is of cannl typ wit rspct to a ow ld b if t following tr conditions ar satisd: (i) G = E((@G) ), (ii) E((@G) 0 ) (@G) 0, E((@G) \ (@G) + ) \ G = ;, (iii) j(b G )(x)j > 0 on (@G) [ (@G) +. In particular, tis implis tat all stramlins x (), x G, lav G in nit tim. Hnc turning points wit kb; IR d k = 0 and priodic caractristics ar xcludd. For an illustration of cannl typ problms s Figur 5.1, wras Figur 5. sows som situations not allowd. On t otr and, boundary layrs will appar in t cas G = at (@) + and (@) 0. W av t following rsult of [1, Torm.3] wic givs a localization of t boundary layrs R ". Lmma 5.1 Lt and i, i = 1;, b simply connctd domains of cannl typ, and assum tat 1, (@ 1 ) (@ ) (@) and tat (@ ) is sucintly smoot. For givn numbrs s > 0 and r IN 0 tr xist C i (r; s; i ), i = 1;, and C( ; ) suc tat if dist( 1 ; (@ ) + ) C 1 "j ln "j; dist( 1 ; (@ ) 0 ) C p "j ln "j; 14

15 Violats r b = 0 Violats (i) Violats (ii) Violats (iii) Figur 5.: Not allowd situations. tn n o ku; W r+1; ( 1 )k C kf; W r+1; ( )k + " s kf; L ()k : So w dnot by R " fx : dist(x; (@) + ) C 1 "j ln "j; dist(x; (@) 0 ) C p "j ln "jg (5.1) t boundary layr rgion of a domain of cannl typ. Furtrmor w dn R + " fx R " : dist(x; (@) + ) C 1 "j ln "jg R 0 " fx R " : dist(x; (@) 0 ) C p "j ln "jg (5.) R c " R + " \ R 0 " 5. Gnration of t anisotropic ms in t boundary layr T mss ar constructd as introducd in Subsction 4.. W coos 1; = ; = g 1 ("); and ; = ; = g ("); (5.3) wit g 1 (") = O(1); g (") = " if R + " n R c "; g 1 (") = O(1); g (") = p " if R 0 " n R c " ; g 1 (") = p "; g (") = " if R c "; (5.4) 15

16 R + " b R c " R 0 " Figur 5.3: Gnration of t anisotropic ms, s xampl (4.1) (4.) wit 1 = 0, =. and obsrv tat g (") = o(g 1 (")) and g 1 (") O(1): (5.5) Not tat by construction a condnsd ms occurs around t cornrs of (@) + [ (@) 0 wr t layrs intrsct, compar Figur 5.3. Outsid R " w doubl ; in -dirction (prpndicularly to (@) + and (@) 0, rspctivly) until ;. W s asily tat t numbr of lmnts is of t ordr j ln "j 1. In rgard of lacking Sobolv norm stimats of u in R ", w assum t following ypotsis to b satisd: (H.6) kd ; u; L ()k p mas() [(g 1 (")) 1 + (g (")) + (g1 (")) 1 (g (")) ] K(f) wit g 1, g as in (5.4). T manifold wit = 0 corrsponds to (@) + for R + " n R c " and to (@) 0 lswr in R ". Rmark 5. As in Siskin mss [11, 7] w could omit t transition layr wr w doubl t prvious ms sizs; our fortcoming analysis is not actd. Howvr, w xpct a mor rgular baviour of t discrt solution and bttr algbraic proprtis of t rlatd systm of quations wit our approac. 5.3 Error stimats Wit I (u) as in (4.4), w split t rror as follows: jjj u U jjj "; I (u) = I (u) + nr " R " I (u): (5.6) In viw of Lmma 5.1 w can considr t lmnts in t rst sum as in Sction 3 and it rmains to trat t anisotropic lmnts R ". Lmma 5.3 T rror trm I (u) for R " is minimal (up to multiplicativ constants) for t following coic of : wit P an ; = p if (P an " 1 + (P an ) + ( an ) ) ( P ~ an ) ( ) " ; = min ; B " 1 + (P an ) + an 1 + (P an ) + ( an ) and an dnd in (4.9). Hnc tr olds q if 0 P an 1 + (P an I (u) C r+ K (f) 1; 1 an ; "(1 + P + an ) C r+ K (f)(" 1; 1 ; + C 1; ; + B 1; ) ) + ( an ) ; (5.7) ~ P an ; (5.8) (5.9) 16

17 Proof T rlations (4.5), (5.3), (5.5) as wll as assumption (H.6) imply I (u) = C E;; an (+) kd ++ u; L ()k C jj=r 1 jj=1 jj=1 jj=r 1 jj=1 jj=1 E an ;; (g 1 (") ) ( 1+ 1 )+1 (g (") ) ( + )+1 g ( ) 1 + g ( + + ) + g ( ) K (f) C r+ K (f) E;; an g ;; ; (5.10) jj=1 jj=1 1 g ( + + ) g ;; g g +1 + g g +1 + g g +1 : (5.11) Exprssing g ;; via (5.3) in trms of, 1;, and ;, and using ; = o( 1; ), 1; O(1), w nd g ;; 1; 1 ; ; g ;; 1; ; ; g ;; 1; 3 ; ; jj=1 jj=1 tat mans wit (4.6) tat jj=1 jj=1 I (u) C r+ K (f) 1; 1 ; Ean jj=1 jj=1 wit E an from (4.8), and tus w gt wit t sam argumnts as in Subsction 4.3 t xprssions (5.7) (5.8) for and (5.9) for I (u). Not tat tis rsult dos not old for gnral anisotropic mss or gnral convctiondiusion-raction problms bcaus t assrtion is mainly basd on assumption (H.6) and a ms satisfying (5.3). As a rsult of t analysis in Lmmata 3.8 and 5.3 w propos t dsign of t numrical damping paramtrs as in (5.7) (5.8) in all cass. Tat mans, as wll as t local numbrs P and ar dpndnt only on ;, wic is quivalnt to t radius of t innr circl. Using (5.6) w can summariz t rror stimats as follows. Corollary 5.4 Undr t assumptions (H.1) : : : (H.6), u H r+1; (), 1 r k, and using t anisotropically rnd boundary layr ms (5.3) (5.4) and t paramtr dsign (5.7) (5.8) w gt t almost uniform (wit rspct to ") rror stimat jjj u U jjj "; C r j ln "jk (f)(1 + C 1; ; + B 1; ): (5.1) Rmark 5.5 T paramtrs ar vry small in t boundary layr, but = 0 dos not giv t optimal rsult: In parabolic boundary layrs w would gt instad jjj u U jjj "; C r j ln "jk (f) p "(1 + (C + " 1 B ): Rmark 5.6 W conjctur tat t analysis of tis sction can b rnd in ordr to avoid t factor j ln "j in (5.1) if w usd a sarpr stimat on t xponntial dcay of t solution tan in Lmma 5.1 and Assumption (H.6). i T rst autor was supportd by DFG (Grman Rsarc Founda- Acknowldgmnt. tion), No. La 767-3/1. 17

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