NESTED NEWTON STRATEGIES FOR ENERGY-CORRECTED FINITE ELEMENT METHODS U. RÜDE1, C. WALUGA 2, AND B. WOHLMUTH 2 Abstract. Energy-corrected fnte eement methods provde an attractve technque to dea wth eptc probems n domans wth re-entrant corners. Optma convergence rates n weghted L 2 -norms can be fuy recovered by a oca modfcaton of the stffness matrx at the re-entrant corner, and no pouton effect occurs. Athough the exstence of optma correcton factors s estabshed, t remans open how to determne these factors n practce. Frsty, we show that asymptotcay a unque correcton parameter exsts and that t can be formay obtaned as mt of eve dependent correcton parameters whch are defned as roots of an energy defect functon. Secondy, we propose three nested Newton type agorthms usng ony one Newton step per refnement eve and show oca or even goba convergence to ths asymptotc correcton parameter. Key words. corner snguartes, energy-corrected fnte eement methods, optma convergence rates, pouton effect, re-entrant corners 1. Introducton. Eptc parta dfferenta equatons wth snguar souton components are of speca nterest n many appcatons, e.g., n fracture mechancs or n heterogeneous porous meda fow. In the presence of snguar souton components, standard fnte eement methods on unform meshes show poor resuts compared to the best-approxmaton n the L 2 -norm. Ths effect s commony referred to as pouton, and s the reated to the fact that the souton of the dua probem has, n genera, no H 2 -reguarty. In ths work, we consder the Lapace probem n a two-dmensona doman Ω wth a re-entrant corner as a prototype for a systematc approach to pouton effects. We refer to [21] for a numerca study of more genera settngs ncudng second order fnte eements, egenvaue probems and heterogeneous coeffcents. It s we-known, see, e.g., [17, 18, 22, 26], that n the presence of corners wth nteror ange π < θ < 2π the souton w, n genera, have snguar components of type r kπ/θ sn(kπφ/θ, k N, even when the data are smooth. Here, r denotes the dstance to the snguar corner and φ the ange. Thus, the snguar components of the souton are not smooth,.e., s := r π/θ sn(πφ/θ H α (Ω, α 1 + π/θ. As a consequence, ony reduced convergence rates are obtaned by standard fnte eement methods on a sequence of quas-unform meshes,.e., the estmates (u R u 0 = O(h π/θ and u R u 0 = for the Rtz projector R are, n genera, sharp wth respect to powers of the mesh-sze h. Athough the convergence order n the H 1 -norm s the same as the order of the best approxmaton, ths does not hod for the L 2 -norm, where a gap of 1 π/θ n the convergence order can be observed [6, 7, 30]. For a weghted L 2 -norm wth weght r 1 π/θ, the gap s even bgger. The noda nterpoant shows then O(h 2 convergence whe such a weght does not recover any addtona convergence rate for O(h 2π/θ 1 Department of Computer Scence 10, Unversty Erangen-Nuremberg, Cauerstraße 6, D 91058 Erangen, Germany 2 Insttute for Numerca Mathematcs, Technsche Unverstät München, Botzmannstrasse 3, D 85748 Garchng b. München, Germany. 1
the Rtz projector,.e., typcay a gap of 2(1 π/θ can be seen. Standard technques to mprove the convergence are to dea wth weghted Soboev space norms n combnaton wth graded meshes [1, 2, 10, 12, 13, 15] and/or enrchment of the fnte eement space [3, 6, 9, 11, 14, 20, 25, 30]. Aternatve technques are frst order systems east squares methods that provde the fexbty to seect approprate weghts n the norms, see, e.g., [4, 5, 8, 16, 23]. Most strateges requre more than a oca modfcaton and am to mprove the fnte eement approxmaton at the snguarty. However, the quantty of nterest s often not a goba standard norm but a functona or weghted norm whch excudes or reaxes the nfuence of the neghborhood of the re-entrant corner. Exampes are the stress ntensty factor that can be evauated as a ne ntegra at a fxed dstance from the re-entrant corner, the egenvaues and the fux at some gven nterface not ncudng the re-entrant corner. To obtan mproved error reducton rates for such quanttes, an accurate representaton of the souton at the re-entrant corner s not requred. Ths observaton s our motvaton to focus on energy correcton schemes [21, 28, 29] that do not enrch the fnte eement spaces assocated wth a sequence of unformy refned meshes. The basc dea was orgnay ntroduced for fnte dfference schemes n [33]. In the recent contrbuton [19], a mathematcay rgorous anayss for fnte eement methods s presented, and t was shown that a carefu modfcaton of the orgna Gaerkn method can drastcay mprove the convergence. Namey, for conformng ow-order fnte eement spaces, f π < θ < 3π/2, then O(h 2 convergence on a sequence of unformy refned meshes can be observed n a sutabe weghted L 2 -norm, assumng that a suffcenty accurate correcton parameter s known. If addtonay the nta mesh restrcted to eements touchng the re-entrant corner s symmetrc, the same argument aso hods for 3π/2 θ < 2π. The rest of ths paper s organzed as foows: Frsty, n Secton 2, we sketch the energy corrected fnte eement method and motvate the need for an effcent agorthm to determne ts parameter. In Secton 3, we ntroduce a eve-dependent non-near energy defect functon and reca that ts unque root defnes a sutabe eve-dependent parameter. Based on the propertes of the energy defect we estabsh goba convergence of a Newton agorthm and gve reabe stoppng crtera. Secton 4 s devoted to the proof that the roots of these functons converge, and we provde convergence rates. Addtonay, we show that the mt vaue defnes the unque eve ndependent parameter n energy-corrected fnte eement methods. We dscuss, n Secton 5, severa nested Newton type strateges to approxmate the roots and show that the asymptotc correcton parameter can be approxmated wth amost no extra cost. Numerca resuts that ustrate the convergence of the dfferent strateges can be found n Secton 6. Here we aso nvestgate the dependency of the asymptotc correcton factor on the ange of the re-entrant corner and the number of attached eements. Addtonay, we appy the approach to a doman wth severa re-entrant corners and more genera boundary condtons. 2. Energy corrected fnte eement method. In ths secton, we sketch the dea of energy-corrected fnte eement methods. For smpcty we restrct ourseves to a smpe mode probem and to a bounded poygona doman Ω R 2 wth one 2
re-entrant corner. We consder the numerca souton of the Posson probem u = f n Ω, u = 0 on Ω. (2.1 The standard bnear form assocated wth (2.1 s gven by a(v, w := v w dx, Ω v, w H 1 (Ω, and (, denotes the usua L 2 -scaar product. 2.1. Defnton of the energy-corrected fnte eement method. To defne the energy correcton, we ntroduce a (v, w := v w dx, ω where ω B k0h s a unon of eements n T. The sequence T forms a nested herarchy of unformy refned smpca meshes wth mesh-sze h, and B k0h s a ba wth radus k 0 h and center at the re-entrant corner. Asymptotcay, t s essenta that k 0 s fxed and bounded ndependenty of the eve, snce ths guarantees that the number of eements n ω s bounded ndependenty of ; see Subsecton 3.3 for a theoretca dscusson. We note that Agorthm 1 of Subsecton 3.5 gves us for a our numerca exampes k 0 = 1. For gven γ [0, 0.5], we defne the parameter-dependent bnear form by a ec (v, w := a(v, w γa (v, w (2.2 and note that t depends on the mesh dependent subdoman ω and aso on the scaar parameter γ. Obvousy, we have 0.5 a(v, v a ec (v, v a(v, v for v H 1 (Ω. The assumpton γ [0, 0.5] guarantees that a ec (, s unformy coercve on H 1 0 (Ω. A modfed fnte eement formuaton of (2.1 then reads: Fnd u (γ V H 1 0 (Ω such that a ec (u (γ, v = (f, v, v V H 1 0 (Ω. (2.3 Here, V stands for the standard conformng pecewse near fnte eement space assocated wth T. The modfcaton (2.3 does not change the structure of the standard fnte eement stffness matrx and changes ony a sma number of ts coeffcents. Hence, t s cheap and easy to mpement nto exstng codes provded that γ and ω are gven. Moreover, fast hgh performance sovers may proft from usng data structures for unformy refned grds that avod the ogstc overhead of unstructured and adaptve mesh technques. Remark 2.1. We note that for γ = 0 the standard fnte eement souton s recovered. Recang that the Posson equaton modes the norma dspacement of a homogeneous membrane, the effect of the modfcaton wth γ (0, 0.5] can be regarded as a softenng of the matera n ω. We emphasze that the quaty of u (γ s determned by the choce of γ and ω. The choce of ω s motvated by the fact that the modfcaton shoud change the orgna stffness matrx as tte as possbe and shoud not deterorate the convergence order. In [19] t has been shown that such ω and a eve dependent parameter γ exst such that no pouton occurs,.e., there s no gap n the convergence order between the nterpoaton and energy-corrected fnte eement approxmaton. Moreover, second 3
order convergence n a sutaby-weghted L 2 -norm can be recovered. A sutabe correcton parameter γ can be defned by γ := γ, where γ s the root of a non-near energy defect functon that w be ntroduced n Secton 3. In a our numerca exampes of Secton 6 and aso those dscussed n [19, 21], t has been suffcent to choose the correcton doman ω as the unon of eements adjacent to the re-entrant corner. We now ca the modfed fnte eement approach (2.3 energy corrected fnte eement method f ω and the possby eve dependent parameter γ are seected such that optma order convergence rates can be observed on unformy refned meshes. 2.2. Numerca exampe. We start wth an ustratng exampe that shows the performance of an energy corrected fnte eement method. Moreover from ths exampe t w become apparent how mportant the proper seecton of the parameter γ s. To demonstrate the accuracy of the energy-correcton method and to motvate the agorthms proposed n ths paper, we consder here a tranguaton of a poygona doman Ω wth a snge re-entrant corner at x c = (0, 0 and the nteror ange θ = 3π/2 (L-shape. We set non-homogeneous Drchet boundary condtons gven n poar coordnates as u = s := r 2/3 sn(2φ/3 on Ω and zero forcng f = 0 such that the exact souton u = s H 1+π/θ ε (Ω for any ε > 0. Besdes the standard norms, we empoy the weghted L 2 -norm (L 2 ρ and the weghted H 1 -semnorm (Hρ 1 defned by u u h 0,ρ := ρ(u u h 0, u u h 1,ρ := ρ (u u h 0, (2.4 wth the rada weghtng functon defned as ρ := r 1 π/θ. Ths weght s ustrated n Fgure 2.1 aongsde wth the nta tranguaton and the souton. Fg. 2.1. Inta mesh, weghtng functon and souton for the L-shape exampe. We next conduct a convergence study for dfferent vaues of the correcton parameter γ on a seres of unformy refned meshes. A fnte eement formuatons show amost the same quanttatve resuts n the H 1 -norms, but there s a sgnfcant dfference n the performance for the L 2 -norms. Tabe 2.1 presents the errors for the standard FE method when no correcton s used (γ = 0. As s we known, ths shows the pouton effect,.e., t resuts n suboptma convergence rates, n both weghted and standard L 2 -norms. Next, we guess a correcton parameter γ = 0.1 for whch we observe a sgnfcant mprovement of the souton accuracy n the L 2 -norms as shown n Tabe 2.2. However, here the asymptotc behavor remans suboptma. Ths ndcates that the guessed correcton parameter may be consdered good enough for the frst two mesh eves = 1, 2 but that t s not suffcenty accurate for hgher eves. Extendng the resuts 4
L 2 error rate L 2 ρ error rate H1 error rate H 1 ρ error rate 0 3.704e-2-2.362e-2-2.506e-1-1.625e-1-1 1.434e-2 1.37 8.248e-3 1.52 1.622e-1 0.63 8.677e-2 0.91 2 5.545e-3 1.37 3.002e-3 1.46 1.039e-1 0.64 4.595e-2 0.92 3 2.150e-3 1.37 1.129e-3 1.41 6.620e-2 0.65 2.414e-2 0.93 4 8.370e-4 1.36 4.336e-4 1.38 4.199e-2 0.66 1.259e-2 0.94 5 3.273e-4 1.35 1.688e-4 1.36 2.657e-2 0.66 6.535e-3 0.95 Tabe 2.1 Standard and weghted L 2 errors for the L-shape doman wth γ = 0.0. L 2 error rate L 2 ρ error rate H1 error rate H 1 ρ error rate 0 3.016e-2-1.784e-2-2.541e-1-1.645e-1-1 9.722e-3 1.63 4.664e-3 1.94 1.652e-1 0.62 8.801e-2 0.90 2 3.140e-3 1.63 1.257e-3 1.89 1.060e-1 0.64 4.653e-2 0.92 3 1.019e-3 1.62 3.564e-4 1.82 6.754e-2 0.65 2.439e-2 0.93 4 3.339e-4 1.61 1.086e-4 1.71 4.284e-2 0.66 1.270e-2 0.94 5 1.108e-4 1.59 3.597e-5 1.59 2.711e-2 0.66 6.577e-3 0.95 Tabe 2.2 Standard and weghted L 2 errors for the L-shape doman wth γ = 0.1. of [19], we w show n Secton 3.4 that on each mesh eve there exsts a sutabe nterva of energy correcton parameters. Asymptotcay there addtonay exsts a unque parameter γ whch works for a eves. We w desgn effcent agorthms to approxmate ths mt vaue accuratey enough, whch n ths exampe s gven by γ = 0.11917674.... Usng ths asymptotcay correct parameter for computng the resuts n Tabe 2.3 yeds the optma convergence rates predcted by the theory for both, weghted and standard, L 2 -norms. L 2 error rate L 2 ρ error rate H1 error rate H 1 ρ error rate 0 2.911e-2-1.700e-2-2.557e-1-1.659e-1-1 9.147e-3 1.67 4.229e-3 2.01 1.666e-1 0.62 8.887e-2 0.90 2 2.882e-3 1.67 1.057e-3 2.00 1.069e-1 0.64 4.696e-2 0.92 3 9.084e-4 1.67 2.646e-4 2.00 6.815e-2 0.65 2.459e-2 0.93 4 2.862e-4 1.67 6.619e-5 2.00 4.323e-2 0.66 1.279e-2 0.94 5 9.021e-5 1.67 1.656e-5 2.00 2.735e-2 0.66 6.620e-3 0.95 Tabe 2.3 Standard and weghted L 2 errors for the L-shape doman wth γ = γ. However, the queston on how to compute the parameter γ or how to fnd a sutabe eve dependent parameter nterva remans open. Unfortunatey, there s so far no anaytca formua known to determne these quanttes, whch depend on the ange and on the nta mesh at the re-entrant corner but not on the goba mesh and not on the souton. The man contrbuton of ths paper s to deveop and anayze Newton type agorthms for the approxmaton of γ. Moreover we show that the computed approxmatons on each eve defne an energy corrected fnte eement method. The three agorthms proposed n Secton 5 requre each one step of a Newton teraton per refnement eve of the mesh to determne such a correcton parameter. They dffer n whether the anaytc exact energy of the domnatng snguar functon must be known and n how many fnte eement systems must be soved per step. 3. The energy defect functon. For the ease of presentaton, we assume agan that Ω R 2 has one re-entrant corner x c = (0, 0, that a part of the postve x- axs startng at the orgn s n Ω and that a x Ω can be represented n poar coordnates as x = (r cos φ, r sn φ wth φ [0, θ]. Let s := r π/θ sn(πφ/θ, then s s 5
a souton of the Drchet boundary vaue probem: Fnd u such that Moreover a(s, v = 0 for a v H 1 0 (Ω. u = 0 n Ω, u = s on Ω. (3.1 In terms of the bnear form a ec (, gven n (2.2, we defne a fnte eement approxmaton s (γ V such that the nhomogeneous Drchet boundary condtons s (γ(p = s(p are satsfed for a vertces of T beng on Ω and a ec (s (γ, v = 0, v V H 1 0 (Ω. (3.2 We reca that for γ [0, 0.5], the bnear form a ec (, s unformy eptc, and thus a unque souton exsts. Moreover, we have (s s (γ 0;Ω\B Ch, ch 2π/θ (s s (γ 2 0 2C 0 h 2π/θ, (3.3 where B denotes a ba wth a fxed and postve dameter and center at the re-entrant corner. The frst bound foows from the reguarty of s on Ω \ B, the goba L 2 - estmate s s (γ 0 = O(h and Wahbn-type arguments for the fnte eement error on subdomans, [32]. The second equvaence foows from the best approxmaton propertes and the propertes of the modfed bnear form a ec (,. We refer to [19] for detas. Now we defne on each eve the energy defect functon g (γ, for γ [0, 0.5], as g (γ := a(s, s a ec (s (γ, s (γ. (3.4 The man dfference compared to [19] s that we defne the energy defect functon n terms of the snguar functon s and not wth respect to a cut-off of s havng homogeneous Drchet boundary vaues on Ω. Observng that s (γ restrcted to Ω does not depend on γ, we get that s (γ V H 1 0 (Ω, where the prme stands for the dervatve wth respect to γ. A straghtforward computaton then shows that g (γ = 2a ec (s (γ, s (γ + a (s (γ, s (γ = a (s (γ, s (γ 0, (3.5 and thus g ( defned by (3.4 s, n contrast to the defnton gven n [19], a monotoncay ncreasng functon on [0, 0.5]. Moreover, t s easy to see that s (γ satsfes the varatona probem: a ec (s (γ, v = a (s (γ, v, v V H 1 0 (Ω. (3.6 3.1. Agebrac representaton of the energy defect functon. For smpcty of notaton, we suppress n the agebrac representaton the eve ndex and ndcate the γ dependence by an upper ndex. We decompose the non-trva degrees of freedom of the agebrac representaton of s (γ nto two bocks. Then s (γ V can be dentfed wth the vector s γ havng the two bock components s γ I and sγ R. The vaues of s γ I are the noda vaues of s (γ at the nner vertces, and the vaues of s γ R are the noda vaues of s (γ at a subset of boundary vertces. More precsey, et θ p be the ange of the poar coordnates of the boundary vertex p, then we ncude p n the subset of boundary nodes f and ony f θ p (0, θ, see Fgure 3.1. 6
We assembe the matrces wth respect to the reduced space Ṽ := {v V ; v (x = 0, x Ω R } wth Ω R := {x Ω, x = (r x, 0 or x = (r x cos θ, r x sn θ}, and n := dm(ṽ. By A we denote the standard stffness matrx on eve assocated wth Neumann boundary condtons, by A D the stffness matrx assocated wth Drchet condtons and by B the matrx assocated wth a (,. A three stffness matrces A, A D, B R n n have a two tmes two bock structure assocated wth the degrees of freedom n the two bocks I and R. In ths agebrac notaton, the coeffcent vector s γ of the modfed fnte eement souton s (γ satsfes (A D γ Bs γ =: (( AII A IR 0 Id γ ( B 0 0 0 ( s γ = I s γ R ( 0, (3.7 s R, where s R stands for the vector obtaned by noda nterpoaton of the snguar functon s on part of the boundary. Havng (3.7, g (γ defned by (3.4 and g (γ defned n (3.5 can be expressed equvaenty by g (γ = a(s, s s R(A RR A RI (A II γb 1 A IR s R, g (γ = s RA RI (A II γb 1 B(A II γb 1 A IR s R, (3.8a (3.8b wth A RI := A IR. We note that the matrx B s symmetrc and postve sem-defnte and of ow rank N, where N s the number of nteror vertces contaned n ω, see aso Fg. 3.1. Fg. 3.1. The entres n the bock component I are marked wth fed crces and the entres n the bock component R are marked wth empty squares, (N = 4 for k = 1 The matrx A II s postve defnte, and moreover a straghtforward computaton shows that wth q = 1 for k 0 > 1 and q < 1 for k 0 = 1. x Bx qx A II x (3.9 3.2. Reaton to snguar enrchment. As an aternatve to the prevous rank-n modfcaton of the stffness matrx, one coud consder a rank-1 modfcaton havng the same oca effect. Ths can be accompshed by frst enrchng Ṽ wth one bass functon φ E compacty supported n Ω. More precsey, we requre that the support of φ E s bounded away from the outer boundary, supp φ E T = f T ( Ω \ Ω R. Secondy, we consder the orgna varatona probem on the enrched space and appy statc condensaton to obtan ( s γ (( AII A IR 0 Id γ ( S 0 0 0 7 I s γ R = ( 0 s R, (3.10
where the sub-matrx S has the form S := A IE A 1 EE A EI and γ = 1. Here A EE := a(φ E, φ E s a scaar. A IE = A EI, and A EI s a row vector whose entres are obtaned by evauatng a(, φ E wth the bass functons from the I-bock. Now we set γ [0, 1] as parameter. For γ = 0 we obtan the standard formuaton and for γ = 1 the enrched form as descrbed above. For γ (0, 1 we get by constructon a rank-one modfcaton actng as a oca softenng of the matera. Moreover f φ E s supported n ω, then S has the same sparsty pattern as B n (3.7,.e. t has at most N 2 non-zero entres. If φ E s a cut-off of the snguar functon beng supported n a subdoman ndependenty of h and γ = 1, we obtan the condensed form of a snguar enrchment. In ths case, however, the number of non-zero entres n S grows as the mesh-sze tends to zero. Lemma 3.1. If the modfcaton n ts agebrac form s gven by (3.10 wth S = zz, then the energy defect functon has a unque root f β := z y αs R A RIyy A IR s R 0. Here α 1 := a(s, s a(s (0, s (0 and y := A 1 II z. Moreover the unque root s then anaytcay gven by γ = 1 β. Proof. The proof foows by an appcaton of the Sherman Morrson formua n (3.8a and the fact that γ s characterzed by g (γ = 0. Recang that the Sherman Morrson formua as a speca case of the Sherman Morrson Woodbury formua reads as (M + wv 1 = M 1 M 1 wv M 1 1 + v M 1 w for qute genera matrces M and vectors w, v. Settng M := A II and w := γz and v := z, we get that γ s defned by a(s, s s R (A RR A RI ( A 1 II II ( γ zz A 1 II 1 + z A 1 II ( γ A IR s R = 0. z A 1 Observng that a(s (0, s (0 = s R (A RR A RI A 1 II A IRs R, a straghtforward computaton gves γ = β 1. Based on Lemma 3.1 the computaton of γ woud cost the souton of two unmodfed dscrete boundary vaue probems. However, t does not provde the we-posedness and makes no statement about the convergence of such a γ. Thus n the rest of ths paper, we do not dscuss ths opton any further but focus on the modfcaton defned by (2.2 and provde effcent agorthms to approxmate sutabe correcton parameters. Remark 3.2. Equaton (3.8a shows that the non-nearty n g ( stems from the term (A II γb 1. Recang that B s a ow rank matrx, one can use a ow rank representaton of (Id γa 1/2 II BA 1/2 II 1 usng the Sherman Morrson Woodbury formua to rewrte g ( as a ratona functon n γ,.e., g (γ = P (γ/q (γ where P and Q are poynomas of degree at most N. The coeffcents of these poynomas can be computed numercay by sovng N tmes a dscrete boundary vaue probem. In [19] t has been shown that optma convergence order can be observed f the parameter γ on each eve s seected such that g (γ Ch 2 wth C fxed and moderate. In the rest of ths secton, we frsty provde exstence resuts and secondy 8
propose a Newton agorthm ncudng a reabe stoppng crtera for the seecton of ω and γ on each eve. 3.3. Exstence of ω. The propertes of g ( depend on the choce of ω. To determne a sutabe ω, we foow the nes of [19, 29, 28, 33]. Settng ω 1 := {T T, x c T } and then recursvey enargng the neghborhood of x c by we defne ω k+1 := {T T, T ω k }, γ k := a(i s, I s a(s, s I s I s dx, ω k where I s the standard noda nterpoaton operator. Then t s obvous that γ k+1 < γ k. Consderng the numerator n more deta, we obtan by ntegraton by parts and from the reguarty of s the upper bound a(i s, I s a(s, s a(i s s, I s s + 2 a(i s s, s (s I s 2 0 + 2 s n (s I s dτ C(h 2π/θ Ω + h 2 C 2 1h 2π/θ, where n the ast step we used that s = I s = 0 on Ω R Ω. The denomnator can be bounded from beow n terms of the trange nequaty by I s 0; ω k s 0; ω k (s I s 0; ω k s 0; ω k (s I s 0 (C 2 k π/θ C 1 h π/θ. From now on we fx ω := ω k0 wth k 0 N arge enough such that (C 2 k π/θ 0 C 1 2 > 2C1. 2 Ths choce guarantees γ k0 < 0.5. 3.4. Exstence of an nterva for γ. We proceed n two steps. In Lemma 3.3, we w show that g ( has a unque root n (0, 0.5. In Lemma 3.4, we w estabsh ower and upper bounds for g (. These two premnary resuts aow us to specfy a cosed nterva J such that for γ J, we obtan an energy-corrected fnte eement method. Lemma 3.3. There exsts a coarse eve 0 N such that for 0 there s a unque γ (0, 0.5 wth g (γ = 0. Proof. The proof s smar to the proof of [19, Lemma 5.2], see aso [28, Lemma 3]. In a frst step, we show exstence and n a second step unqueness. Snce g ( s contnuous, we start wth the evauaton of g ( at γ = 0 and at γ = 0.5 and deduce g (0g (0.5 0. For γ = 0, we fnd n terms of eq. (3.3 and the fact that s = 0 g (0 = a(s, s a(s (0, s (0 = a(s s (0, s s (0 + 2a(s, s s (0 2C 0 h 2π/θ s + 2 n (s I sdτ 2C 0 h 2π/θ + C 3 h 2 Ω h 2π/θ (C 3 h 2(1 π/θ 2C 0. 9
Thus for 0 arge enough, we have C 3 h 2(1 π/θ 0 C 0 and g (0 C 0 h 2π/θ, 0. (3.11 Notng that s (γ I s V H 1 0 (Ω and thus a ec (s (γ, s (γ I s = 0, we get for γ = 0.5 g (0.5 = a(s, s a(i s, I s + 0.5 a (I s, I s + I s s (0.5 2 h ;0.5 (0.5 γ k0 a (I s, I s + I s s (0.5 2 h ;0.5 > 0, where v 2 h ;γ := a(v, v γa (v, v. Due to the contnuty of g (, there exsts a root γ (0, 0.5 for arge enough. To guarantee unqueness, t s suffcent to sharpen the monotoncty of (3.5 and show g (γ 0. Assumng that a (s (γ, s (γ = 0, then s (γ restrcted to ω s equa to zero and thus s (γ = s (0. Now g (γ = 0 and g (γ = g (0 yeds a contradcton to g (0 < 0, and thus g ( has one unque root. Lemma 3.4. There exst two constants 0 < α β < and a eve 0 such that for 0 and γ [0, 0.5] αh 2π/θ g (γ βh 2π/θ. (3.12 Proof. We start wth the upper bound n (3.12. Usng (3.5, the trange nequaty yeds g (γ = s (γ 2 0;ω ( s 0;ω + (s s (γ 0 2. The upper bound for g (γ s now easy to see by notng that s 0;ω Ch 2π/θ and recang (3.3. We emphasze that C depends on k 0 but not on h. To show the ower bound n (3.12, we proceed n two steps. We frst consder g (γ and then g (0. The second dervatve s gven by g (γ = 2a (s (γ, s (γ. Settng v = s (γ n (3.6, we get g (γ = 2a ec (s (γ, s (γ 0 (3.13 and thus g (γ g (0 0. The proof of a ower bound for g (0 s based on the agebrac representaton of g (γ and on g (γ. Usng (3.8, (3.9, the fact that g (γ = 0 and the equaty (A II γb 1 = A 1 II + γa 1 II B 1 1 2 (Id γb 2 A 1 II B 1 2 1 B 1 2 A 1 II we can bound g (0 n terms of g (0 by g (0 = a(s, s + s R(A RR A RI A 1 II A IRs R, γ [0, 0.5] = g (γ + γ s RA RI A 1 II B 1 2 (Id γ B 1 2 A 1 II B 1 2 1 B 1 2 A 1 II A IRs R γ (Id γ B 1 2 A 1 II B 1 2 1 g (0 γ g 1 γ (0 1 3 g (0 g (0. These premnary consderatons show n terms of (3.11 that there exsts a eve ndependent constant such that the ower bound n (3.12 s satsfed. Now we can set J := [γ τh 2(1 π/θ, γ + τh 2(1 π/θ ] [0, 0.5] wth τ > 0 and reasonaby sma. Then we fnd for γ J that g (γ τβh 2 and for γ J an energy-corrected fnte eement method s obtaned. 10
3.5. Newton agorthm. Our theoretca fndngs aow us to formuate a gobay convergent Newton agorthm. Each Newton step requres the souton of one f- Agorthm 1 Determne ω and cacuate γ on eve 0 k 0 := 1 whe g (0.5 0 do k 0 k 0 + 1 end whe σ := 0.01 k := 0; γ 0 := 0.5 whe g (γ k > σh 2 do γ k+1 := γ k a(s,s aec(s (γ k,s (γ k a (s (γ k,s (γ k k k + 1 end whe nte eement system (3.2 and some extra O(1 cost evauatons snce s (γ s (0.5 H 1 0 (Ω. The parameter σ can be confgured for the probem at hand. Too sma vaues may requre more Newton steps, whe for arger vaues, the approxmaton of γ may be too poor. In a our tests, settng σ = 0.01 worked very we. Theorem 3.5. Agorthm 1 termnates and defnes an energy-corrected FE method. Proof. The theoretca resuts of Secton 3.3 guarantee the exstence of a fnte eve ndependent k 0 such that g (0.5 > 0, and thus the frst whe statement termnates. To show that the second whe statement termnates, we frsty estabsh that the sequence g (γ k s strcty decreasng. Snce g (γ 0, see (3.13, and g (0.5 > 0 a our γ k satsfy 0 < γ γ k 0.5 and thus g (γ k 0. A straghtforward computaton shows n terms of the equvaence (3.12 that g (γ k (1 α β g (γ k 1 (1 α β k g (0.5. Snce g (0.5 Ch 2π/θ, the number of requred Newton steps s at most O(. Remark 3.6. For a our numerca resuts k 0 s equa to one. If N + 1 s the number of eements n ω, then ony 3N 2 entres of the stffness matrx have to be modfed. Typcay N = 2, 3, 4, 5, 6, 7, hence ess than 20 entres have to be modfed; see aso Fg. 3.1 where N = 4 Remark 3.7. A nested Newton varant mproves the performance sgnfcanty. In partcuar, f g (γ 0 = O(h 2 then the number of requred Newton steps w be evendependent. Ths s the case, e.g., for γ 0 := γ 1. 4. Convergence of the eve dependent correcton factor. The resuts of the prevous sectons ndcate that on each eve we have to sove a non-near probem to determne an accurate enough approxmaton for γ. Ths can be done by Agorthm 1 where each step requres the souton of a fnte eement probem. Thus t s for hgher eves qute expensve and makes the scheme unattractve for practca appcatons. We are then nterested n desgnng nested Newton schemes whch can 11
be easy embedded n a fu mutgrd method and whch requre ony one Newton step per refnement eve. To guarantee goba convergence for such a scheme we have to provde theoretca resuts for the energy defect functon. 4.1. Propertes of the energy defect functon. In ths subsecton, we consder the propertes of g ( gven by (3.4 as a functon of γ n more deta. Reca, that dfferent from [19], we do not work wth homogeneous Drchet boundary condtons but wth a boundary vaue probem wth a homogeneous rght hand sde. As we w see, n ths case g ( s strcty ncreasng. Lemma 4.1. For the constants 0 < α β < and 0 of Lemma 3.4 we have for 0 and γ, γ [0, 0.5] 0 g (γ 2 1 γ g (γ. (4.1a g ( γ 4β α g (γ. (4.1b Proof. The proof of (4.1a s based on the equaty n (3.13 and the upper bound (3.9. Let us denote the agebrac representaton of s (γ wth dsγ then ds γ R = 0. Moreover we have g (γ = 2(sγ I Bds γ I and (3.5 yeds g (γ = (s γ I Bs γ I. In terms of (3.9, we then get (1 γ(ds γ I Bds γ I (dsγ I (A II γbds γ I = a ec(s (γ, s (γ and therefore (1 γ ( (ds γ I Bds γ I = a (s (γ, s (γ = (s γ I Bds γ I ( (s γ I Bs γ I 1 2 ( (s γ I Bs γ 1 2 I,.e., 1 ( 2 (ds γ I Bds γ 1 2 I g (γ 2 ( (s γ I 1 γ Bs γ 1 ( 2 I (s γ I Bs γ 1 2 I = 2 1 γ (sγ I Bs γ I = 2 1 γ g (γ. Fnay, the proof of (4.1b foows drecty from (3.12 and (4.1a. 4.2. Convergence of γ. The choce γ = γ resuts n a fary expensve agorthm f the γ are not pre-computed. The man theoretca resut of ths secton s to estabsh convergence of γ. Ths observaton then aows us to formuate a nested one-step Newton agorthm for the approxmaton of γ. To nk γ wth γ +1, we frsty ntroduce an auxary quantty γ, = 1, 2, and secondy reate γ to γ. Let Ω Ω be defned by Ω := ω 1, = 1, 2, 3, see Fgure 4.1 for the case = 1 and = 2. Assocated wth Ω, we defne dscrete soutons s (γ V and s (γ V such that both are gven by nterpoaton on Ω \ Ω,.e., s (γ(p = s (γ(p = s(p for a vertces of T beng n Ω \ Ω and by varatona equaty on Ω, and Ω \ Ω 3, respectvey,.e., a(s (γ, v γa (s (γ, v = 0, v V, v Ω H 1 0 (Ω, (4.2a a( s (γ, v = 0, v V, supp v Ω \ Ω 3. (4.2b To obtan a we-defned s (γ on Ω, we set t equa to s (γ on Ω 3. 12
Ω 2 Ω 1 Fg. 4.1. The constructon of Ω, = 1, 2. Smary to (3.2, (3.4 and γ, we defne now γ, = 1, 2, as the unque souton of g (γ := a Ω (s, s a Ω (s (γ, s (γ + γa (s (γ, s (γ = 0, (4.3 where the dscrete souton s (γ V on Ω s defned by (4.2a. Here we use the notaton a X (, for the bnear form a(, wth the ntegra restrcted to the subdoman X Ω. We pont out that the resuts of Secton 3 aso appy for the energy defect functons g (, = 1, 2. Lemma 4.2. Let γ 1 and γ+1 2 be the unque correcton parameter on eve and + 1 such that (4.3 hods for = 1 and = 2, respectvey. Then, we have γ 1 = γ 2 +1. Proof. Wthout oss of generaty, we have assumed that x c s the orgn of the coordnate system, and thus the near mappng F (x = 2x maps Ω 2 onto Ω 1. Moreover for x Ω 2, we fnd that F (x Ω 1 and s F = 2 π/θ s. By constructon of Ω 1 and Ω 2, the boundary nodes on eve + 1 of doman Ω 2 are mapped by F onto the boundary nodes on eve of doman Ω 1, and ω +1 s mapped onto ω. The defnton (4.2a of s (γ now yeds s 2 +1(γ(x = 2 π/θ s 1 (γ(2x, x Ω 2, and thus n terms of (4.3, we get g 1 (γ = 22π/θ g 2 +1 (γ. Lemma 4.3. Let γ and γ, = 1, 2, be the correcton parameters on eve such that g (γ = 0 and (4.3 hod, respectvey. Then, we have γ γ Ch 2(1 π/θ. Proof. In a frst step, we provde an upper bound for γ γ. Usng the fact that s (γ s (γ H0 1 (Ω, we fnd n terms of (3.2 that a(s (γ, s (γ γ a (s (γ, s (γ a(s (γ, s (γ γ a (s (γ, s (γ. Now the defnton of γ yeds ĝ := a(s, s a(s (γ, s (γ + γ a (s (γ, s (γ g (γ = 0. 13
Decomposng Ω nto Ω and Ω o := Ω \ Ω, we can rewrte ĝ as ĝ = a Ω (s, s a Ω (s (γ, s (γ + γ a (s (γ, s (γ +a Ω o (s, s a Ω o (s (γ, s (γ + (γ γ a (s (γ, s (γ = g (γ + a Ω o (s, s a Ω o (I s, I s + (γ γ a (s (γ, s (γ = a Ω o (s, s a Ω o (I s, I s + (γ γ a (s (γ, s (γ. To obtan an upper bound for γ γ, we reca that (g (γ = a (s (γ, s (γ and thus get n terms of Lemma 3.4 that a (s (γ, s (γ ch2π/θ > 0. On the other hand, notng that dst(x, x c = O(1 for a x Ω o, we fnd a Ω o (s, s a Ω o (I s, I s Ch 2. These two observatons yed γ γ Ch 2(1 π/θ. (4.4 The proof of the ower bound for γ γ requres the use of s (γ and appes smar arguments. We note that s (γ s (γ restrcted to Ω s n H0 1 (Ω, = 1, 2. Then the defnton (4.2a yeds that a Ω ( s (γ, s (γ γ a ( s (γ, s (γ a(s (γ, s (γ γ a (s (γ, s (γ. Now, we use the defnton of γ, the fact that g (γ = 0 and fnd n terms of s (γ = s (γ on Ω 3 and s (γ = I s on Ω o that 0 = g (γ a Ω (s, s a Ω ( s (γ, s (γ + γ a ( s (γ, s (γ = a(s, s a(s (γ, s (γ + γ a (s (γ, s (γ + a Ω o (I s, I s a Ω o (s, s +a(s (γ, s (γ a( s (γ, s (γ + (γ γ a (s (γ, s (γ = a Ω o (I s, I s a Ω o (s, s ( s (γ s (γ 2 0 + (γ γ a (s (γ, s (γ. Appyng the defnton (4.2b, we get ( s (γ s (γ 2 0 = ( s (γ s (γ 2 0;Ω. o 3 In a next step, we can further decompose Ω o 3 nto Ω o and Ω \ Ω 3. On Ω \ Ω 3, both dscrete functons s (γ and s (γ are dscrete harmonc. Moreover, s (γ s (γ s equa to zero on Ω 3. As a consequence, the H 1 -semnorm on Ω \Ω 3 can be bounded by the H 1/2 00 -semnorm on Ω Ω whch can be bounded by the H 1 -semnorm on Ω o, see, e.g., [31]. Usng s (γ = I s on Ω o, we have due to (3.3 ( s (γ s (γ 0 C ( s (γ s (γ 0;Ω o C ( (I s s 0;Ω o + (s s (γ 0;Ω o Ch. Fnay, we reca that a (s (γ, s (γ ch 2π/θ γ γ Ch 2(1 π/θ. > 0 whch mpes Havng the upper bound (4.4 and the ower bound, the dstance between γ and γ can be bounded. Theorem 4.4. Let ω be such that the root γ [0, 0.5]. Then, we have for π < θ < 2π γ γ +1 Ch 2(1 π/θ. 14
Moreover γ converges to γ (0, 0.5] wth γ γ Ch 2(1 π/θ and g (γ Ch 2. Proof. Combnng the resuts of Lemmas 4.2 and 4.3, we get ( γ γ +1 = γ γ 1 + γ+1 2 γ +1 C h 2(1 π/θ + h 2(1 π/θ +1 Ch 2(1 π/θ. Usng the trange nequaty and the boundedness of a geometrc seres, t can be easy seen that γ defnes a Cauchy sequence and converges wth the gven rate. To prove the bound for g (γ, we use the fact that there exsts a ξ [0, 0.5] such that g (γ = g (γ + (γ γ g (ξ Ch 2(1 π/θ h 2π/θ, whch foows by Tayor expanson and Lemma 3.4. 5. Nested Newton agorthms. Athough Agorthm 1 converges gobay, t does not expot the convergence of γ. In ths secton, we present severa nested Newton strateges for the approxmatve computaton of γ. We assume that we start wth a coarse nta mesh and use unform refnement. 5.1. A nested Newton teraton usng the exact energy: Agorthm 2. Our frst agorthm s a smpe Newton strategy based on the observaton that γ s the root of g (. We seect γ a 0 [0, 0.5] and for = 0, 1,... we compute γ+1 a := mn(0.5, max(0, γtra, a γtra a := γ a g (γ a g (Agorthm 2 (γa. Reca that the ndex stands for the refnement eve, and on each eve ony one Newton step s carred out. There s no need for reteraton, and there s amost no extra computatona cost provded that we know a(s, s. Theorem 5.1. Let us assume that ω = ω k0 wth k 0 fxed such that γ < 0.5 f k 0 > 1 or γ 0.5 f k 0 = 1. Then there exsts a constant C a < such that wth C(q, γ := [0, 0.5]. γ+1 a γ C(q, γ γ a γ 2 + C a h 2(1 π/θ, (5.1 q 1 qγ and q as n (3.9. Thus we have convergence for a γ a 0 Proof. Let C 1 := A II γ 1 B and C 2 := A II γ 2 B, then we have for γ 1, γ 2 [0, 0.5], the reaton C1 1 = C2 1 + (γ 1 γ 2 C2 1 B 1 2 (Id (γ1 γ 2 B 1 2 C 1 2 B 1 2 1 B 1 2 C 1 2 = C2 1 + (γ 1 γ 2 C2 1 BC 1 2 +(γ 1 γ 2 2 C2 1 1 BC 2 2 = C 1 2 + (γ 1 γ 2 C 1 2 BC 1 2 +(γ 1 γ 2 2 C 1 2 BC 1 2 2 ( (γ 1 γ 2 (C 1 2 =0 2 BC 1 2 2 C 1 2 ( Id (γ 1 γ 2 (C 1 2 2 BC 1 2 2 1 C 1 2 2 BC2 1 2 BC2 1 = C2 1 + (γ 1 γ 2 C2 1 BC 1 2 + (γ 1 γ 2 2 C2 1 B ( C 2 (γ 1 γ 2 B 1 BC 1 2 = C2 1 + (γ 1 γ 2 C2 1 BC 1 2 + (γ 1 γ 2 2 C2 1 BC 1 1 BC 1 2. 15
In terms of ths eementary equaty and by means of (3.8a and (3.8b, we obtan g (γ 1 = g (γ 2 + (γ 1 γ 2 g (γ 2 + (γ 1 γ 2 2 s RA RI C 1 2 BC 1 1 BC 1 2 A IRs R, (5.2 g (γ 1 g (γ 2 + (γ 1 γ 2 g (γ 2 + (γ 1 γ 2 2 B 1 2 C 1 1 B 1 2 g (γ 2. In contrast to a standard Tayor expanson, the quadratc term n (γ 2 γ 1 2 s weghted by the frst dervatve and not by a second one. Settng γ 1 = γ, γ 2 = γ a and usng (3.8b, we fnd g (γ g (γ a g (γa = (γ γ a + (γ γ a 2 s R A RIC2 1 BC 1 1 BC 1 2 A IRs R s R A RIC2 1 BC 1 2 A. (5.3 IRs R The defnton of γtra a and (5.3 resut n γ a γ γ a tra γ = γ a γ g (γ a g (γ g (γa g (γ g (γa (γ a γ 2 B 1 2 (AII γ B 1 B 1 2 + g (γ g (γa Due to (3.9 the frst term on the rght can be bounded, and Theorem 4.4 and Lemma 3.4 yed a bound for the second term. Atogether we get γ a +1 γ γ a tra γ q (γ a γ 2 + Ch 2 h 2π/θ 1 qγ C(q, γ max(γ, 0.5 γ γ a γ + Ch 2 h 2π/θ. Under the assumptons on γ and k 0, we have that the frst term on the rght s a contracton, and goba convergence s obtaned. 5.2. A nested Newton teraton on two-eves: Agorthm 3. The man dsadvantage of Agorthm 2 s that the determnaton of γ+1 a requres the exact evauaton of g (γ a whch depends on a(s, s and s (γ a. The unknown energy a(s, s can possby be evauated anaytcay or up to order h 2 accurate by quadrature formuas for Ω s ns ds on the edges of Ω. Our man nterest s the formuaton of an agorthm whch does not requre ths evauaton. A frst step nto ths drecton s to propose an agorthm whch does not requre the expct knowedge of a(s, s. To ths end we defne an aternatve characterzaton of the correcton parameter by requrng approxmatey that the energy defect functon on two consecutve eves must concde: g (γ = g 1 (γ. We then set γ0 b = γ1 b = 0 and defne γ+1 b := mn(0.5, max(0, γtra b, = 1, 2,... wth γtra b := a(s (γ b, s (γ b a(s 1(γ b, s 1(γ b a (s (γ b, s (γ b a 1(s 1 (γ b, s 1(γ b (Agorthm 3. Agorthm 3 s motvated by the observaton that γtra b can be equvaenty wrtten as γtra b = γ b g (γ b g 1(γ b g (γb (5.4 g 1 (γb.. Ths can be nterpreted as one Newton step wth start terate γ b g (γ = g 1 (γ. 16 apped for sovng
Before we consder the convergence of the sequence γ b deveop a reaton between g (γ and g +1 (γ. gven by Agorthm 3, we Lemma 5.2. The energy defect functon on eve 1 s reated to the energy defect functon on eve by g 1 (γ = 2 2π/θ g (γ + O(h 2, (5.5a g 1(γ = 2 2π/θ g (γ + O(h 1+π/θ. (5.5b Proof. We frst use the resuts and notaton of Secton 4 to reate g (γ to g 2 (γ and g 1 (γ to g 1 1 (γ va g (γ = g 2 (γ + (a Ω 2 (s, s a Ω 2 (I s, I s + s (γ s 2 (γ 2 h ;γ, g 1 (γ = g 1 1(γ + (a Ω 1 (s, s a Ω 1 (I s, I s + s 1 (γ s 1 1(γ 2 h 1 ;γ, where we reca that v 2 h ;γ := a(v, v γa (v, v. Usng the same arguments as n the proof of Lemma 4.3, we fnd that the second and thrd terms on the rght are of order h 2. Takng nto account the equaty g1 1 (γ = 22π/θ g 2 (γ yeds (5.5a. To obtan a smar reaton for the dervatves, we have to consder g (γ, g 1 (γ and (g 2 (γ, (g 1 1 (γ n more deta. Startng wth the trva equates g (γ (g 2 (γ = a (s (γ, s (γ a (s 2 (γ, s 2 (γ, g 1(γ (g 1 1 (γ = a 1 (s 1 (γ, s 1 (γ a 1 (s 1 1(γ, s 1 1(γ, we have to show that the dfferences on the rght are of order h 1+π/θ. Wthout oss of generaty, we restrct ourseves to the term D := a (s (γ, s (γ a (s 2 (γ, s2 (γ. Usng the defnton of s (γ and s 2 (γ, we get D a (s (γ s 2 (γ, s (γ s 2 (γ + 2 a(s (γ, s (γ s 2 (γ ( Ch 2 + 2 a (s (γ, s (γ s 2 (γ C h 2 + h π/θ h Ch 1+π/θ and thus (5.5b hods. Theorem 5.3. Let us assume that ω = ω k0 wth k 0 fxed such that γ < 0.5 f k 0 > 1 or γ 0.5 f k 0 = 1. Then there exsts a constant C b < such that γ b +1 γ C(q, γ γ b γ 2 + C b h 2(1 π/θ, (5.6 wth C(q, γ as n Theorem 5.1. Thus we have convergence for a γ b 0 [0, 0.5]. Proof. The proof s based on the equvaent representaton of γ b tra by (5.4. We foow the nes of the proof of Theorem 5.1, use (5.5a and (5.5b and get for h sma 17
enough γtra b γ = γ b γ g (γ b g 1(γ b g (γb g 1 (γb = γ b γ + (22π/θ 1g (γ b + O(h2 (2 2π/θ 1g (γb + O(h1+π/θ γ b γ + g (γ b ( + C g (γb h 2 h 2π/θ C(q, γ (γ b γ 2 + Ch 1 π/θ C(q, γ (γ b γ 2 + C b h 2(1 π/θ. + g (γ b h 1+π/θ ( h 1 π/θ h 2π/θ + h 2 + γ b γ h 2π/θ We note that n the ast step, we have used 1 π/θ 2π/θ and that γ b γ s bounded. Let (5.6 hod, then the frst term on the rght s by constructon a contracton, and thus goba convergence can be observed. 5.3. An nexact nested Newton teraton: Agorthm 4. Athough, Agorthm 3 does not requre the vaue a(s, s, we have to sove the fnte eement equaton on both eves, and 1, wth the gven vaue γ b. Thus we propose a further smpfcaton of the agorthm where we reuse the resuts of the prevous computatons by repacng γ b n the fnte eement approxmaton on eve 1 by γ 1 b. We set γ0 c = γ1 c = 0 and defne γ+1 c := mn(0.5, max(0, γc tra, = 1, 2,... wth γtra c := a(s (γ c, s (γ c a(s 1(γ 1 c, s 1(γ 1 c a (s (γ c, s (γ c a 1(s 1 (γ 1 c, s 1(γ 1 c (Agorthm 4. Theorem 5.4. Under the assumptons of Theorem 5.3, there exsts a τ > 0 sma enough and a eve 0 such that for γ c 0 γ, γ c γ 0+1 τ, we have γ c γ τ, 0 and moreover ( γ+1 c γ C c h 2(1 π/θ + γ c γ 2 + γ 1 c γ 2, 0 (5.7 wth C c <, and thus oca convergence s guaranteed. Proof. The proof s technca but essentay foows by the arguments of the proofs of Theorems 5.1 and 5.3. Thus we do not work out a detas on the constants but ony sketch the man aspects. We start by reformuatng the defnton of Agorthm 4. Usng (3.4 and (3.5, we can rewrte γtra c n terms of g 1(γ 1 c, g (γ c, and g 1 (γc 1, g (γc, obtanng γ c tra = g 1(γ c 1 γc 1 g 1 (γc 1 g (γ c + γc g (γc g (γc g 1 (γc 1 = γ c + g 1(γ c 1 g (γ c + (γc γc 1 g 1 (γc 1 g (γc g 1 (γc 1. Usng the notaton of the proof of Theorem 5.1 on eve 1, and settng γ 1 = γ c, γ 2 = γ 1 c n (5.2, we can reformuate the numerator as γ c tra = γ c g (γ c g 1(γ c + (γc γc 1 2 s R A RIC 1 2 BC 1 1 BC 1 2 A IRs R g (γc g 1 (γc 1. 18
Tayor expanson of g ( around γc 1 wth a sutabe ξ [0, 0.5 + τ] yeds n terms of (5.5b that g (γ c = g (γ c 1 + (γ c γ c 1g (ξ = 2 2π/θ g 1(γ c 1 + O(h 1+π/θ + (γ c γ c 1g (ξ. Now due to (3.12 and (4.1b, we get for a sutabe σ < ndependent of g (γ c g 1(γ c 1 = (2 2π/θ 1g 1(γ c 1 + O(h 1+π/θ + (γ c γ c 1g (ξ (1 2 2π/θ σh 1 π/θ σ γ c γ c 1 g 1(γ c 1. For 0, 0 arge enough and for γ c γc 1 2τ, τ sma enough, we fnd g (γc g 1 (γc 1 σg 1 (γc 1 wth σ dependng on 0, τ and θ but not on 0. We reca that for γ c 3/5, we have B 1/2 C1 1 B1/2 B 1/2 (A II 3/5 B 1 B 1/2 A 1/2 II (A II 3/5 A II 1 A 1/2 II = 5/2 Id. These premnary consderatons yed by means of the agebrac representaton (3.8b s R A RIC 1 2 BC 1 1 BC 1 2 A IRs R g (γc g 1 (γc 1 1 σ = 1 σ s R A RIC 1 2 BC 1 1 BC 1 2 A IRs R g 1 (γc 1 s R A RIC2 1 BC 1 1 BC 1 2 A IRs R s R A RIC2 1 BC 1 2 A 5 IRs R 2 σ. Usng the trange nequaty, we obtan the upper bound γtra c γ γ c γ g (γ c g 1(γ c 5 + g (γc g 1 (γc 1 2 σ (γc γ 1 c 2. (5.8 To further bound the frst term on the rght, we foow the nes of the proof of Theorem 5.1 and use g (γ c g 1(γ c = (22π/θ 1g (γ c +O(h2 as we as g (γc g 1 (γc 1 = (2 2π/θ 1g (γc + O(h1+π/θ + g 1 (γc g 1 (γc 1. For γc γc 1 2τ and τ sma enough, we obtan by means of the propertes of the energy defect functon γ c γ g (γ c g 1(γ c g (γc g 1 (γc 1 = γ c (2 2π/θ 1g (γ c γ + O(h2 γ c γ (2 2π/θ 1g (γc + O(h1+π/θ + (γ c γc 1 g (2 2π/θ 1g (γ c (2 2π/θ 1g (γc + O(h1+π/θ + (γ c γc 1 g γ c γ g (γ c ( g (γc + C g (γ c (h 1 π/θ C(γ c γ c 1 2 + C Fnay, (5.8 yeds ( γ c γ (h 1+π/θ 1 (ξ + γ c γ c 1 + h 2(1 π/θ + γ c γ c 1 + h 2(1 π/θ γ c +1 γ C c ( h 2(1 π/θ + γ c γ 2 + γ c 1 γ 2 1 (ξ + Ch 2(1 π/θ provded that the assumptons are satsfed, and τ s sma enough, and 0 s arge enough. Moreover f 0 s arge enough we obtan γ c +1 γ τ, and thus (5.7 foows by nducton. 19.
Remark 5.5. In contrast to Theorems 5.1 and 5.3, Theorem 5.4 does not guarantee goba but ony oca convergence. Athough a three upper bounds (5.1, (5.6 and (5.7 have the same structure, there s one characterstc dfference. In (5.1 and (5.6 the constants n front of the quadratc error terms can be more precsey specfed, and thus convergence for a admssbe start terates s gven. Startng wth Agorthm 3 on coarse eves and then swtchng to Agorthm 4 guarantees goba convergence. Remark 5.6. Athough Agorthm 3 and Agorthm 4 can be apped wthout the expct knowedge of a(s, s, we st requre s to set the boundary condtons of our auxary probems. However, n any probem setup where the snguar component s causes the domnatng error contrbuton, the same agorthms can be used. 6. Numerca resuts. Uness mentoned otherwse, we consder the probem (3.1 where we know the exact souton that s gven by the snguar functon u = s. Startng from coarse meshes T 0, we generate a sequence of meshes T by unform mdpont-refnements. For our numerca tests, we often consder the symmetrc (crcuar L-shape geometry and the st-doman wth nteror anges θ = 3π/2 and θ = 2π, respectvey; cf. Fgure 6.1. Fg. 6.1. Meshes T 0 and T 1 for n = 6 eements attached to the reentrant corner and nteror anges θ = 3π/2 and θ = 2π, respectvey. A mpementatons are based on the Python nterface of the DOLFIN (v. 1.2.0 fnte eement envronment [24]. 6.1. A comparson of the nested Newton agorthms. In the foowng numerca experments, we compute the errors n weghted L 2 -norms defned n (2.4. For the standard fnte eement method wthout energy correcton, we expect a suboptma asymptotc convergence rate of 2π/θ,.e., 4/3 for the L-shape and 1 for the st doman. Ths behavor can ndeed be observed n Tabes 6.1 and 6.2 for the seres of fnte eement soutons obtaned wthout energy correcton; see aso [19]. Next, we compare the dfferent varants of the energy corrected method, as proposed n Sectons 3 and 5, namey: the exact Newton on each eve (Agorthm 1, the nested teraton n terms of the exact energy (Agorthm 2, the approach on two eves (Agorthm 3, and the nexact method (Agorthm 4. The errors for subsequenty refned meshes of the L-shaped and st-doman are dspayed n Tabes 6.1 and 6.2, respectvey. A four agorthms successfuy recover the optma asymptotc convergence rates. Furthermore, n terms of the absoute error, the three nested agorthms reach amost the same or even sghty better resuts than when usng Agorthm 1 to approxmate the root γ. The underned vaues ndcate that wth energy correcton approxmatey two eves of refnement can be saved n case of the L-shaped doman, as compared wth the unmodfed fnte eement souton. In the case of the st doman, the error on eve 3 wth correcton s aready smaer than the error on eve 6 for the uncorrected souton. 20
uncorrected Agorthm 1 Agorthm 2 Agorthm 3 Agorthm 4 error rate error rate error rate error rate error rate 0 2.362e-2-1.774e-2-1.696e-2-1.706e-2-1.706e-2-1 8.248e-3 1.52 4.458e-3 1.99 4.272e-3 1.99 4.283e-3 1.99 4.105e-3 2.06 2 3.002e-3 1.46 1.115e-3 2.00 1.068e-3 2.00 1.070e-3 2.00 1.071e-3 1.94 3 1.129e-3 1.41 2.788e-4 2.00 2.672e-4 2.00 2.678e-4 2.00 2.676e-4 2.00 4 4.337e-4 1.38 6.973e-5 2.00 6.686e-5 2.00 6.701e-5 2.00 6.701e-5 2.00 5 1.688e-4 1.36 1.743e-5 2.00 1.673e-5 2.00 1.676e-5 2.00 1.676e-5 2.00 6 6.621e-5 1.35 4.361e-6 2.00 4.186e-6 2.00 4.195e-6 2.00 4.195e-6 2.00 Tabe 6.1 L 2 ρ errors obtaned wth uncorrected vs. energy corrected methods for n = 6 and θ = 3π/2. uncorrected Agorthm 1 Agorthm 2 Agorthm 3 Agorthm 4 error rate error rate error rate error rate error rate 0 4.245e-2-2.587e-2-2.456e-2-2.472e-2-2.472e-2-1 1.874e-2 1.18 6.397e-3 2.02 6.251e-3 1.97 6.234e-3 1.99 8.819e-3 1.49 2 8.890e-3 1.08 1.587e-3 2.01 1.561e-3 2.00 1.557e-3 2.00 1.605e-3 2.46 3 4.347e-3 1.03 3.945e-4 2.01 3.892e-4 2.00 3.887e-4 2.00 4.351e-4 1.88 4 2.152e-3 1.01 9.817e-5 2.01 9.702e-5 2.00 9.692e-05 2.00 9.695e-5 2.17 5 1.071e-3 1.01 2.444e-5 2.01 2.417e-5 2.00 2.415e-05 2.00 2.415e-5 2.01 6 5.343e-4 1.00 6.088e-6 2.01 6.024e-6 2.00 6.019e-6 2.00 6.019e-6 2.00 Tabe 6.2 L 2 ρ errors obtaned wth uncorrected vs. energy corrected methods for n = 6 and θ = 2π. In Fgure 6.2 we pot the evouton of the correcton parameters γ wth the refnement eves for the dfferent agorthms. We pont out that a three nested Newton agorthms proposed n Secton 5 converge wth the same order but quanttatvey faster to γ than the mesh-dependent root γ of the energy defect functon. Agorthm 2 and Agorthm 3 produce amost the same curves whereas the cheapest method, Agorthm 4, shows oscatons n the pre-asymptotc range; cf. aso Remark 5.5. 0.130 0.125 0.120 0.115 0.110 Agorthm 1 Agorthm 2 0.105 Agorthm 3 Agorthm 4 0.100 0 2 4 6 8 10 12 0.38 0.36 0.34 0.32 0.30 0.28 0.26 Agorthm 1 Agorthm 2 Agorthm 3 Agorthm 4 0.24 0 2 4 6 8 10 12 Fg. 6.2. Pot of γ obtaned by the dfferent nested Newton methods, dependng on eve for θ = 3π/2 (eft and θ = 2π (rght. The dashed nes ndcate nonnear fts to γ = c 0 + c 1 h 2 2π/θ, startng from eves 2 and 4, respectvey, to excude pre-asymptotca effects. 6.2. The nfuence of the doman and the coarse mesh on γ. In ths subsecton, we consder numercay the nfuence of the nteror ange θ at the reentrant corner and of the number of eements touchng the re-entrant corners on γ. A tests are setup wth k 0 = 1, and thus ω s the unon of the n eements T T such that x c s a vertex of T. The asymptotc parameter γ n = γ n (θ s regarded as a functon on θ [π, 2π]. Here soscees tranges wth ange θ/n are used; the effects 21
of the eement shapes and asymmetry near the corner are dscussed n [19]. To obtan an approxmaton of γ that s suffcenty accurate for many practca computatons and to save an expct computaton, we propose a nonnear ft whch s constructed as foows: For each n {3,..., 12} and 60 sampes of θ n [π, 2π], we appy Agorthm 3 to compute parameters γ on a seres of 7 meshes, whch are 1 successvey refned accordng to the Bursch sequence (.e., 1, 2, 1 3, 1 4, 1 6,... n order to reduce the cost per evauaton; cf. Fgure 6.3. Fg. 6.3. Seres of meshes refned accordng to the Bursch sequence for n = 6 and θ = 7π/4. Gven ths data, and assumng an asymptotc expanson for γ(h = γ + c 1 h 2 2π/θ + o(h 2 2π/θ, we use a Rchardson extrapoaton of the form γ (θ, n γ + (γ +1 γ /(1 ( h +1 h 2 2π/θ (6.1 on the ast eve to emnate the domnatng error term n the asymptotc expanson and obtan mproved vaues for the correcton parameters. Havng a coser ook at the numerca resuts obtaned by ths procedure, we fnd that furthermore γ (θ, n can be accuratey approxmated by a ft n the form of γ ft (θ, n = σ 1,n (exp( 2(θ π 1 + σ 2,n (θ π. In Tabe 6.3, we st the coeffcents σ 1,n and σ 2,n obtaned for Drchet and Neumanntype snguartes for typca numbers of eements attached to the snguarty. Drchet corner Neumann corner n σ 1,n σ 2,n σ 1,n σ 2,n 3 0.0998183980437 0.1896155427030 0.1033975735530 0.1917937440240 4 0.0555624819392 0.1280415576990 0.0559892102051 0.1282280329260 5 0.0415019850858 0.1072128902000 0.0425309204472 0.1076059179200 6 0.0363481781425 0.0979881012415 0.0369403570671 0.0981918095083 7 0.0328888599638 0.0925971779024 0.0319701432127 0.0922167915794 8 0.0313092655216 0.0894450110842 0.0303097852573 0.0890798605638 9 0.0304135897967 0.0874557266743 0.0296697187321 0.0871982940151 10 0.0289942470411 0.0857622163158 0.0290198736762 0.0858256530158 11 0.0279010670390 0.0844853256090 0.0273173027417 0.0842839957719 12 0.0279439846719 0.0838604929991 0.0268036813107 0.0834159415925 Tabe 6.3 Coeffcents obtaned from the nonnear ft Numerca tests ndcate, that the straghtforward mpementaton of ths ft of the correcton parameters n a fnte eement code aready sgnfcanty mproves the souton n presence of corner snguartes; cf. e.g. [21] for an appcaton to egenvaue probems n non-convex domans. Moreover, n convergence studes wth randomy chosen vaues of n and θ, we have aways observed a substantay better souton compared to fnte eements wthout energy correcton. In addton, the ft aso provdes a good nta guess for the prevousy dscussed nested Newton agorthms. Fgure 6.4 ustrates the nfuence of n and θ on γ (θ, n. As the rght pot ndcates, the correcton parameter assumes a maxma vaue at γ (2π, 3 = 1/2, and n the 22
numerca agorthms ths vaue s obtaned sharpy. Whether ths s mere chance or not needs to be further nvestgated. Moreover we see that for n, γ (θ, n converges and that for θ [3π/2, 2π], we obtan amost near dependence for a n. γ 0.5 0.4 0.3 0.2 0.1 n=3.0 n=4.0 n=5.0 n=6.0 n=7.0 0.0 1.0 1.2 1.4 1.6 1.8 2.0 θ/π 3 4 5 6 7 8 n 9 10 11 12 1.2 1.0 0.50 0.44 0.39 0.33 0.28 0.22 0.17 0.11 0.06 0.00 2.0 1.8 1.6 1.4 θ/π γ 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 Fg. 6.4. Pot of γ over θ and n. 6.3. An exampe wth deazed cracks. Fnay, we consder the doman Ω = [ 2, 2] [ 1, 1] that has seven deazed cracks,.e., re-entrant corners wth θ = 2π, as depcted n Fgure 6.5 (eft. The probem s defned as u = 0, n Ω, u = 1 4 cos(πx 2 + 1 on Γ 1 := { 2} ( 1, 1, u = 1 4 cos(πx 2 on Γ 2 := {2} ( 1, 1, u n = 0 on Ω\(Γ 1 Γ 2. Such probems may e.g. arse n heat conducton studes n materas wth cracks. Gven a tranguaton of the doman, our mpementaton automatcay detects a corners and extracts the patch of eements attached to the corner. Ths tme we use the cheaper method Agorthm 4 to compute a seres of correcton parameters on meshes, whch are agan successvey refned accordng to the Bursch sequence. Usng the Rchardson extrapoaton (6.1, we obtan suffcenty accurate vaues for the correcton parameters γ = 0.28033007... at the respectve corners. Fg. 6.5. Left: Inta mesh and weghtng functon; Rght: Pot of the souton on mesh eve 3. 23