UNIFIED ERROR ANALYSIS

Transcription

1 UNIFIED ERROR ANALYSIS LONG CHEN CONTENTS 1. Lax Equivalnc Torm 1 2. Abstract Error Analysis 2 3. Application: Finit Diffrnc Mtods 4 4. Application: Finit Elmnt Mtods 4 5. Application: Conforming Discrtization of Variational Problms 5 6. Application: Prturbd Discrtizations 7 7. Application: Nonconforming Finit Elmnt Mtods 8 8. Application: Finit Volum Mtods Application: Finit Diffrnc Mtods (rvisitd) Application: Suprconvrgnc of t linar finit lmnt mtod Exrcis 14 Rfrncs 14 Lt X, Y b two Banac spacs and L L (X, Y ) b a linar and boundd oprator. W considr t problm of approximating an oprator quation: givn y Y, find x X suc tat (1) Lx = y. W assum (1) is wll dfind, i.., for any y Y, tr xists a uniqu solution to (1). Tn L is a on-to-on linar and boundd map and so is L 1 by t opn mapping torm. W sall study t convrgnc analysis in tis nots and rfr to Inf-sup conditions for oprator quations for conditions on t wll-posdnss of (1). 1. LAX EQUIVALENCE THEOREM W ar intrstd in t approximation of L. Suppos L L (X, Y ), for H, 0, is a family of discrtization of L. W considr t problm: givn y Y, find x X, suc tat (2) L x = y. W also assum (2) is wll dfind. So L 1 is a linar and boundd map. T norm L 1, owvr, could dpnd on t paramtr. T uniform bounddnss L 1 will b calld t stability of t discrtization. Namly, tr xists a constant C indpndnt of suc tat (3) L 1 Y X C, for all H. Dat: Oct, Last updatd January 18,

2 2 LONG CHEN T consistncy masurs t approximation of L to L. W call t discrtization is consistnt if, for all x X, (4) Lx L x Y 0 as 0. T convrgnc of t discrtization is dfind as (5) x x X 0 as 0, wr x and x ar solutions to (2) and (1), rspctivly. W follow t book [4] to prsnt t Lax quivalnc torm. Torm 1.1 (Lax quivalnc torm). Suppos t discrtization L of L is consistnt, i.., (4) olds for all x X, tn t stability (3) is quivalnt to t convrgnc (5). Proof. W writ t rror as x x = L 1 (L x L x ) = L 1 (L x Lx). Hr w us t fact L x = Lx = y. If t scm is stabl, tn x x X C Lx L x Y 0 as 0. On t otr and, for any y Y, lt x = L 1 y and x = L 1 y. T convrgnc x x mans for any y Y L 1 y L 1 y as 0. By t uniformly bounddnss principl, w conclud L 1 is uniformly boundd. 2. ABSTRACT ERROR ANALYSIS T Lax quivalnc torm may not b rady to us for t rror analysis of discrtizations sinc discrt spacs X, Y ar usd to approximat X, Y, rspctivly. T quation may not b solvd xactly, i.., L x Lx. Instad, it solvs approximatly in Y, i.., L x = y for som y Y. In som cass X X but in gnral X and X ar diffrnt spacs so tat vn x x dos not mak sns. T norm X could b an approximation of X tc. W now rfin t analysis to andl ts cass. Hr w follow closly Tmam [3]. Lt us introduc two linar oprators I : X X, and Π : Y Y. W considr t discrt problm: givn y Y, find x X suc tat (6) L x = Π y in Y. T following diagram is not commutativ L X Y I Π. X Y and t diffrnc is dfind as t consistncy rror T (uniform) stability is t sam as bfor L Π Lx L I x Y. L 1 Y X C, for all H. On t convrgnc, vn x x may not b wll dfind as in gnral X is not a subspac of X. T rror is rvisd to I x x.

3 UNIFIED ERROR ANALYSIS 3 Rmark 2.1. T oprator Π is in fact part of t discrtization and to rflct to tis fact t discrtization will b dnotd by (L, Π ), wil t oprator I is introducd for t rror analysis. For a givn Π, I is not uniqu and a suitabl coic of I is t art of numrical analysis. W sall prov if t scm is consistnt and stabl, tn t discrt rror I x x is convrgnt. Torm 2.2. Suppos t scm (L, Π ) is consistnt wit ordr r, i.., tr xists an oprator I suc tat Π Lx L I x Y C r, and stabl L 1 Y X C for all H, tn w av t discrt convrgnc wit ordr r: I x x X C r. Proof. T proof is straigtforward using t dfinition, assumptions, and t idntity: I x x = L 1 L (I x x ) = L 1 (L I x Π Lx). If w rally want to control x x, on way is to dfin anotr linar oprator P : X X to put a discrt function x into X in a stabl way, i.., P x X C x X. Tn by t triangl inquality, w av x P x X x P I x X + P (I x x ) X x P I x X + C I x x X. T additional rror trm x P I x X masurs t approximation proprty of X by t subspac P X and will b calld approximability. Torm 2.3. Assum consistncy: Π Lx L I x Y C 1 r, stability: L 1 Y X C 2 H, approximability: x P I x X C 3 s, tn w av convrgnc x P x X C 2 (C 1 r + C 3 s ). Rmark 2.4. As t stability of L 1 is not usd, t domain of t oprator L could b just a subspac of X. Strictly spaking w sould writ L : dom(l) X Y. W lav a biggr spac X in t analysis suc tat t mbdding P : X X is asir to construct. And wit oprators I and Π, t spac X and Y do not av to b Banac spacs. Anotr way is to mbd bot X and X into a largr spac,.g. X + X and dfin a nw norm. Similar assumption is mad for Y and Y. Also assum L can b dfind on X + X. W can brak t consistnc rror Π Lx L I x into tr parts prturbation of t data: Π Lx Lx = L x Lx = y y consistnc rror: Lx L x approximation rror: L (x I x)

4 4 LONG CHEN T first trm masurs t approximation of t data in t subspac Y and t tird on masurs t approximation of t function. T middl on is t traditional consistnc rror introducd in Sction APPLICATION: FINITE DIFFERENCE METHODS Considr t finit diffrnc mtod for solving t Poisson quation u = f wit omognous Diriclt boundary condition u Ω = 0. W assum Ω can b partitiond into qual siz squars wit sid. T numbr of intrior nods is N. T stting is (X, X ) = (Y, Y ) = (C( Ω),,Ω ); (X, X ) = (Y, Y ) = (R N, l ); I = Π as t nodal intrpolation; P : X X can b dfind using t bilinar finit lmnt spac on t uniform grid or t linar finit lmnt spac on t uniform triangular grid; L = and L = is t 5-point stncil matrix. Not tat t domain of L is a subspac of X. T consistncy rror is ( u) I u I l = max 1 i N ( u) i ( u I ) i C(u) 2, wic can b asily analyzd by t Taylor xpansion. Hr C(u) = d D4 i u,ω. T stability 1 l l may b provd using t discrt maximum principl; s Finit Diffrnc Mtods. T approximation proprty is givn by t intrpolation rror stimat u u I,Ω C 2 D 2 u,ω. Hr follow t convntion, w us t sam notation u I for P u I. T radrs ar ncouragd to nail down t diffrnc and t connction of t vctor u I and its corrsponding finit lmnt function P u I. T maximum norm is too strong. Latr on w sall sow a corrct spac and stability for t finit diffrnc mtod. 4. APPLICATION: FINITE ELEMENT METHODS Considr t finit lmnt mtod for solving t Poisson quation u = f wit omognous Diriclt boundary condition u Ω = 0. T stting is (X, X ) = (H0 1 (Ω), 1 ), and (Y, Y ) = (H 1 (Ω), 1 ); (X, X ) = (V, 1 ), and (Y, Y ) = (V, 1,). Hr t dual norm f, v f 1, = sup, for f Y. v V v 1 I : H 1 0 (Ω) V is arbitrary; Π = Q : H 1 (Ω) = (H 1 0 (Ω)) V, i.., Q f, v := f, v, v V. Wn Q is rstrictd to L 2 (Ω), it is t L 2 projction. Not tat (H 1 0 (Ω)) V, t oprator Q can b also tougt as t natural mbdding of t dual spac and tus is usually omit in t notation. P : V H 1 0 (Ω) is t natural mbdding sinc now V H 1 0 (Ω);

5 UNIFIED ERROR ANALYSIS 5 L = : H0 1 (Ω) H 1 (Ω) and L = Q LP : V V. Wn Q rstrictd to L 2 (Ω), it is t adjoint of P in t L 2 -innr product and L = Q LQ. Using ts notation, w now formulat t finit lmnt mtods for solving t Poisson quation wit omognous Diriclt boundary condition. Givn an f H 1 (Ω), find u V suc tat L u = Q f in V. T corrct (comparing wit finit diffrnc mtod) stability for L 1 is L 1 1, 1 : (7) u 1 Q f 1,, and can b provd as follows u 2 1 = L u, u = Q f, u Q f 1, u 1. Comparing wit t finit diffrnc mtod, now t consistncy rror is masurd in a muc wakr norm. W can control t wakr dual norm Q f 1, by a strongr on f 1. Indd implis (Q f, u ) = f, u f 1 u 1, (8) Q f 1, f 1. Lt us dnotd by v = I u. Using t fact L = Q LP and inquality (8), t consistncy rror is Q Lu L v 1, = Q L(u v ) 1, L(u v ) 1 u v 1. By t stability rsult (7), w obtain u v 1 u v 1. Sinc v is arbitrary, by t triangl inquality, w obtain t Céa lmma (9) u u 1 2 inf v V u v 1. T approximation proprty can b provd using Brambl-Hilbrt lmma. For xampl, for t linar finit lmnt mtod and u H 2 (Ω), inf u v 1 C u 2. v V Tn Torm 2.3 will giv optimal rror stimat for finit lmnt mtods. Rmark 4.1. Hr w us only t stability and consistncy dfind for t Banac spacs. If w us t innr product structur and t ortogonality (or mor gnral t variational formulation), w could improv t constant in (9) to APPLICATION: CONFORMING DISCRETIZATION OF VARIATIONAL PROBLEMS W now gnraliz t analysis for Poisson quation to gnral lliptic quations and sow t connction wit t traditional rror analysis of variational problms. T oprator L : V V is dfind troug a bilinar form: for u, v V Lu, v := a(u, v). W considr t conforming discrtization by coosing V V and t discrt oprator L : V V V bing t rstriction of t bilinar form: for u, v V L u, v = a(u, v ).

6 6 LONG CHEN If w dnotd by P : V V and Q : V V as t natural mbdding. Tn by dfinition L = Q LP : V V. T wll known Lax-Milgram torm (Lax again!) says if t bilinar form satisfis: corcivity: continuity: α u 2 V a(u, u). a(u, v) β u V v V. Tn tr xists a uniqu solution to t variational problm: givn f V find u V suc tat (10) a(u, v) = f, v for all v V. Sinc V V, it also implis t xistnc and uniqunss of t solution u V (11) a(u, v ) = f, v for all v V. Tis stabliss t wll posdnss of (10) and (11). T corcivity condition can b rlaxd to t so-calld inf-sup condition; s Inf-sup conditions for oprator quations. T continuity of t bilinar form a(, ) implis t continuity of L and L. T corcivity can b usd to prov t stability of L 1 in a straigtforward way: u 2 V a(u, u ) = f, u f V u V. For t conforming discrtization, w av t ortogonality (borrowd t nam wn a(, ) is an innr product but in gnral a(, ) may not b vn symmtric) i.., a(u u, v ) = 0 for all v V. Lu = L u in V, wic can b intrprtd as consistncy rsult: Lu = f in V or in a lss prcis notation L u = Lu in V wit L u undrstood as Q Lu. Not tat t consistncy is masurd in t wak norm V but not in V. T rsidual Lu L u = 0 only in V but 0 in V. In our stting, t consistnc mans: for any v V Lv = L v in V. In t oprator form, it is simply L = Q LP. T traditional rror analysis is as follows u u 2 V 1 α a(u u, u u ) (corcivity) = 1 α a(u u, u v ) (ortogonality) β α u u V u v V (continuity) wic implis t Céa lmma u u V β α inf u v V. v V

7 UNIFIED ERROR ANALYSIS 7 Using our framwork of stability and consistncy, t stimat is lik u v V 1 α L (u v ) V (stability) = 1 α Lu L v V (ortogonality) = 1 α L(u v ) V (consistncy) β α u v V (continuity). Combind wit t triangl inquality, w gt ( u u V 1 + β ) inf u v V. α v V 6. APPLICATION: PERTURBED DISCRETIZATIONS W continu t study of t conforming discrtization but wit prturbation in t data and/or t bilinar form. W first considr t quadratur of t rigt and sid, i.., f, v f, v. T stting is as bfor xcpt Π : V V dfind as Π f, v = f, v, and solv t quation L u = Π f in V. T stability of L 1 is uncangd. W only nd to stimat t consistncy rror Π L u L v V = Π f L v V. By t triangl inquality, w av By t dfinition Π f L v V Q f L v V + Π f Q f V u v V + Π f Q f V. Π f Q f, w f, w f, w Π f Q f V = sup = sup. w V w w V w W tn nd up wit a vrsion of t first Strang lmma u u V f, w f, w inf u v V + sup. v V w V w V Tat is, a prturbation of t data in V is includd. W tn considr t prturbation of t bilinar form wic includs t cas of using numrical quadratur to comput t bilinar form a(, ) or a non-conforming discrtization of L. W ar solving t quation L u = Π f in V. To gt similar stimat, w nd to assum L is dfind on V + V. T spac V is now ndowd by a possibly diffrnt norm. W assum t stability and continuity of L wit rspct to t norm. Corcivity: for all v V L v, v α v 2. Continuity: for all u V + V, v V L u, v β u v.

8 8 LONG CHEN Tn it is straigtforward to prov t stability of L : (V, V ) (V, ). For t consistncy rror, Π f L v V Π f L u V + L (u v ) V f, w L u, w sup + u v. w V w Trfor w nd up wit t scond Strang lmma u u inf u v + C v V sup w V f, w L u, w. w 7. APPLICATION: NONCONFORMING FINITE ELEMENT METHODS W apply t analysis to t nonconforming finit lmnt discrtization, i.., V V = H 1 0, of t modl Poisson quations using t Crouzix-Raviart (CR) lmnt as an xampl. Givn a sap rgular ms T, dnot all t dgs in Ω and on Ω by E and E, rspctivly. On ac E, lt [ ] dnot t jump across t dg, i.. for ac picwis H 1 function v, [v] = v τ1 v τ2, wr τ 1 and τ 2 ar t triangls saring t dg. For t boundary dg E, [v] = v. W tn dfin t Crouzix-Raviart lmnt as { } V CR = v v L 2 (Ω), v τ P 1 τ T, and [v] = 0 E E. It can b vrifid tat a function v V CR is uniquly dtrmind by its avrag on ac dg v ds wic is known as t dgr of frdom and tus t dimnsion of t spac V CR, dnotd by NE, is t numbr of t intrior dgs of t triangulation T and As bfor, t oprator L : V V is dfind troug t bilinar form: Lu, v := a(u, v) := ( u, v). Sinc t finit lmnt spac V CR is not a subspac of V, w sall us a modifid bilinar form a (, ), tat is a (u, v ) = ( u, v ) := u v dx u, v V CR. τ T Namly t gradint oprator is applid lmnt-wis. T corrsponding discrt oprator L : V CR (V CR ) is dfind as: for all u, v V tat L u, v := a (u, v ). Not tat t bilinar form a (, ) is also wll dfind on V V and indd a (u, v) = a(u, v) wn u, v H 1 (Ω). Trfor t oprator L can b also dfind from V (V). On t spac V CR + V, w us t norm inducd by t bilinar form a (, ): v 1, = a 1/2 (v, v) v VCR + V. τ

9 UNIFIED ERROR ANALYSIS 9 On can asily obtain corcivity and continuity of t bilinar form by sowing 1, is indd a norm on V (but not on V). Tn for any v, u V + V, it olds tat v 2 1, = a (v, v), a (u, v) u 1, v 1,. L 1 Consquntly t discrt oprator L and is uniformly boundd. Givn an f L 2 (Ω), t nonconforming discrt problm is to find a u V CR suc tat L u = Q f. W mpasiz tat as V V, diffrnt from t conforming lmnt mtod, tr is no ortogonality a (u u, v ) 0 or quivalntly in t oprator form L u L u v V CR, in (V CR ). It is inconsistnt. But t consistncy rror can b controlld as follows. Lmma 7.1. Assum tat t triangulation T is quasi-uniform, and t solution u as H 2 rgularity. Tn, it olds for any w V CR tat (f, w ) L u, w C u H 2 w 1,. Proof. By intgration by parts, w driv tat (f, w ) L u, w = E u n [w ]ds. Lt u n and w b t avrag of u n and w on rspctivly. Tn, u n[w ]ds = ( u n u n )[w w ]ds u n u n L 2 () [w w ] L 2 (). For ac, w dnot τ b t union of t two triangls saring, and av t trac torm wit scaling wic yilds v 2 L 2 () C( 1 v 2 L 2 (τ ) + v 2 H 1 (τ ) ) v H1 (τ ). Sum ovr all dgs to complt t proof. u n[w ]ds C u H 2 (τ ) w H 1 (τ ). T approximation rror can b stablisd using t following canonical intrpolation I : V V CR using t d.o.f. I v ds = v ds E. Tn tr xists a constant indpndnt of t ms siz suc tat for any τ T, v I v H 1 (τ) C m 1 v H m (τ) v H m (τ), m = 1, 2. T inquality follows from t scaling argumnt, t trac torm, and t fact tat t oprator I prsrv linar function on τ.

10 10 LONG CHEN Combining t approximation and consistncy, togtr wit t scond Strang lmma, w av t optimal ordr convrgnc for CR non-conforming approximation u to t solution of Poisson quation u u 1, C u H APPLICATION: FINITE VOLUME METHODS W considr t vrtx-cntrd finit volum mtod and rfr to Finit Volum Mtods for a dtaild dscription and proof of corrsponding rsults. Simply spaking, if w coos t dual ms by conncting an intrior point to middl points of dgs in ac triangl, t stiffnss matrix is idntical to tat from t linar finit lmnt mtod. So w sall us t stting for t finit lmnt mtod and trat t vrtx-cntrd linar finit volum mtod as a prturbation. T only diffrnc is t rigt and sid, i.., Π : L 2 (Ω) V. Hr w nd to srink t spac Y from H 1 (Ω) to L 2 (Ω) to av t volum intgral f dx wll b dfind. To dfin suc Π, lt us introduc t picwis constant spac on t dual ms B and dnotd by V 0,B. W rwrit t linar finit lmnt spac as V 1,T. Troug t point valus at vrtics, w dfin t following mapping N Π : V 1,T V 0,B as Π v = v (x i )χ bi. For any f L 2 (Ω), w tn dfin Π f V as Π f, v = (f, Π v ), for all v V. T analysis blow follows closly to Hackbusc [2]. Dnotd by u G as t standard Galrkin (finit lmnt) approximation and u B is t box (finit volum) approximation. T quivalnc of t stiffnss matrics mans Trfor by t stability of L 1, w av By t dfinition L u G = Q f, L u B = Π f. u G u B 1 Q f Π f 1,. Q f Π f, v = (f, v Π v ). Dnot t support of t at basis function at vrtx x i as ω i. Not tat b i ω i and t oprator I Π prsrv constant function in t patc ω i and tus (f, v Π v ) bi f bi v Π v ωi C f bi v 1,ωi. Summing up and using t Caucy Scwarz inquality, w gt t first ordr convrgnc u G u B 1 C f. Furtrmor, if t dual ms is symmtric in t sns tat w us barcntrs as a vrtx of control volums. Tn w av v = Π v, τ and tus, lt f b t L 2 projction f to t picwis constant function in ac triangl, (f, v Π v ) = (f f, v Π v ) C 2 f 1 v 1. τ

11 UNIFIED ERROR ANALYSIS 11 W tn obtain t suprconvrgnc or t suprclosnss of u G and ub : (12) u G u B 1 C 2 f 1. Wit suc rlation, w can obtain optimal L 2 rror stimat for u B and quasi-optimal L rror stimat in two dimnsions. By t triangl inquality, Poincaré inquality, and t optimal ordr convrgnc of u u G, u u B u u G + u G u B C 2 u 2 + u G u B 1 C 2 ( u 2 + f 1 ), and similarly wit discrt mbdding v C log v 1, u u B u u G + u G u B C 2 u 2 + C log u G u B 1 C log 2 ( u 2 + f 1 ). 9. APPLICATION: FINITE DIFFERENCE METHODS (REVISITED) W rvisit t finit diffrnc mtod but trat it as a prturbation of t linar finit lmnt mtod. So w will us t stting for finit lmnt mtods in Sction 4. T rlation btwn t linar finit lmnt and 5-point stncil on t uniform grids is L u, v = u A v = 2 u v. Hr w us boldfac small lttrs to dnotd t vctor formd by t function valus at vrtics. Mor prcisly for u V, w dfin u R N suc tat (u ) i = u (x i ).W us boldfac capital lttrs for matrics. For xampl, is t stncil matrix and A is t matrix for P1 finit lmnt mtod. W rscal t stiffnss matrix A from t linar finit lmnt mtod to gt t fiv-point stncil matrix. Hr uniform grids w man t tr dirctional triangulation obtaind by using diagonals wit t sam dirction of uniform rctangular grids; s Fig. 1. FIGURE 1. A uniform ms for a squar W coos I as t nodal intrpolation and dfin Π : C(Ω) V as Π f, v = 2 f I v = N 2 f(x i )v (x i ). It can b intrprt as using tr vrtics as t numrical quadratur. T quation for FDM can b rwrittn as L u = Π f.

12 12 LONG CHEN By t first Strang lmma, w only nd to stimat t prturbation Π f Q f 1, wic is lft as an xrcis, cf. Exrcis 1 in Sction 11. T approximation rror u u I 1 u 2 is t intrpolation rror studid in Introduction to Finit Elmnt Mtods. In summary, t first ordr rror stimat u I u 1 ( u 2 + f 1 ) can b asily provd witout ccking t point-wis truncation rror and t smootnss of t solution u can b rlaxd to u H 2 (Ω) only. If w assum tat t truncation rror in l norm is of scond ordr, i.., ( u) I u I l C 2, tn w can prov a scond ordr convrgnc by xamining t consistncy rror Π f L u I 1, mor carfully: Π f L u I, v = 2 (f I + u I ) v In t last stp, w av usd t bound f I + u I C(u) 2 v 1. N 2 v (x i ) N 2 v (x i ) C v L 1 C v L 2 C v 1. Trfor Π f L u I 1, C(u) 2 and consquntly u I u 1 2 u 4,. In 2-D, using t discrt mbdding torm, w can obtain t narly optimal maximum norm stimat u I u (1 + log ) u I u 1 C(u, f)(1 + log ) 2. Hr w considr Diriclt boundary condition and assum t truncation rror for intrior nods is scond ordr. For Numann boundary condition or cll cntrd finit diffrnc mtods, t boundary or nar boundary stncil is modifid and t truncation rror could b only first ordr O(); s Finit Diffrnc Mtods. W sall sow tat t boundary stncil can b just first ordr wil t scm is still scond ordr masur in H 1 -norm. Lt us labl t boundary (or nar boundary) nods from 1 : N b and intrior nods from N b + 1 : N and assum t truncation rror for boundary nods is only O() and for intrior nods is still O( 2 ). Tn Π f L u I, v = 2 (f I + u I ) v f I + u I ( Nb = 2 v (x i ) + N i=n b +1 2 v (x i ) C(u) 2 ( ) v L 1 ( Ω) + v L 1 (Ω) C(u) 2 v 1. ) N 2 v (x i )

13 UNIFIED ERROR ANALYSIS 13 In t last stp, w av usd t following inqualitis v L 1 ( Ω) v L 2 ( Ω) v 1,Ω v 1. T trick is to bound t boundary part by t L 1 norm of t trac and tn us t trac torm to cang to H 1 -norm of t wol domain. By t sam lin of t abov proof, to gt t first ordr convrgnc i.. u I u 1, t truncation rror of t stncil on t boundary (or an intrfac) can b only O(). Namly t scm is not consistnt point-wisly but still convrgs! 10. APPLICATION: SUPERCONVERGENCE OF THE LINEAR FINITE ELEMENT METHOD W sall rvisit t linar finit lmnt mtod and provid a rfind analysis on t consistncy rror. In som cass w may av a sarpr stimat of t consistncy rror in t wak norm 1,. For xampl, for 1-D Poisson problm, using intgration by parts and noting tat (u u I ) τ = 0, w gt, for any v V (Q Lu L u I, v ) = (u u I, v ) = τ T (u u I, v ) τ = 0, wic implis t discrt rror is zro. Namly u (x i ) = u(x i ) at ac grid points x i. In ig dimnsions, wn t ms is locally symmtric and t function is smoot noug, w may stablis t following strngtnd Caucy-Scwarz inquality (13) Q Lu L u I, v = ( u u I, v ) C 2 u 3 v 1. Tn consquntly w obtain t suprconvrgnc rsult u u I 1 Q Lu L u I 1, C 2 u 3. Rougly spaking, t symmtry of t ms will bring mor canclation wn masuring t consistncy rror in a wak norm. For xampl, (13) olds wn t two triangls saring an intrior dg forms an O( 2 ) approximat paralllogram for most dgs in t triangulation; S [1] for dtaild condition on triangulation and a proof of (13). W sall giv a simpl proof of suprconvrgnc rsult on uniform grids by using t rlation to t finit diffrnc mtod. Rwrit Q Lu = Q f and divid t consistncy rror into two parts Q f L u I 1, Q f Π f 1, + Π f L u I 1, = I 1 + I 2. To stimat I 1, w not Q f, v = N ( ω i fφ i )v (x i ) and 2 f(x i ) is a quadratur for t intgral ω i fφ i. It is only xact for constant f rstrictd to on simplx. But for uniform grids, du to t configuration of ω i, t support of t at function φ i, t linar functional E i (f) = fφ i dx 2 f(x i ) ω i prsrv linar polynomials in P 1 (ω i ). Namly E i (f ) = 0 for f P 1 (ω i ). Trfor E i (f) = E i (f f ) f f f 2,. Rmark For finit volum approximations, t corrsponding diffrnc is E i (f) = f dx 2 f(x i ). b i

14 14 LONG CHEN For uniform grids, w can cos b i as t dual grid wic contains t vrtx as t cntr and b i = 2 and tus E i (f) can prsrv linar polynomial in P 1 (b i ). Trfor w can stimat t I 1 as N Q f Π f, v = E i (f)v (x i ) C f 2 N 2 v (x i ) C f 2 v 1. And consquntly Q f Π f 1, C(f) 2. T scond ordr stimat of part I 2, i.., Π f L u I 1, C(u) 2 as bn analyzd in Sction 9. In conclusion, w av provd t suprconvrgnc for t linar finit lmnt mtod on uniform grids u I u 1 C(u, f) 2, by stimating t consistncy rror in a wakr norm. In our simplr proof w nd strongr smootnss assumption wic can b rlaxd to u H 3 (Ω); s [1]. C(u, f) = D 4 u + D 2 f, 11. EXERCISE 1. Us tr vrtics quadratur rul, i.. for a triangl τ formd by tr vrtics x i, i = 1, 2, 3, g(x)dx 1 3 g(x i ) τ, 3 τ to comput t rigt and sid (f, φ i ) and sow it can b simply writtn as Mf I, wr M is a diagonal matrix. Prov t optimal first ordr approximation in H 1 -norm for t linar finit lmnt mtod of Poisson quation using tis quadratur. 2. Sow 1, dfind in Sction 7 is a norm dfind on V CR. 3. Prsnt rror stimat for t cll-cntrd finit diffrnc mtod (Sction 4 in Finit Diffrnc Mtods) wit diffrnt tratmnt of t Diriclt boundary condition. REFERENCES [1] R. E. Bank and J. Xu. Asymptotically xact a postriori rror stimators, Part I: Grids wit suprconvrgnc. SIAM J. Numr. Anal., 41(6): , [2] W. Hackbusc. On first and scond ordr box scms. Computing, 41(4): , [3] R. Tmam. Numrical Analysis. D. Ridl Publising Company, [4] G. Zang and Y. Lin. Functional analysis: I. Bijing Univrsity, 1987.