CONVERGENCE OF AN IMPLICIT FINITE ELEMENT METHOD FOR THE LANDAU-LIFSHITZ-GILBERT EQUATION SÖREN BARTELS AND ANDREAS PROHL Abstract. Te Landau-Lifsitz-Gilbert equation describes dynamics of ferromagnetism, were strong nonlinearity, nonconvexity are ard to tackle: so far, existing scemes to approximate weak solutions suffer from severe time-step restrictions. In tis paper, we propose an implicit fully discrete sceme and verify unconditional convergence.. Introduction Te penomenological Landau-Lifscitz-Gilbert equation LLG describes dynamics of ferromagnetism; let α denote te damping parameter, ten te magnetization m :, T Ω S 2, for S 2 = {x R 3 x = }, solves. m t = α m m m + m m, supplemented by initial and boundary conditions, m = m W,2 Ω; S 2, and n m = on, T Ω. A proper definition of weak solutions is given below. Limiting equations are te Heisenberg equation α, and eat flow for armonic maps α, see [, Propositions 5., 5.2],.2 m t = m m α, m t = m + m 2 m α. Te construction of convergent scemes for. is a nontrivial task, due to te nonconvex sideconstraint m = a.e. in, T Ω, wic is difficult to realize in a numerical approximation sceme. A first explicit sceme is proposed in [2], were also weak sub- convergence towards weak solutions is verified; tis program is continued in [3], were k = o α 2 + n 2 is identified to be sufficient for stability and convergence; sarpness of tese restrictions is evidenced by computational studies in [3]. From tis background, we look for an implicit sceme exempted from restricting requirements for numerical parameters, and iger flexibility wit respect to small coices of α >. Te construction of our discretization is based on a reformation of. by Gilbert see, e.g. [5], m t α m m t = + α 2 m m. Given te lowest order finite element space V W,2 Ω; R 3 subordinate to a triangulation T of Ω and a time-step size k >, our approximation sceme reads as follows: Algoritm.. Let m V. Given j and m j V determine m j+ d t m j+, φ + α m j d tm j+, φ = + α 2 m j+/2 V from m j+/2, φ φ V. 99 Matematics Subject Classification. 35K55, 65M2, 65M5, 68U, 94A8. Key words and prases. ferromagnetism, Landau-Lifsitz-Gilbert equation, nonconvexity, finite elements, convergence. Supported by Deutsce Forscungsgemeinscaft troug te DFG Researc Center Mateon Matematics for key tecnologies in Berlin
Here,, denotes a discrete version reduced integration of te inner product in L 2 Ω; R 3, : W,2 Ω; R 3 V is a discrete version of te Laplace operator, and we use d t ϕ j := k ϕ j ϕ j for j and ϕ j+/2 := 2 ϕ j+ + ϕ j for j and a sequence {ϕ j } j ; we refer te reader to Section 2 for details. Remark.. Te linear second term in Algoritm. is motivated by te identity m j d tm j+ = m j+/2 k 2 d tm j+ dt m j+ = m j+/2 d t m j+. It is well-known, tat weak solutions to. solve m t = div m m + α m + m 2 m in distributional sense; cf. [, 4]. Corresponding relations need not old for discretizations, due to competition of local and nonlocal aspects inerent to fully discrete finite-element based metods. Lemma 6. below sows tat solutions of Algoritm. satisfy.3 d t m j+, φ + α m j+/2 j+/2, φ = α m 2 m j+/2, φ m j+/2 m j+/2, φ + Corr, for all φ V and a correcting term Corr. Lemma 3. below states conservation of m j = at te nodes of te triangulation T and verifies a discrete energy law for solutions to Algoritm.. Tis indicates tat te forcing correction term Corr serves to balance te damping effect of te implicit Euler metod wit employed reduced integration and local averaging tools in Algoritm.. Te unconditional stability of Algoritm. allows to prove subconvergence to a weak solution of.. Te remainder of tis paper is organized as follows: Preliminaries are stated in Section 2. Our main result is Teorem 3. wic verifies unconditional convergence for Algoritm.; a simple fixed-point iteration is proposed in Algoritm 4., wose convergence is establised for k = O 2, uniformly for values α C, in Section 4. We discuss numerical experiments wit finite-time blowup in Section 5, allowing for direct comparison wit results for values α = O in [3], and study te limiting case α. Section 6 proves.3 and illustrates difficulties in te construction of convergent implicit finite element scemes. 2. Preliminaries Trougout tis paper we assume tat T is a quasiuniform regular triangulation of te polygonal or polyedral bounded Lipscitz domain Ω R n into triangles or tetraedra for n = 2 or n = 3, respectively. We define te lowest order finite element space V W,2 Ω; R 3 by V = { φ CΩ; R 3 : φ K P K; R 3 K T }, were P K; R 3 denotes te set of polynomials of total degree less or equal tan one restricted to te element K T. Given te set of nodes { x l : l L } of te triangulation T, te nodal interpolation operator I : CΩ; R 3 V satisfies I φx l = φx l for all l L. Given functions f, g L 2 Ω; R m and letting, denote te inner product in R m we set f, g = f, g dx. For continuous functions φ, Z CΩ; R 3 we define φ, Z = Ω Ω I φ, Z dx = l L 2 β l φx l, Zx l,
for certain weigts β l >, l L. If for eac l L we denote by ϕ l CΩ te nodal basis function wic is T -elementwise affine and satisfies ϕ l x l = and ϕ l x m = for all m L \ {l} ten we ave β l = Ω ϕ l dx. We define φ 2 = φ, φ and notice tat φ 2 L 2 φ 2 n + 2 φ 2 L 2, for all φ V. We define a discrete Laplace operator : W,2 Ω; R 3 V by 2. φ, χ = φ, χ χ V. It is well known tat tere exists a constant c > suc tat for all φ V tere olds 2.2 φ L 2 c φ L 2, were is te maximal mes-size in T, i.e., = max{diamk : K T }. Coosing χ = φ in 2. and using 2.2 we observe tat for all φ V tere olds 2.3 φ 2 = φ, φ φ L 2 φ L 2 c φ L 2 φ. Given φ V and a node x l for some l L we obtain from using χ = ϕ l φ x l in 2. tat φ x l 2 = β l φ, χ = β l φ, χ 2.4 = β l φ x m, φ x l ϕ m, ϕ l c 2 2 φ L φ x l, m L: K T, x m,x l K were we used 2.2, tat given a node x l te cardinality of te set {m L : K, x m, x l K} is bounded -independently, and tat ϕ m L 2 cβ /2 l for all m L. 3. Unconditional Convergence We first recall te definition of a weak solution to LLG. Trougout tis section we abbreviate Ω T =, T Ω. Definition 3.. Let m W,2 Ω; S 2, ten m is called weak solution to LLG, if for all T > m W,2 Ω T ; R 3 suc tat m = almost everywere in Ω T ; 2 for all φ C Ω T ; R 3 olds m t, φ dxdt + α m m t, φ dxdt = + α 2 m m, φ dxdt ; Ω T Ω Ω T 3 m = m in te sense of traces; 4 for almost all T, T tere olds mt 2 dx + α 2 + α 2 Ω,T Ω m t 2 dxdt 2 Ω m 2 dx. Te following lemma provides discrete counterparts of and 4. We remark tat wellposedness of Algoritm., i.e., te existence of a unique sequence {m j } j tat solves Algoritm., can be deduced from a classical argument, see e.g. [, Section 3]. Lemma 3.. Suppose tat m x l = for all l L. Ten te sequence {m j } j produced by Algoritm. satisfies for all j i m j+ x l = l L, ii 2 d t m j+ 2 L + α 2 + α 2 d tm j+ 2 =. 3
Proof. Verification of i follows from coosing φ = ϕ l m j+/2 x l V for l L in Algoritm.. In order to verify ii, we first coose φ = m j+/2 and find 2 d t m j+ 2 L + α m j+/2 2 d t m j+, m j+/2 =. Coosing φ = d t m j+ yields α + α 2 d tm j+ 2 = α m j+/2 m j+/2, d t m j+. A combination of te two identities proves ii and finises te proof of te lemma. Definition 3.2. For x Ω and t [t j, t j+ define Mt, x := t t j m j+ k x + t j+ t m j k x, M t, x := m j x, M+ t, x := m j+ x, Mt, x := m j+/2. Given any T > equation ii in Lemma 3. may be rewritten as 2 M+ T 2 L + α 2 + α 2 M t 2 dt 2 M 2 L 2. Tis bound yields te existence of some m W,2 Ω; R 3 wic is te weak limit as k, of a subsequence suc tat M m in W,2 Ω T, M, M +, M m in L 2 Ω, M, M +, M m in L 2 Ω T. Since M t, x l = for every l L and almost all t, T, tere olds for every K T, M 2 L 2 K C M 2 L 2 K = C 2 M M L 2 K 2C M L 2 K, wic implies M in L 2 Ω T ; R 3, and ence m = almost everywere in Ω T. Algoritm. may be written as follows: taking φ t := I φt,, for φ C Ω T ; R 3, tere olds 3. M t, φ dt + α M M t, φ dt = + α 2 M M, φ dt. Effects of reduced integration are controlled using te fact tat for all χ, η V tere olds χ, η χ, η C χ L 2 η L 2. Tis implies tat for almost all t, T we ave Mt, φ M t, φ C Mt L 2 φ L 2 and allows to prove tat M t, φ dt m t, φ dt. Using tat for χ V and η CΩ; R 3 tere olds χ, η = χ, I η and employing a triangle inequality and standard estimates for nodal interpolation results in Mt, M φ M t, Id ± I M φ C Mt L 2 M φ L 2. Tis yields tat M M t, φ dt 4 m m t, φdt.
Te only troublesome limit is for te last term in 3.. We write M M, φ = M φ, M = Id I M φ, M + I IdM φ, M + M φ, M =: I + II + III. Control of I uses te bound χ L 2 c χ L 2 and estimates for nodal intepolation, I C 2 D 2 M φ L 2 K M L 2 K C M L 2 φ L M L 2. K T A similar argumentation proves II C K T D 2 M φ L 2 K M L 2 K C M L 2 φ L M L 2. We use tat given any Z, χ W,2 Ω; R 3 tere olds Z, Z χ = Z, Z χ to verify A combination of te last four assertions sows III = M φ, M = M φ, M. T M M, φ dt m φ, m dt = Tis proves our main teorem. m φ, m dt. Teorem 3.. Suppose m x l = for all l L and let {m j } j solve Algoritm.. Assume tat m m in W,2 Ω for. For k, tere exists m W,2 Ω T ; R 3 suc tat M subconverges to m in W,2 Ω T and m is a weak solution of LLG. 4. Solving te nonlinear system In te numerical experiments reported below we employ te following fixed-point iteration to solve te nonlinear system in Algoritm.: Algoritm 4.. Set m j+, := m j and l :=. i Compute m j+,l+ V suc tat for all φ V tere olds 4. k mj+,l+, φ + α k mj mj+,l+ + α2 m j+,l+ 4, φ + α2 4 m j, φ + α2 4 ii If m j+,l+ ε ten stop and set m j+ iii Set l := l + and go to i. m j+,l+ m j+,l, φ m j m j+,l+, φ = k mj, φ + + α2 m j 4 m j, φ := m j+,l+. Te following lemma sows tat te iteration converges provided tat k c 2 / + α 2, for an, k, α-independent constant factor c > tat only depends on te geometry of T. Lemma 4.. Suppose tat γ := 5 + α 2 c 2 2 k/4 < and m j x m = for all m L. Ten, for all l tere exists a unique solution m j+,l+ to 4.. For all l tere olds m j+,l+ Θ γ γ mj+,l, 5
provided tat Θ := +ρ ρ > for ρ := + α2 c 2 k 2 /4. Moreover, for all l and all φ V tere olds dt m j+,l+, φ + αm j d tm j+,l+, φ + α 2 m j+/2,l+ were d t m j+,l+ = k m j+,l+ m j+/2,l+, φ Θ 5 + α2 c 2 4 2 m j+,l+ φ, m j and mj+,l+/2 = 2 mj+,l+ + m j. Proof. We abbreviate µ = + α 2 /4. For φ = m j+,l+ from below by te left-and side of 4. is bounded k mj+,l+ k mj+,l+ 2 µmj m j+,l+, m j+,l+ 2 µ mj L m j+,l+ m j+,l+ k µ 5c 2 2 m j+,l+ 2, were we used m j L = and m j+,l+ c 2 5 2 m j+,l+. Terefore, te bilinear form defined by te left-and side of 4. is positive definite on V V if γ < and ten 4. admits a unique solution. Let m L be suc tat m j+,l L = m j+,l x m. Coosing φ = φ m m j+,l x m in te equation for m j+,l proves k mj+,l x m 2 µ m j x m m j+,l x m m j+,l x m + k mj x m m j+,l x m + µ m j x m m j x m m j+,l x m = µ m j+,l x m m j+,l x m + k mj+,l x m + µ m j x m m j+,l x m Tere olds φ x m c 2 2 φ L for all φ V and ence m j+,l L + kµc 2 2 = Θ. kµc 2 2 Subtraction of two subsequent equations in te fixed-point iteration yields k mj+,l+ µ[m j+,l+, φ + α k mj [mj+,l+ µ[m j+,l+ for all φ V. Coosing φ := m j+,l+ m j+,l+ ], φ ] m j+,l, φ µm j+,l [m j+,l ], φ ] m j, φ µm j [m j+,l+ ], φ = kµθ [m j+,l sows ] + kµ [m j+,l+ ]. Using φ c 2 5 2 φ for all φ V we deduce te first estimate of te lemma. In order to verify te second estimate we notice tat owing to 4., m j mj =, and te above 6
estimate m j+,l+ L Θ tere olds for all φ V d t m j+,l+, φ + αm j d tm j+,l+, φ µm j+/2,l+ m j+/2,l+, φ = k mj+,l+ µm j+,l+, φ k mj, φ + α k mj mj+,l+, φ α k mj mj, φ m j+,l+, φ µm j+,l+ m j, φ µm j m j+,l+, φ µm j m j, φ =µm j+,l+ µ m j+,l+ Θµc 2 5 2 m j+,l wic finises te proof of te lemma. m j+,l, φ µm j+,l+ L m j+,l + φ + φ m j+,l+, φ 5. Numerical Experiments Te implementation of Algoritms. and 4. was performed in MATLAB wit an assemblation of te stiffness matrices in C. We set ε = 4 for te termination criterion in Algoritm 4., and it terminated after at most 5 iterations in all of our experiments. Te experiments are defined troug te following example wic is taken from [3]. Example 5.. Let Ω = /2, /2 2 and let m : Ω S 2 be defined by {,, for x /2, m x = 2xA, A 2 x 2 / A 2 + x 2 for x /2, were A := 2 x 4 /s for some s >. Te triangulations T l used in te numerical simulations are defined troug a positive integer l and consist of 2 2l+ alved squares wit edge lengt = 2 l. Motivated by Lemma 4. we use k = 2 / + α 2 As discrete initial data we employ te nodal interpolant of m, i.e., we set m = I Tl m in all experiments. Figures and 2 display snapsots of te numerical approximation provided by Algoritm. wit α =, s =, and l = 4. Te plots in Figure display te first two components of te vector field M at te nodes of te triangulation after an appropriate rescaling and at various times. Figure 2 sows a zoom towards te origin and reveals tat in tis experiment regularity of te exact solution cannot be expected. At time t.529 te vector at te origin points in anoter direction tan all surrounding vectors resulting in a large maximal W, norm. Figures 3 and 4 sow similar snapsots for α = /64, s =, and l = 4. Owing to te significantly smaller stabilization corresponding to te small value of α, te numerical solution is even less regular tan in te previous experiment and fails to become stationary for times t /2. For fixed α = and s = 4 we used l = 4, 5, 6 in Example 5.. In Figure 5 we displayed te energy EMt = Mt, 2 dx 2 Ω and te W, semi-norm Mt, = Mt L as functions of t for t, 6/ for l = 4, 5, 6. For eac l = 4, 5, 6 te function Mt L assumes te maximum value 2 2 among functions φ V wit φ x m = for all nodes x m. We observe tat for decreasing mes-size te blowup time te time at wic Mt L assumes its maximum approaces t.3. In order to study te dependence of blow-up beaviour on te parameter α we ran Algoritm. in Example 5. for fixed l = 5, s =, and for α =, /4, /6, /64, /256. Te plot in Figure 6 7
.5.5.5.4.4.4.3.3.3.2.2.2.......2.2.2.3.3.3.4.4.4.5.5.5.5.4.3.2...2.3.4.5.5.4.3.2...2.3.4.5.5.4.3.2...2.3.4.5.5.5.5.4.4.4.3.3.3.2.2.2.......2.2.2.3.3.3.4.4.4.5.5.5.5.4.3.2...2.3.4.5.5.4.3.2...2.3.4.5.5.4.3.2...2.3.4.5 Figure. Numerical approximation Mt, in Example 5. wit s =, l = 4, and α = for t =,.9,.295,.529,.588,.646..5.5.5....5.5.5.5.5.5..5..5..5....5.5..5.5..5..5.5.5.5.5..5.5..5..5.5.5.5.5..5.5..5..5.5.5.5.5.5....5.5.5.5.5.5..5..5..5....5.5..5.5..5..5.5.5.5.5..5.5..5..5.5.5.5.5..5.5..5..5.5.5 Figure 2. Nodal values Mt, x m for nodes x m close to te origin in Example 5. wit s =, l = 4, and α = for t =,.9,.295,.529,.588,.646. 8
.5.5.5.4.4.4.3.3.3.2.2.2.......2.2.2.3.3.3.4.4.4.5.5.5.5.4.3.2...2.3.4.5.5.4.3.2...2.3.4.5.5.4.3.2...2.3.4.5.5.5.5.4.4.4.3.3.3.2.2.2.......2.2.2.3.3.3.4.4.4.5.5.5.5.4.3.2...2.3.4.5.5.4.3.2...2.3.4.5.5.4.3.2...2.3.4.5.5.5.5.4.4.4.3.3.3.2.2.2.......2.2.2.3.3.3.4.4.4.5.5.5.5.4.3.2...2.3.4.5.5.4.3.2...2.3.4.5.5.4.3.2...2.3.4.5.5.5.5.4.4.4.3.3.3.2.2.2.......2.2.2.3.3.3.4.4.4.5.5.5.5.4.3.2...2.3.4.5.5.4.3.2...2.3.4.5.5.4.3.2...2.3.4.5 Figure 3. Numerical approximation Mt, in Example 5. wit s =, l = 4, and α = /64 for t =,.2,.297,.492,.687,.78,.37,.664,.254,.2347,.2738,.33. 9
.5.5.5....5.5.5.5.5.5..5..5..5....5.5..5.5..5..5.5.5.5.5..5.5..5..5.5.5.5.5..5.5..5..5.5.5.5.5.5....5.5.5.5.5.5..5..5..5....5.5..5.5..5..5.5.5.5.5..5.5..5..5.5.5.5.5..5.5..5..5.5.5.5.5.5....5.5.5.5.5.5..5..5..5....5.5..5.5..5..5.5.5.5.5..5.5..5..5.5.5.5.5..5.5..5..5.5.5.5.5.5....5.5.5.5.5.5..5..5..5....5.5..5.5..5..5.5.5.5.5..5.5..5..5.5.5.5.5..5.5..5..5.5.5 Figure 4. Nodal values Mt, x m for nodes x m close to te origin in Example 5. wit s =, l = 4, and α = /64 for t =,.2,.297,.492,.687,.78,.37,.664,.254,.2347,.2738,.33.
indicates tat te blow-up time approaces te time t.6 for decreasing α. Te experimental values for α = /64 and α = /256 almost coincide. We remark tat te results of our experiments for α =, /4, /6 are similar to te results obtained in [3] wit an explicit sceme. Te implicit sceme of tis article allows to use smaller values for α wic lead to too restrictive conditions on te time step size for te explicit sceme of [3]. For te triangulations employed ere and for α = te total runtimes of te explicit sceme using reduced integration and te implicit sceme are comparable. 6. Proof of.3 Lemma 6.. Assume tat m x l = for all l L and let {m j } j solve Algoritm.. Tere olds for all φ V, d t m j+, φ + α m j+/2 m j+/2 for a correcting term Corr = Corr A + Corr B, wit Corr A := α j+/2 m 2 +α m j+/2 j+/2, φ = α m 2 m j+/2, φ m j+/2, φ + Corr, 2, m j+/2, φ + α2 + α 2 d t m j+, [ m j+/2 2 φ ] and Corr B = 3 i= Corr B i given in te proof below.,, [ m j+/2 2 ]φ Proof. Given any Z V coose φ = I m j+/2 Z in Algoritm., ten te properties of, imply j+/2 m d t m j+ 6., Z + α m j+/2 m j+/2 d t m j+, Z = + α 2 m j+/2 m j+/2 m j+/2, Z. Owing to a b c = a, c b a, b c for all a, b, c R 3, te second term on te left-and side in 6. may be rewritten as α m j+/2, d t m j+ m j+/2, Z α m j+/2 2 d t m j+, Z dt m j+ 2 m j+/2, Z α m j+/2 2 d t m j+, Z, = α 2 and te first term on te left-and side vanises owing to Lemma 3. below. We again use te above vector identity to recast te rigt-and side of 6. as 6.2 + α 2 m j+/2 + α2 m j+/2 2 m j+/2, Z. We proceed independently wit arising two terms: intermitting te Lagrange interpolant for te, m j+/2 m j+/2, Z nonlinear term in te first case to benefit from 2. yields Id j+/2 ± I m, Z m j+/2, m j+/2 Id = j+/2 I m + Id I m j+/2, Z m j+/2, m j+/2 m j+/2, Z m j+/2, Z m j+/2, m j+/2 j+/2, m, were te first two terms on te rigt-and side are referred to as Corr B. For te last term, we resume j+/2 m, m j+/2, Z m j+/2 = m j+/2 2 m j+/2 j+/2, Z + m 2 2, m j+/2, Z.
8 6 4 2 E[ M t ] =/8 M t, =/8 E[ M t ] =/6 M t, =/6 E[ M t ] =/32 M t, =/32 E[ M t ] =/64 M t, =/64 8 6 4 2..2.3.4.5.6 t Figure 5. Energy and W, semi-norm for decreasing mes-sizes in Example 5. wit α = and s = 4. 9 8 7 6 5 E[ M t ] α = M t, α = E[ M t ] α = /4 M t, α = /4 E[ M t ] α = /6 M t, α = /6 E[ M t ] α = /64 M t, α = /64 E[ M t ] α = /256 M t, α = /256 4 3 2..2.3.4.5.6.7.8.9. t Figure 6. Energy and W, semi-norm in Example 5. for l = 5, s =, and α =, /4, /6, /64, /256. 2
Similarly, we account for effects of reduced integration and local averaging inerent to te sceme for te second term in 6.2, Id j+/2 ± I m 2 Id Z, m j+/2 = j+/2 I m 2 Z, m j+/2 + Id I m j+/2 2 Z, m j+/2 m j+/2 2 Z, m j+/2, were te first two terms on te rigt-and side are gatered in Corr B2. Finally, by Algoritm., and a b, c = a c, b, te first term in 6. is identical to for Corr B3 α dt m j+, Z + + α2 α = α = Id I m j+/2 m j+/2 dt m j+, Z + + α2 α Reassembling 6. ten yields to + α 2 m j+/2 2 d t m j+ [ +α + α 2 j+/2 m m j+/2 Z, m j+/2, Z m j+/2, Z m j+/2 Z, m j+/2 + CorrB3, Id I m j+/2 Z, m j+/2. = + α2 m j+/2 Z, m j+/2 j+/2, Z + m 2 2, m j+/2, Z 2 m j+/2, m j+/2 2 Z ] + α Rearranging terms ten yields to te assertion. Corr A + + α 2 Corr B. Acknowledgment: Part of te work was written wen S. B. visited Forscungsinstitut für Matematik ETH Züric in January 25. S. B. gratefully acknowledges ospitality of te Department of Matematics at te University of Maryland at College Park. References [] F. Alouges, A. Soyeur, On global weak solutions for Landau-Lifsitz equations: existence and nonuniqueness, Nonl. Analysis, Teory, Met. & Appl. 8, pp. 7 84 992. [2] F. Alouges, P. Jaisson Convergence of a finite elements discretization for te Landau Lifsitz equations, downloadable at maery.mat.u-psud.fr/alouges/ 24. [3] S. Bartels, J. Ko, A. Prol, Numerical approximation of Landau-Lifsitz equations and finite time blow up of weak solutions, manuscript 25. [4] B. Guo, M.-C. Hong, Te Landau-Lifsitz equation of te ferromagnetic spin cain and armonic maps, Calc. Var., pp. 3 334 993. [5] M. Kružík, A. Prol, Recent Developments in Modeling, Analysis and Numerics of Ferromagnetism, SIAM Review accepted, also downloadable at: www.fim.mat.etz.c/preprints/ 25. Department of Matematics, Humboldt-Universität zu Berlin, Unter den Linden 6, D-99 Berlin, Germany. Department of Matematics, ETHZ, CH-892 Züric, Switzerland 3