Robust Norm Equivalencies and Preconditioning

Transcription

1 Robust Norm Equvalences and Precondtonng Karl Scherer Insttut für Angewandte Mathematk, Unversty of Bonn, Wegelerstr. 6, Bonn, Germany Summary. In ths contrbuton we report on work done n contnuaton of [1, 2] where addtve multlevel methods for the constructon of precondtoners for the stffness matrx of the Rtz- Galerkn procedure were consdered wth emphass on the model problem ω u = f wth a scalar weght ω. We present an new approach leadng to a precondtoner based on a modfcaton of the constructon n [4] usng weghted scalar products thereby mprovng that one n [2]. Further we prove an upper bound n the underlyng norm equvalences whch s up to a fxed level completely ndependent of the weght ω, whereas the lower bound nvolves an assumpton about the local varaton the coeffcent functon whch s stll weaker than n [1]. More detals wll be presented n a forthcomng paper. 1 Prelmnares 1.1 Rtz -Galerkn-Method Let Ω be a bounded doman n R 2 and H0(Ω) 1 = Y be the Hlbert space defned as the closure of C0 (Ω) wth respect to the usual Sobolev norm. Further let A be an ellptc operator defned on H0(Ω) 1 wth an assocated coercve and symmetrc blnear form a(u, v). The Lax-Mlgram Theorem guarantees then a unque soluton u Y of a(u, v) = (f, v) := f v dx, v Y, for any f L 2(Ω). Defne the Rtz -Galerkn approxmaton u h V h Y by a(u h, v) = (f, v), v V h. If ψ 1,, ψ N s a bass of V h, u h s obtaned by the equatons: N N α a(ψ, ψ k ) = (f, ψ k ), 1 k N, u h := α ψ. =1 These equatons are solved teratvely n the form Ω =1

2 374 K. Scherer u (ν+1) = u (ν) ω Cr (ν), ν = 0, 1, 2, (1) ( ) where r (ν) := Au (ν) b wth stffness matrx A A ψ := a(ψ, ψ k ) and b := {(f, ψ k )}. Further ω denotes a relaxaton factor and C a precondtoner matrx. The goal s to acheve κ(ca) κ(a) or at least of order O(1) ndependent of N. A basc fact s: If C s the matrx assocated to operator C : V V satsfyng γ (u, C 1 u) a(u, u) Γ (u, C 1 u), u V, (2) then κ(ca) Γ/γ. Thus C can be taken as a dscrete analogue of C or an approxmatve nverse of B = C 1. In the theory of Addtve Mult-level-Methods an approach to construct the blnear form wth assocated B s to assume a herarchcal sequence of subspaces V 0 V 1 V J := V h Y X := L 2(Ω), (3) and construct bounded lnear projectons Q j : V V j wth β 0 a(u, u) d 2 j Q ju Q j 1u 2 X β 1 a(u, u), (4) wth Q 0u := 0 and sutable coeffcents {d j}. β 0, β 1 are constants not dependng on the d j, u V h or J. Then defne the postve defnte operator B = C 1 by (u, B u) := d 2 j Q ju Q j 1u 2 X, u V. (5) j=1,k 2 A Dffuson Problem as a Model Problem 2.1 Spectral Equvalences Let T 0 be an ntal coarse trangulaton of a regon Ω R 2. Regular refnement of trangles leads to trangulatons T 0 T 1 T J = T. Each trangle n T k s geometrcally smlar to a trangle of T 0. We defne then the {V j} J j=1 n (3) as spaces of pecewse lnear functons wth respect to these trangulatons. Also ts elements have to satsfy Drchlet boundary condtons. In partcular there exsts a nodal bass ψ (j) k for V j = span{ψ k }. In the followng we consder the model problem a(u, v) := ω ( u, v) (6) Ω whch correspond s to the dfferental operator A = ω. Observe that n case u, v V j, j = 0, 1,, J the blnear form a(u, v) reduces n vew of v = const. on T T j to

3 wth average weghts Robust Norm Equvalences and Precondtonng 375 a(u, v) = a j(u, v) := ( u, v) (7) := 1 µ(t) Ths leads to weghted norms v 2 j,ω := T ωdx, µ(t) = area of T. T T v 2, v ω := v J,ω. (8) Instead of the usual orthogonal projectons Q j : V V j we defne now n contrast to [1] operators Q ω j : V V j wth level-dependng weghts by and A ω j : V j V j for u V j by (Q ω j u, v) j,ω = (u, v), v V j (9) (A ω j u, v) j,ω = a(u, v), v V j. (10) Then the followng modfcaton of a well-known result n the theory of multlevel methods (cf. surveys [3, 5] of J. Xu and H. Yserentant) can be proved. Theorem 1. Suppose that there exsts a decomposton u = J u k for u V wth u k V k and postve defnte operators B ω k : V k V k satsfyng (Bk ω u k, u k ) k,ω K 1 a(u, u), (11) then C ω := J (Bω k ) 1 Q ω k satsfes λ mn (C ω A) 1/K 1. If the operators B ω k further satsfy a( w k, w l ) K 2 l=0 (Bk ω w k, w k ) k,ω, w k := (Bk ω ) 1 Q ω kau (12) then λ max (C ω A) K 2,.e. the operator C ω s spectrally equvalent to A. The proof wll be gven n a forthcomng paper by M. Grebel and M.A. Schwetzer. For the dffuson problem (6) we can choose the operator Bk ω B k u k := 4 k u k for u k V k, hence now smply as C u := 4 k Q ω ku. (13) Ths has several advantages over the approach n [1] whch uses drect norm equvalences lke n (4). For spectral equvalence of C wth A one needs to prove the upper nequalty (11) n the form

4 376 K. Scherer 4 k (u k, u k ) k,ω K 1 a(u, u), (14) only for some decomposton u = J u k. However (12) has to be verfed n the form a( w k, w l ) K 2 4 k (w k, w k ) k,ω, (15) l=0 for any decomposton v = J w k, w k V k. These weghted Jackson- and Bernsten nequaltes wll be verfed n the next sectons n a robust form,.e. the constants depend only weakly from the dffuson coeffcent ω. Another advantage of the above theorem s that (13) leads to a practcal form for the precondtonng matrx C n (1), namely one shows that the operator C above s spectrally equvalent to the operator C := 4 k Mk ω, Mk ω v := N k (k) (v, ψ (1, ψ (k) ) ψ (k) ) k,ω where {ψ (k) } N k =1 denotes the nodal bass of V k for k 1. Thus the operators Mk ω u replace the operators Q ω k defned as n (9). The reason for ths s that one can show (up to an absolute constant) (Q ω ku, u) N k (k) (u, ψ ) 2 ( = u, (1, ψ (k) ) k,ω (k) (u, ψ N k (1, ψ (k) )ψ (k) ) k,ω ) = (Mk ω u, u). Detals as well as the realzaton of ths condtoner n optmal complexty wll be presented n the forthcomng paper by M. Grebel and M.A. Schwetzer. We remark that t can be modfed stll further to obtan a precondtoner Ĉu := N k (k) (u, ψ ) a(ψ (k), ψ (k) ) ψ(k). 2.2 A Weghted Bernsten-Type Inequalty Accordng to (15) we consder here arbtrary decompostons u = w k, w k V k (16) of an element u V J. In the followng we employ the a orthogonal operators Q a j : V J V j defned by a(q a ju, v) = a(u, v), u V J, v V j, so that the elements v j := Q a ju Q a j 1u, v 0 := Q a 0u satsfy

5 Robust Norm Equvalences and Precondtonng 377 u = v j, a(v k, v j) = δ j,k, a(u, u) = =0 a(v j, v j). (17) =0 We ntroduce then the followng assumpton on the weght ω : there exsts a constant C ω ndependent of j and a number ρ < 2 such that for all T T j sup{ω U/ : U T k, U T } C ω ρ k j, j k. (18) Lemma 1. Under the above assumpton on the weght ω there holds the hybrd Bernsten type nequalty v j a 6 6 C 1 C2C ω (2/ρ) j/2 (2ρ) k/2 w k k,ω (19) where C 1 := max T T0 dam(t) max T T0 µ(t), and C 2 := max T T0 dam(t)/ µ(t) are constants whch depend on the ntal trangulaton T 0 only. k=j Proof. In vew of the representaton u = j 1 w k we have by (17) a(v j, v j) = a(v j, u) = a(v j, w k ). (20) By ntegraton by parts we obtan, keepng n mnd that w k, v j are constant on U T k and T T j, respectvely, a(v j, w k ) = = U T k ω U U T U ω U ( v j, w k ) = k=j U T k ω U w k ( v j, n ) = w k ( v j, n ) ω U w k ( v j, n ), where S k (T) denotes the boundary strp along T consstng of trangles U T k, U T. Applyng the Cauchy-Schwarz nequalty gves ( a(v j, w k ) ω U w k 2) 1/2 ( ω U v j 2) 1/2. Concernng the frst double sum we note that w k 2 dam(u)[b b b 2 3] 12 C 2C 1 2 k where we have used dam(u) C 2C 1 2 k µ(u) and the formula w k 2 = µ(u) 12 [b2 1 + b b (b 1 + b 2 + b 3) 2 ] U U w k 2, for lnear functons v on U wth vertces b 1, b 2, and b 3. It follows that (21)

6 378 K. Scherer w k 2 T U S k (T) w k 2 12C 2C 1 2 k w k 2 k,ω. (22) For the second factor n (21) note that by assumpton (19) and by the fact that µ(s k (T))/µ(T) 6 2 j k (cf. [5]) ω U v j 2 C ω ρ k j v j 2 3C 12 k C ω ρ k j 18C 12 k C ω (ρ/2) k j Insertng ths and (22) nto (21) nequalty (19) follows by (20). T U v j 2 v j 2 Wth the help of ths lemma the Bernsten-type nequalty (15) can be establshed. It mproves the correspondng ones n [1, 2] n that assumpton (18) s weaker and at the same tme more smple than those made there. Theorem 2. Consder a sequence of unformly refned trangulatons T j and the respectve sequence of nested spaces V j of lnear fnte elements. Then, under assumpton (18) on the weght ω wth ρ < 2 n (6) there holds the upper bound for w j gven n (16). a(u, u) 432C 2 1C 2C ω 2 ( 2 ρ) 2 Proof. By summng the estmate (18) accordng to (17) we get a(u, u) = 2 2j w j 2 j,ω (23) v j 2 a 216C1 2 C 2C ω (2/ρ) j( ) 2 (2ρ) k/2 w k k,ω. (24) From ths an upper bound for a(u, u) follows by applcaton of a Hardy nequalty to the latter double sum. If quanttes s j, c j are defned by s j := k=j a k, s 1 := 0, c j := wth a k 0 and b > 1 such an nequalty reads ( b j s 2 j k=j j b l, c J+1 := 0 l=0 ) 1/2 b b 1 ( b j a 2 j) 1/2. Applcaton of ths wth a k := (2ρ) k/2 w k k,ω and b = 2/ρ to yelds (2/ρ) j( ) 2 (2ρ) k/2 2 w k k,ω ( 2 ρ) 2 k=j and after nserton nto (24) the bound (23) for a(u, u). 2 2j w j 2 j,ω

7 2.3 A Weghted Jackson-Type Inequalty Robust Norm Equvalences and Precondtonng 379 The goal here s to establsh nequalty (11),.e. to prove 4 k v k k,ω K 1 a(u, u), u V J. (25) By Theorem 2.1 we can employ a partcular decomposton of u. We choose u = v k, v k := Q a ju Q a j 1u as n (17). (26) The basc dea s as n [1] to prove a local estmate for v j j,ω on subdomans U Ω by modfyng the dualty technque of Aubn-Ntsche. The followng result gves an estmate whch mproves the correspondng one n [1] n that the constant does not depend on the weght ω. Lemma 2. Let U = supp ψ (j 1) l be the support of a nodal functon n V j 1. There holds ) v j j,ω,u dam(u) ( v j j,ω,u + 18C R v j j,ω,u, (27) where C R s an absolute constant. Proof: We gve only a rough dea of t. For trangles S T j wth T U consder the Drchlet problems φ S = v j on S, φ S S = ψ (j 1) l S. Then v j 2 = v j φ S on U. Partal ntegraton on each S U gves v j 2 j,ω,u = ω S S, v j) S U S( φ ω S v j( φ S, n S ) S U S. The rest of the proof conssts n a careful estmate of both terms on the rght hand sde. Concernng detals we refer agan to the forthcomng paper wth by M. Grebel and M.A. Schwetzer. We menton only that the constant C R above arses from the well-known regularty result φ S 2,2,S C R v j 2 0,S. Now by the assumpton made on the trangulatons there holds dam(u) C 02 j wth a constant C 0 dependng only on the ntal trangulaton. Then choose j 0 as the smallest nteger wth 2 j 0 = 27 3C RC 0 and the second term on the rght hand sde n (27) s (2/3) v j j,ω,u for all j j 0. If we nsert ths, square and multply the resultng nequalty wth the factor 4 j, the summaton wth respect to U and j j 0 yelds

8 380 K. Scherer Theorem 3. There holds the Jackson-type nequalty j=j 0 4 j v j 2 j,ω 9C 2 0 a(v j, v j) U 9C0 2 j=j 0 U j=j 0 a(v j, v j) (28) for j 0 = log 2 (27 3C RC 0). If one solves at frst the Rtz-Galerkn equatons up to level j 0 1 the precondtonng to the levels j j 0 would be robust under condton (18) on the weght ω. Another possblty would be to establsh a bound of the remanng sum on the left hand sde up to level j 0 1. Here one has to use a dfferent argument at the expense of a dependence of the correspondng constant on ω. However one can acheve ths under a condton whch s weaker than (18). Corollary 1. Under the condton (18) on the weght ω the dscretzed verson of the operator C n (13) yelds a robust precondtonng n (1) for the dffuson problem. 3 Concludng Remarks The results represented here are concerned wth the classcal addtve mult-level method for solvng Rtz-Galerkn equatons wth pecewse lnear elements by precondtonng. The proofs gven or ndcated here for the necessary norm equvalences smplfy and mprove those n [1]. They show that for the dffuson problem a smple modfcaton (13) of the classcal precondtoner makes t robust for a large class of dffuson coeffcents ω. It covers all pecewse constant functons ndependent of the locaton of jumps, ther number or ther frequency. In partcular we do not requre the jumps to be algned wth the mesh on any level,.e. no mesh must resolve the jumps. However the constants n the Jackson- and Bernsten type nequaltes nvolve the heght of the maxmal jump. If we assume that m ω := mn x Ω ω(x) = 1, M ω := max x Ω ω(x) = ǫ 1 a bound for the constant C ω n assumpton (13) s gven by ǫ 1. For most practcal purposes t s therefore necessary to assume that M ω s not too bg. By the form of (13) one sees that even sngulartes of maxmal heght ρ J and exponental growth lmted by ρ are admssble. References [1] M. Grebel, K. Scherer, and M.A. Schwetzer. Robust norm-equvalences for dffuson problems. Math. Comp., 76: , [2] K. Scherer. Weghted norm-equvalences for precondtonng. In Doman Decomposton Methods n Scence and Engneerng, volume 40 of Lect. Notes Comput. Sc. Eng., pages Sprnger, Berln, [3] J. Xu. Iteratve methods by space decomposton and subspace correcton. SIAM Rev., 34(4): , [4] H. Yserentant. Two precondtoners based on the mult-level splttng of fnte element spaces. Numer. Math., 58(2): , [5] H. Yserentant. Old and new convergence proofs for multgrd methods. In Acta Numerca, 1993, pages Cambrdge Unv. Press, Cambrdge, 1993.