The Highest Superconvergence of the Tri-linear Element for Schrödinger Operator with Singularity

Transcription

1 J Sci Comput (06) 66: 8 DOI 0.007/s Th Highst Suprconvrgnc of th Tri-linar Elmnt for Schrödingr Oprator with Singularity Wnming H Zhimin Zhang Rn Zhao Rcivd: May 04 / Rvisd: 8 January 05 / Accptd: 5 March 05 Publishd onlin: March 05 Springr Scinc+Businss Mdia Nw York 05 Abstract In this papr, th ignvalus for Schrödingr oprator with singularity ar analyzd. A spcial picwis uniform rctangular partition is constructd and it has bn provn that, undr this partition, th tri-linar rctangular finit lmnt mthod has th highst possibl suprconvrgnc rat for ignvalu. Kywords Schrödingr oprator Richardson xtrapolation Th highst suprconvrgnc Mathmatics Subjct Classification 65N5 65N0 65N5 Introduction Richardson xtrapolations for FEMs for lliptic problm hav bn prsntd sinc 970 s (s,.g. [,9,0,,8, 6], for an incomplt list of rfrncs). Intrstd radrs ar rfrrd to th survy articl by Blum [8]. In this papr, w ar mainly concrnd with Richardson xtrapolation of finit lmnt mthods for scond-ordr lliptic ignvalu problm. Th first author is supportd in part by th National Natural Scinc Foundation of China (57) and th Zhjiang Provincial Natural Scinc Foundation of China undr Grant (No. Y5A00040); and th scond author is supportd in part by th US National Scinc Foundation through grant DMS B Wnming H h_wnming@aliyun.com Zhimin Zhang zzhang@math.wayn.du Rn Zhao rzhao@math.wayn.du Dpartmnt of Mathmatics, Wnzhou Univrsity, Wnzhou 005, Zhjiang, Popl s Rpublic of China Dpartmnt of Mathmatics, Wayn Stat Univrsity, Dtroit, MI 480, USA

2 J Sci Comput (06) 66: 8 Considr th following Hlmholtz problm: u = λρu, in = (0, ), u = 0, on, u dx =, () whr ρ H 5 (). By applying Richardson xtrapolation to th bi-linar lmnt for th problm (), Lin t al. (s [4]) had th following highst suprconvrgnc rsult: λ h λ ch4, () whr λ h = 4λh/ λ h and u H 5 () is assumd. Schrödingr typ oprators ar of grat importanc in th fild of partial diffrntial quations. In particular, th spctrum of Schrödingr oprators attracts trmndous attntion for th study of th non-rlativistic Born Oppnhimr approximation of th Schrödingr oprators for lctrons moving in a lattic of atoms. Modling such lctrons is a critical part of th implmntation of th Dnsity Functional Thory in Quantum Chmistry [7,6,]. In addition, Hamiltonian systms with tru invrs squar potntials aris in rlativistic quantum mchanics from th squar of th Dirac oprator on an lctron [9]. Thrfor, it is crucial to xplor quations involving such oprators in diffrnt aras of physics and to invstigat approximations of solutions to such quations. Many work hav bn don on approximating Schrödingr typ oprators numrically. For instanc, th problm of optimal approximation of ignfunctions of Schrödingr oprators with isolatd invrs squar potntials, as wll as th solutions to quations involving such oprators, wr considrd (s [0,]). Howvr, to our bst knowldg, no suprconvrgnc rsult of th FEM has bn publishd for Schrödingr oprators. Th purpos of this work is to fill in this blank and to obtain th highst possibl suprconvrgnc rsult for ignvalus of th linar Schrödingr oprators. Th rst of th papr is organizd as follows. In Sct., w provid som prliminaris and stat th main thorm of this papr. In Sct., w dcompos th proof of our main rsults into svral stps and prsnt a proof for ach stp. In th last sction, w complt th proof of th main thorm. Prliminaris For th sak of simplicity and clarity, w only considr th tri-linar lmnt for th following Schrödingr problm: { Δu + φ(x)u = λu, in = ( a, a), () u = 0, on. Hr a > 0,φ(x) C ( \{(0, 0, 0)}), and thr xists a positiv constant c and β such that φ(x) c. It is obsrvd that u(x) is singular at point (0, 0, 0) and dos not blong x β to th Sobolv spac H 5 (). Dfinition (S []) Suppos p (, + ) and assum J is a positiv intgr, ε< is a positiv constant. Dnot a family of wight functions by ω ={ω α = ω α (x), x, α J}, (4)

3 J Sci Comput (06) 66: 8 whr ω α ar wight functions. W dfin th wightd Sobolv spac W J,p (, ω) ={u L p (, ω α ) D α u L p (, ω α ), α J}. (5) Hr, W J,p (, ω) is a normd linar spac whn quippd with th norm u W J,p (,ω) = /p D α u p ωα p (x)dx, (6) α J whr ω α (x) = x α.walsodfin u W J, (,ω) = α J ss sup x D α u(x)ω α (x), (7) whr ssntial suprmum mans th lowst uppr bound ovr xcluding substs of of Lbsgu masur zro. Rmark Th rgularity of ignfunctions u for Schrödingr oprators with priodic potntials was studid in [0], whr it was provd that u W, ( \ κ) whr κ is a st of all singular points and ρ(x) dnots th distanc btwn x and κ and W, ( \ κ) ={v : ρ α α v L ( \ κ), α Z+ }. (8) Howvr, for th problm (), u(x) dos not blong to wightd Sobolv spac W, ( \ {(0, 0, 0)}) for th rason that thr xist th singularitis nar cornrs or dgs, which hav bn discussd and tratd in [,,7 ]. In this articl, w will only discuss th singularity of u(x) at th point (0, 0, 0) and assum that u W 5, (, ω). Th wak form of () istofindu H0 (, φ) satisfying { A(u,v)= λ(u,v), v H0 (, φ), (9) (u, u) =, whr ( [ ] ) H u u (u,φ)= + φ(x)u (x) dx, (0) x i= i x i and [ ] u(x) v(x) A(u,v)= + φ(x)u(x)v(x) dx, (u,v)= u(x)v(x)dx. () x i= i x i In this papr, a mor gnral bilinar form A E (, ) is dfind by [ ] u(x) v(x) A E (u,v)= + φ(x)u(x)v(x) dx, () x i x i E i= whr E is a boundd domain and functions u,v H (E,φ).DfinB(y, r) by B(y, r) ={x : y x r}. () Throughout this papr, standard notations for th classical Sobolv spacs and thir norms arusd.thlttrc or C dnots a gnric constant which is indpndnt of u and N but may not b th sam at ach occurrnc, whr N dnots th numbr of nods that th partition T N, prsntd in Sct.., nds in x i (i =,, ) dirction.

4 4 J Sci Comput (06) 66: 8. A Spcial Rctangular Partition for th Problm() Gradd msh has bn widly usd in finit lmnt mthod (s [5,6,4,9,0]). For xampl, Hunsickr t al. (s [9]) proposd a gradd partition to obtain th optimal convrgnc of th linar lmnt for Schrödingr oprators with priodic potntials. In this subsction, w construct a spcial gradd partition which is th tnsor-product of a on-dimnsional gradd picwis uniform partition. Assum that l is a positiv intgr and N dnots th numbr of partitions in [0, a] satisfying W dscrib th partition as follows. (i) For 0 p l, w dfin b p = and (ii) Dcompos [ a, a] into l N l+. (4) a 4p l i=0 4i +(N l +) a(n l +) l l l, if 0 p l, i=0 4i +(N l +) l, if p = l, (5) b p, if p = 0, a p = p b j, if p. j=0 [ a, a] = (6) l D p, () p=0 whr D p = { [ a 0, 0] [0, a 0 ], if p = 0, [ ap, a p ] [ ap, a p ], if p. (8) (iii) W gt T Nx by splitting D p into p+ intrvals with grid siz l i=0 4i +(N l +) l qually if 0 p l, and (N l + ) uniform partitions with grid siz a l l qually if p = l. Similarly, w construct T i=0 4i +(N l +) l Nx and T Nx. Dnot th rsulting partitions in x i ( i ) dirction as T Nxi. Th final rctangular msh is dfind by T N = T Nx T Nx T Nx. W choos a =, l =, N = to dmonstrat th ida of th msh construction. Th graph for x dirction is shown in Fig., and th gradd graph is dpictd in Fig.. Th following rsults hold tru for T N. a p a0 0 a0 Fig. T Nx :graddmshinx dirction on [, ]

5 J Sci Comput (06) 66: Fig. T N :graddmshon[, ]. It is obvious that th partition siz of T N is diffrnt in diffrnt rgions. St, if p =, p = p D j, if 0 p l, and i= j=0 (9) T N,p ={ : p \ p, T N }. (0) W gt h p = b p p c 4(p l) p. () Hr h p dnots th grid siz of T N,p. Not th fact that th distanc btwn th singular point (0, 0, 0) and p \ p is qual to a p = p j=0 b j, and th combination of (5)and () implis and for any p l, h 0 c 4l, () h 4 p a p c6(p l) 4p (l p) = c 4l. () It is worth to point out that () and() will play a critical rol in th proof of our main rsults.

6 6 J Sci Comput (06) 66: 8. Main Rsult Assum that T N is obtaind by splitting vry sgmnt of T N into ight qual rctangular partitions. Lt and S N ={v C() : v (P ), T N } S N ={v C() : v (P ), T N } b th associatd tri-linar finit lmnt spacs for T N and T N, rspctivly. Dnot S0 N = S N H0 () and SN 0 = S N H0 (, φ). W obtain th tri-linar finit lmnt solution (λ N, u N ) R + S0 N and (λ N, u N ) R + S0 N for problm () from { A(u N,v)= λ N (u N,v), v S0 N, (4) (u N, u N ) =, and { A(u N,v)= λ N (u N,v), (u N, u N ) =. v S N 0, (5) Using Richardson xtrapolation, w obtain th following numrical approximation λ N of λ by λ N = 4λ N λ N Th main rsult of this papr stats as follows.. (6) Thorm Lt W J,p (, ω) b dfind as (5). Assum u W 5, (, ω). Thn Th following corollary is a dirct rsult from (4)and(7). Corollary Undr th sam assumption of Thorm, λ N λ c 4l l. (7) λ N λ cn 4 ln N. (8) Rmark Not that () is th highst ordr suprconvrgnt rsult. For any rctangular partition T h with grid siz h, whav h cn, (9) hnc for any linar lmnt, thr dos not xist ε>0 indpndnt of N such that λ h λ cn 4 ε. Combining th abov analysis implis that (8) is th bst possibl rror bound in trms of N up to a factor of ln N.

7 J Sci Comput (06) 66: 8 7 Main Analysis In th sction, w will tak th following stps to prov Thorm. (i) W will first show that λ N λ can b writtn into λ N λ = 4λ N (u I N u, u N ) λ N (u I N u, u N ) + 4A(I N u u, u N ) A(I N u u, u N ) + 4λ N (u R N u, u N u N ) λ N (u R N u, u N u N ), (0) whr u N, u N, R N u S0 N and R N u S0 N (x) ar rspctivly dfind by u N = u N (u, u N ), u N = u N (u, u N ), () and and A(R N u,v)= A(u,v), v S N 0, () A(R N u,v)= A(u,v), v S0 N. (ii) Th following convrgnc stimation will b provd scondly. λ λ N + u u N L () + u R N u L () + u N u N L () + u I N u L () c l l. (iii) W thn giv a proof of th following stimation: 4A(I N u u, u N ) A(I N u u, u N ) + 4(u I N u, u N ) (u I N u, u N ) c 4l l. (4) (iv) Basd on (0), ()and(4), w complt th proof of (7).. Analysis I W now giv a proof of (0). Not that from (9), ()and(), w hav () λ = λ(u, u N ) = A(u, u N ) = A(R N u, u N ) = λ N (R N u, u N ), (5) and th combination of (4), ()and() will giv λ N = λ N (u, u N ) = λ N (R N u, u N ) + λ N (u R N u, u N ) = λ N (R N u, u N ) + λ N (u R N u, u N ) + λ N (u R N u, u N u N ) =λ N (R N u, u N )+λ N (u I N u, u N )+λ N (I N u R N u, u N )+λ N (u R N u, u N u N ) = λ N (R N u, u N )+λ N (u I N u, u N )+ A(I N u R N u, u N )+λ N (u R N u, u N u N ) = λ N (R N u, u N ) + λ N (u I N u, u N ) + A(I N u u, u N ) + λ N (u R N u, u N u N ). Th abov two qualitis imply that λ N λ can b dcomposd into λ N λ = λ N (u R N u, u N u N ) + λ N (u I N u, u N ) + A(I N u u, u N ).

8 8 J Sci Comput (06) 66: 8 Similarly, λ N λ = λ N (u R N u, u N u N ) + λ N (u I N u, u N ) + A(I N u u, u N ). From th abov two idntitis, togthr with (6), w gt th dsird rsult (0).. Analysis II Now w procd to prov th convrgnc stimation (). Firstly, w introduc th following convrgnc stimat. Lmma Undr th assumption that u W, (, ω), u I N u H (,φ) + u u N H (,φ) c l l. (6) Proof Assum that I N u, th modifid dgr trilinar Lagrang intrpolant associatd to th msh T N, satisfis { u(x), if x is a nod satisfying x = 0, I N u(x) = (7) 0, if x = 0. Lt p and T N,p b dfind by (9) and(0), rspctivly. W first obsrv th following dcomposition: u I N u H (,φ) = (u I N u) L () + φ(x)(u I N u) (x)dx. (8) Now, w nd to stimat th two trms on th right-hand sid of (8). Assum that =. For th first trm, w can split it as (u I N u) L () = l (u I N u) L ( p \ p ). (9) p=0 By (), w hav (u I N u) L ( 0 ) u H ( 0 ) c u W, ( 0,ω) 0 x dx ch 0 c 4l. (40) Dnot th volum of p by V p and by () forall p l, (u I N u) L ( p \ p ) cv p h p u W ( p\ p ) ca p h p a 4 p u W ( p\ p,ω) ch p a p c8(p l) p 4(l p) c l. Insrting th abov two stimats into (9), w obtain (u I N u) L () c l l, (4) Assum that =. For th scond trm in (8), w dcompos it as l φ(x)(u I N u) (x)dx = φ(x)(u I N u) (x)dx. (4) p \ p p=0

9 J Sci Comput (06) 66: 8 9 Thn ()givs φ(x)(u I N u) (x)dx u L ( 0,ω) φ L ( 0 ) c u L ( 0,ω) ch 0 0 c u L ( 0,ω) c 4l c 4l, (4) and () implis, for all p l, φ(x)(u I N u) (x)dx p \ p ca p V p u I N u L ( p \ p ) ca p a p h4 p u W ( p\ p ) ca p h 4 p a 4 p u W, ( p \ p,ω) ch 4 p a p u W, ( p \ p,ω) c 4l. (44) Combining (4) and th abov two stimats, w gt φ(x)(u I N u) (x)dx c 4l l. (45) Thus, insrting (4)and(45) into(8), w hav This implis th dsird rsult in (6). u I N u H (,φ) c l l. (46) According to th stimats for th rrors in ignvalu and ignvctor approximation prsntd by Babuska t al. (s [,4]),wnowgivaproofof() basd on Lmma. Similar to (46), w hav Th following rsult is showd by Babuska t al. (s [,4]): u I N u L () c l l. (47) λ λ N + u u N L () + u R N u L () + u N u N L () u u N H (,φ). (48) By (6)and(48), w gt λ λ N + u u N L () + u R N u L () + u N u N L () c l l. This, togthr with (47), givs th dsird rsult in ().. Analysis III Now w ar rady to giv a proof of (4) and th following stps will b usd to approach (4). (i) Assum that k(i) = i + ifi = or,andk(i) = ifi =. Suppos D i = x i, T N satisfis \ 0,andlth x k(i) and h dnot th grid siz of T N in x k(i) dirction and, rspctivly. W shall prov th following two xpansion idntitis (a) and (b):

10 0 J Sci Comput (06) 66: 8 (u I N u)(x) (a) x i + F(x k(i) ) (b) v(x) dx = ( x i h x k(i) ) D k(i) D i u(x)d i v(x)dx [ D 4 k(i) D i u(x)d i v(x) + 4D k(i) D i u(x)d i D k(i) v(x) (u I N u)(x)v(x)dx = i= (h x i ) Di u(x)v(x)dx ] dx, (49) + o(h 4 ) [ u H () v H () + u H 4 () v L ()], (50) whr x = (x,, x,, x, ) is th cntr of, x = (x, x, x ) and B(x i ) = [(x i x i, ) (h x i ) ], F(x i ) = 6 B (x i )(i =,, ). (ii) W thn giv a proof of th following suprconvrgnc stimat: (iii) Basd on (49), (50)and(5), w giv a proof of (4)... Proof of (49) and (50) I N u u N H (,φ) c l l. (5) Ltusfirstconsidr(49). For any v S N, D i (u I N u)(x)d i v(x)dx = B (x k(i) )D k(i) (D i (u I N u)(x)d i v(x))dx = B(x k(i) )Dk(i) (D i (u I N u)(x)d i v(x))dx = B(x k(i) )Dk(i) D i u(x)d i v(x)dx + B(x k(i) )D k(i) D i (u I N u)(x)d i D k(i) v(x)dx Not that K = = and =: K + K. (5) [ F (x k(i) ) (h x ] k(i) ) / Dk(i) D i u(x)d i v(x)dx F(x k(i) )Dk(i) 4 D i u(x)d i v(x)dx + F(x k(i) )Dk(i) D i u(x)d k(i) D i v(x)dx (hx k(i) ) D k(i) D i u(x)d i v(x)dx, (5) K = F (x k(i) )D i D k(i) (u I N u)(x)d i D k(i) v(x)dx (hx k(i) ) D i D k(i) (u I N u)(x)d i D k(i) v(x)dx = F (x k(i) )D i Dk(i) u(x)d i D k(i) v(x)dx = F(x k(i) )D i Dk(i) u(x)d i D k(i) v(x)dx. (54)

11 J Sci Comput (06) 66: 8 Insrting (5) and(54) into(5), w gt th dsird rsult (49). A proof of (50) can b obtaind in a similar way... Proof of (5) W dcompos I N u u N into I N u u N = (I N u R N u) + (R N u u N ). (55) W first stimat th first trm in th right-hand sid. By (), w hav A(I N u R N u, I N u R N u) = A(I N u u, I N u R N u) l l = A p \ p (I N u u, I N u R N u)=: T p. (56) p=0 W first stimat T 0.From(40)and(4) it follows that A0 (I N u u, I N u R N u) c IN u u H ( 0,φ) I N u R N u H ( 0,φ) p=0 c l I N u R N u H ( 0,φ). (57) Nxt w stimat T p ( p l). By(49), w driv (u I N u)(x) (I N u R N u)(x) dx p \ p x i x i ch p a p u W ( p \ p ) I N u R N u H ( p \ p ) ch p a p u W, ( p \ p,ω) I N u R N u H ( p \ p ) c 4(p l) p) 4(p l) I N u R N u H ( p \ p ) c l I N u R N u H ( p \ p ). (58) Not that φ(x)(i N u u)(x)(i N u R N u)(x)dx p \ p ( c I N u R N u H (,φ) φ(x)(i N u u) (x)dx p \ p ) c I N u R N u H (,φ)a p h p a p u W ( p \ p ) c I N u R N u H (,φ)a p h p a p a p u W, (,ω) c I N u R N u H (,φ)h p a p c l I N u R N u H (,φ). (59) Plugging (58)and(59) into th quality (56), w hav T p c l I N u R N u H (,φ). (60)

12 J Sci Comput (06) 66: 8 Hnc, I N u R N u H (,φ) = Furthrmor, by (6), w arriv at l T p c l l I N u R N u H (,φ). (6) p=0 I N u R N u H (,φ) c l l. (6) W now procd to stimat th scond trm of th right-hand sid of (55). By (), on has A(R N u u N, R N u u N ) = A(R N u, R N u u N ) A(u N, R N u u N ) = A(u, R N u u N ) λ N (u N, R N u u N ) = λ(u, R N u u N ) λ N (u N, R N u u N ) = (λ λ N )(u, R N u u N ) + λ N (u u N, R N u u N ). This stimat, togthr with (), givs R N u u N H (,φ) λ λ N (u, R N u u N ) + λ N (u u N, R N u u N ) c l lc l l + c l lc l l c 4l l. Thrfor, R N u u N H (,φ) c l l. (6) Combining th stimats (55), (6)and(6) givs th dsird rsult (5)... Proof of (4) (u I N u) u N x i On obsrvs that x i dx can b dcomposd into (u I N u) u N (u I N u) u N dx = + dx. (64) x i x i 0 0 x i x i W now stimat th two trms of th right-hand sid. At first, by (5), w hav (u I N u) u N dx 0 x i x i ( c u I N u H ( 0 ) u IN u H ( 0 ) + I N u u N H ( 0 ) + u H ( 0 )) ( ) [ ( ) c x dx x dx + l] l c l l l c 4l l. (65) 0 0 To stimat \ 0 (u I N u) \ x i 0 = \ 0 (u I N u) u N x i x i dx, w split it into u N x i dx (u I N u) x i (u N I N u) x i dx + \ 0 (u I N u) I N u dx. (66) x i x i

13 J Sci Comput (06) 66: 8 W first stimat th first itm of th right-hand sid. It follows from (5)that (u I N u) (u N I N u) dx \ x i x i 0 c 4l l. (67) Nxt w stimat th scond itm of th right-hand sid. By (49), w hav th following dcomposition for (u I N u) I N u p \ p x i x i dx for all p l: (u I N u) I N u dx= ( h x ) k(i) D p \ x i x i k(i) D i u(x)d i I N u(x)dx p p \ p + F(x k(i) )Dk(i) 4 D i u(x)d i I N u(x)dx + 4 p \ p p \ p F(x k(i) )D k(i) D i u(x)d i D k(i) I N u(x)dx =: I + I + I. (68) To stimat I, w dcompos it into I = ( h x ) k(i) D k(i) Di u(x)i N u(x)dx p \ p = ( h x ) k(i) D k(i) Di u(x)(i N u u)(x)dx p \ p + ( h x ) k(i) D k(i) Di u(x)u(x)dx =: I, + I,. (69) Not that I, ch p a p u W 4 ( p \ p )ch p a p u W, ( p \ p ) ch 4 p a p u W 4 ( p\ p ) u W ( p \ p ) ch 4 p a p a 4 p u W 4, ( p \ p,ω)a p u W, ( p \ p,ω) ch 4 p a p c 4l, thn w obtain I I, c 4l. To stimat I, w dcompos it into I ch 4 p u H 5 ( p \ p ) I N u H ( p \ p ) Similarly, ch 4 p a/ p a 5 p u W 5, (,ω) a/ p a p u W, (,ω) ch 4 p a p c 4l. I ch 4 p u H 4 ( p \ p ) I N u H () p \ p ch 4 p a/ p a 4 p u W 4, (,ω)a / p u W ( p \ p ) ch 4 p a p a 4 p a p u W (,ω) ch 4 p a p c 4l.

14 4 J Sci Comput (06) 66: 8 Substituting th abov thr stimats into (68), w driv l (u I N u) I N u ( h x ) k(i) D x i x i k(i) Di u(x)u(x)dx p= p \ p c 4l l. Insrting (67)and(70) into(66), w arriv at l ( (u IN u) x i p= p \ p c 4l l. u N ( x i h x k(i) ) ) D k(i) Di u(x)u(x) dx (70) (7) Furthrmor, by (64), (65)and(7), on obsrvs that (u I N u) u N l dx x i x i c 4l l. Similarly, 4 (u I N u) u N l dx x i x i c 4l l. p= p \ p p= p \ p ( ( h x k(i) h x k(i) ) D k(i) D i u(x)u(x)dx ) D k(i) D i u(x)u(x)dx Combining th abov two stimats, w hav [ 4 (u I N u) u N (u I ] N u) u N dx x i x i x i x c 4l l. (7) i W now turn to th stimation of 4 φ(x)(u I N u)(x)u N (x)dx φ(x)(u I N u)(x)u N (x)dx. W split φ(x)(u I N u)u N (x)dx into φ(x)(u I N u)(x)u N (x)dx = φ(x)(u I N u)(x)(u N I N u)(x)dx + φ(x)(u I N u)(x)i N u(x)dx By (6), =: J + J. (7) J c 4l l. (74) To stimat J, on obsrvs that φ(x)(u I N u)(x)i N u(x)dx c x 0 dx u I N u L ( 0 ) I N u L ( 0 ) 0 ch 0 u W, (,ω) u W, (,ω) c 4l. (75)

15 J Sci Comput (06) 66: 8 5 St μ(x) = φ(x)i N u(x) and assum p. W hav th following dcomposition φ(x)(u I N u)(x)i N u(x)dx p \ p = (u I N u)(x)i N μ(x)dx + (u I N u)(x)(μ I N μ)(x)dx p \ p p \ p =: B + B. (76) By (50), B can b dcomposd into B = (u I N u)(x)i N μ(x)dx = p \ p p \ p i= ( h x i ) Di u(x)i N μ(x)dx + o(h 4 ) [ u H () μ H () + u H 4 () μ L ()] p \ p = + p \ p i= p \ p i= p \ p o ( h x i ) Di u(x)(i N μ φu)(x)dx ( x h i ) Di u(x)φ(x)u(x)dx ( h 4 )[ u H () μ H () + u H 4 () μ L ()] =: B, + B, + B,. (77) So w nd to stimat th thr trms in th right-hand sid. Not that μ = φi N u implis I N μ φu L ( p \ p ) I N μ μ L ( p \ p ) + φ(i N u u) L ( p \ p ) ch p a p μ W ( p \ p ) + ch p a p φ L ( p \ p ) u L ( p \ p ) ch p a p a 4 p Combining ()and(78)givs + ch p a p a p a p ch 5 p a p. (78) B, ch p u H ( p \ p ) I N μ φu L ( p \ p ) Similarly, w hav ch p a p ch 5 p a p ch 4 p a p c 4l. B, ch 4 p [ u H ( p \ p ) μ H ( p \ p ) + u H 4 ( p \ p ) μ L ( p \ p )] ch 4 p a p c 4l. Substituting th abov two stimats into (77), on obsrvs that B B, c 4l. (79)

16 6 J Sci Comput (06) 66: 8 By th sam argumnts in th proof of (79), w gt B c 4l. (80) Thus, w driv φ(x)(u I N u)(x)i N u(x)dx p \ p (h x i ) + Di u(x)φ(x)u(x)dx c 4l, p \ p i= whrwhavusd(76), (79) and(80). Insrting (74) and th abov stimat into (7), w arriv at [φ(x)(u I N u)(x)i N u(x)dx l (h x i ) + Di u(x)φ(x)u(x)dx c 4l l. Similarly, + p= p \ p i= [4φ(x)(u I N u)(x)i N u(x)dx l (h x i ) Di u(x)φ(x)u(x)dx c 4l l. p= p \ p i= Finally, substituting th abov two stimats into (7), w gt 4 φ(x)(u I N u)(x)u N (x)dx φ(x)(u I N u)(x)u N (x)dx c 4l l. This stimat, togthr with (7), complts th proof of (4). 4 Proof of Main Thorm Basd on th abov analysis, w ar rady to giv a proof of Thorm. Similar to (), w hav λ λ N + u I N u L () + u u N L () + u R N u L () + u N u N L () c l l. This, togthr with (), givs 4λ N (u I N u, u N u N ) + λ N (u I N u, u N u N ) c [ ] u I N u L () u N u N L () + u I N u L () u N u N L () c 4l l. (8) Combining (0), (4) and(8), w hav th dsird rsult in (7) and this complts th proof of our main thorm.

17 J Sci Comput (06) 66: 8 Tabl Error of bi-linar lmnt ovr T N N λ N λ λ N λ N λ N λ λ N λ λn λ N Numrical Exampl Considr th following Khon Sham problm ( ) u = λu x R, u dx =. (8) x R It is obsrvd that th minimal ignvalu λ for th problm (8), which dnots th groundstat nrgy of th hydrogn atom, is qual to 0.5 (s[5]). Not that th ground stat charg dnsity gos down xponntially (s [,5]), (8) can b approachd by th following problm {( ) x u = λu x (8) u = 0, x, whr is a boundd domain. In this papr, w choos =[, ].Ltλ N and λ N dnot th minimal ignvalus for th problm (4)and(5), rspctivly. St λ N = 4λ N λ N. (84) Th numrical rsults ar shown in Tabl. From th data, it can b concludd that thr xist two constants c and c, indpndnt of N, such that and c N λ N λ c N, c N 4 λ N λ c N 4. bhavs bttr than our tho- Furthrmor, numrical data in Tabl also indicats that λ N rtical rror stimat by a factor ln N. Rfrncs. Agmon, S.: Lcturs on th Exponntial Dcay of Solutions of Scond-ordr Elliptic Oprators. Princton Univrsity Prss, Princton (98). Asadzadh, M., Schatz, A., Wndland, W.: A non-standard approach to Richardson xtrapolation in th finit lmnt mthod for scond ordr lliptic problms. Math. Comp. 78, (009). Babuska, I., Osborn, J.: Estimats for th rrors in ignvalu and ignvctor approximation by Galrkin mthods, with particular attntion to th cas of multipl ignvalus. SIAM J. Numr. Anal. 4, 49 6 (987)

18 8 J Sci Comput (06) 66: 8 4. Babuska, I., Osborn, J.: Finit lmnt-galrkin approximation of th ignvalus and ignvctors of slfadjoint problms. Math. Comput. 5, (989) 5. Bacuta, C., Nistor, V., Zikatanov, L.T.: Improving th rat of convrgnc of high ordr finit lmnts on polyhdra I: a priori stimats. Numr. Funct. Anal. Optim. 6, 6 69 (005) 6. Bacuta, C., Nistor, V., Zikatanov, L.T.: Improving th rat of convrgnc of high ordr finit lmnts on polyhdra II: msh rfinmnt and intrpolation. Numr. Funct. Anal. Optim. 8, (007) 7. Blaha, P., Schwarz, K., Madsn, G.K.H., Kvasnicka, D., Luitz, J.: WIENk., An Augmntd plan Wav + Local Orbitals Program for Calculating Crystal Proprtis., Karlhinz Schwarz, Tchn. Univrsitt Win, Austria (00) 8. Blum, H.: Numrical tratmnt of cornr and crack singularitis, in finit lmnt and boundary lmnt tchniqu from a mathmatical and nginring point of viw. CISM Courss Lct. 0, (988) 9. Blum, H., Lin, Q., Rannachr, R.: Asymptotic rror xpansions and Richardson xtrapolation for linar finit lmnts. Numr. Math. 49, 7 (986) 0. Blum, H., Rannach, R.: Finit lmnt ignvalu computation on domains with rntrant cornrs using Richardson xtrapolation. J. Comput. Math. 8(), (990). Cavalhiro, A.C.: Wightd Sobolv spacs and dgnrat lliptic quations. Bol. Soc. Parana. Mat. 6, (008). Chn, C., Lin, Q.: Extrapolation of finit lmnt approximations in a rctangular domain. J. Comput. Math. 7, 5 55 (989). Costabl, M., Daug, M., Nicais, S.: Analytic rgularity for linar lliptic systms in polygons and polyhdra. Math. Modls Mthods Appl. Sci., 5005 (0) 4. Duran, G.: Error stimats for anisotropic finit lmnts and applications. In: Procdings of th intrnational congrss of mathmaticans, Madrid, Spain (006) 5. Gärding, L.: On th ssntial spctrum of Schrödingr oprators. J. Funct. Anal. 5, 0 (98) 6. Grinr, W.: Quantum Mhcanics: An Introduction. Springr, Hidlbrg (989). Guo, B., Babuska, I.: Rgularity of th solution for lliptic problms on nonsmooth domains in R,Part I: countably normd spacs on polyhdral domains. Proc. R. Soc. Edinb. A, 77 6 (997) 8. H, W., Guan, X., Cui, J.: Th local suprconvrgnc of th trilinar lmnt for th thr-dimnsional Poisson problm. J. Math. Anal. Appl. 88, (0) 9. Hunsickr, E., Li, H., Nistor, V., Uski, V.: Analysis of Schrödingr oprators with invrs squar potntials II: FEM and approximation of ignfunctions in th priodic cas. Numr. Mth. Part. D. E. 0, 0 5 (04) 0. Hunsickr, E., Nistor, V., Sofo, J.: Analysis of priodic Schrödingr oprators: rgularity and approximation of ignfunctions. J. Math. Phys. 49(8), 0850 (008). Krss, G., Joubrt, D.: From ultrasoft psudopotntials to th projctor augmntd-wav mthod. Phys. Rv. B. 59, 58 (999). Li, H.: A-priori analysis and th finit lmnt mthod for a class of dgnrat lliptic quations. Math. Comput. 78, 7 77 (009). Lin, Q.: Fourth ordr ignvalu approximation by xtrapolation on domains with rntrant cornrs. Numr. Math. 58, (99) 4. Lin, Q., Lin, J.: Finit Elmnt Mthods: Accuracy and Improvmnt. Scinc Prss, (006) 5. Lin, Q., Lu, T.: Asymptotic xpansions for th finit lmnt approximation of lliptic problms on polygonal domains. In: Intrnational confrnc on computational mathmatics applid scinc nginring, Vrsaills (98) 6. Lin, Q., Zhu, Q.: Asymptotic xpansion for th drivativ of finit lmnts. J. Comput. Math., 6 6 (98) 7. Schötzau, D., Schwab, C., Wihlr, T.P.: hp dgfem for lliptic problms in polyhdra. I: stability on gomtric mshs. SIAM J. Numr. Anal. 5, 60 6 (0) 8. Schötzau, D., Schwab, C., Wihlr, T.P.: hp dgfem for scond ordr lliptic problms in polyhdra I: stability on gomtric mshs. SIAM J. Numr. Anal 5, 60 6 (0) 9. Schötzau, D., Schwab, C., Wihlr, T.P.: hp dgfem for scond ordr lliptic problms in polyhdra II: xponntial convrgnc. SIAM J. Numr. Anal. 5, (0) 0. Shnk, N.A.: Uniform rror stimats for crtain narrow Lagrang finit lmnts. Math. Comput. 6(07), 05 9 (994). von Ptrsdorf, T., Stphan, E.P.: Rgularity of mixd boundary valu problms in R and boundary lmnt mthods on gradd mshs. Math. Mthods Appl. Sci., 9 49 (990). Zhang, Z., Naga, A.: A nw finit lmnt gradint rcovry mthod: suprconvrgnc Proprty. SIAM J. Sci. Compu. 6, 9 (005)