NUMERICAL APPROXIMATION OF THE LANDAU-LIFSHITZ-GILBERT EQUATION AND FINITE TIME BLOW-UP OF WEAK SOLUTIONS

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1 NUMERICAL APPROXIMATION OF THE LANDAU-LIFSHITZ-GILBERT EQUATION AND FINITE TIME BLOW-UP OF WEAK SOLUTIONS SÖREN BARTELS, JOY KO, AND ANDREAS PROHL Abstract. Te Landau-Lifsitz-Gilbert equation describes magnetic beavior in ferromagnetic materials. Construction of numerical strategies to approximate weak solutions for tis equation is made difficult by its top order nonlinearity and nonconvex constraint. In tis paper, we discuss necessary scaling of numerical parameters and provide a refined convergence result for te sceme first proposed in [1]. As application, we numerically study finite time blow-up in two dimensions for te regime of small damping parameter. 1. Introduction Te Landau-Lifsitz-Gilbert equation (LLG records te excange interaction between magnetic moments in a magnetic spin system on a square lattice. In tis setting, te energy is given by te Hamiltonian H = K S i,j {S i+1,j + S i,j+1 }, i,j were S i,j is te spin vector of unit lengt at site (i, j and K is a positive excange constant. Te dynamics of tis system is given by nearest neigbor interaction: Ṡ i,j = KS i,j (S i+1,j + S i1,j + S i,j+1 + S i,j1. Assigning S i,j = u(i, j, t for u : R 2 R S 2, we ave t u = K 2 u u + O( 3. We adopt a standard usage of K to be inversely proportional to te square of, and arrive at te continuum limit (Heisenberg equation (1.1 t u = u u wit an associated energy given by te Diriclet energy functional. To incorporate te Gilbert damping law, wose origin lies in te observation tat suc systems reac equilibrium and must ave decreasing energy over time, a dissipative term can be added on, resulting in te LLG equation: (1.2 t u = u u λu (u u, λ >. Te Caucy problem for LLG wit natural boundary conditions, ten, is te problem of finding u, given initial data u : Ω R n S 2 satisfying t u = u u λu (u u on Ω (1.3 u ν = on Ω. u(x, = u (x We will refer to te first term as te gyroscopic term and te second as te damping term. Wen only te damping term is present, tis equation is te armonic map eat flow problem. Tere are Date: May 5, 25. Supported by Deutsce Forscungsgemeinscaft troug te DFG Researc Center Mateon Matematics for key tecnologies in Berlin Partially supported by NSF grant DMS

2 several standard forms of (1.2 wic are equivalent for smoot solutions wic we will make use of in tis paper. Te first results from te vector identity ξ (v ξ = v + (v, ξξ, wic olds for ξ a unit vector. From tis, (1.2 can be rewritten as (1.4 t u = u u + λ( u + u 2 u. From (1.2 and (1.4, we can derive te following two additional formulations: (1.5 t u + λu t u = (1 + λ 2 u u (1.6 t u λu t u = (1 + λ 2 ( u + u 2 u. Global weak solutions, even partially regular ones, ave been sown to exist for LLG in two and tree dimensions given initial data wit finite Diriclet energy. Amongst tese is te work of Alouges and Soyeur [2] wo ave made use of te definition of weak solution naturally arising from (1.5 to sow tat energy bounds are sufficient for te existence of suc a weak solution in tree dimensions. Guo and Hong [6] successfully carried troug te argument tat Struwe in [13] employed for te armonic map eat flow to exibit a Struwe solution in two dimensions a partially regular solution tat satisfies an energy inequality and is smoot away from a finite set of point singularities. Recently, Ko [8] in two dimensions and Melcer [1] in tree dimensions independently construct partially regular solutions to LLG smoot away from a locally finite n- dimensional parabolic Hausdorff measure set. Uniqueness of weak solutions in te class of partially regular solutions is, owever, still an open question. For te armonic map eat flow, tere exist nonunique solutions due to te appearance of singularities [5]. Wile tis related question of singularity formation weter singularities develop from smoot initial data in finite time as been demonstrated for te armonic map eat flow, no suc initial data as been produced for LLG. An inquiry into tis problem is a natural start to te problem of nonuniqueness as well as te broader issues regarding te validity of te model and selection criteria for correct solutions. Very little is known about singularities and blow-up dynamics for LLG. As long as u L is bounded, te solution remains regular for all time so singularities in tis case are indicated by loss of control on u L. Te presence of te gyroscopic term precludes te successful application of standard analytical arguments to sow blow-up of solutions suc as convexity arguments, scaling arguments, and constructions of explicit solutions. Te breakdown of tese metods and te subsequent need to understand te contribution of te gyroscopic term ave inspired recent efforts for singularity formation in te limiting case λ =. Sata and Zeng [12] ave produced weak-l 2 initial data wic are locally smoot tat develop a singularity in finite time. However, tese initial data fail to be of finite energy. [7] demonstrates orbital stability about te known explicit armonic maps in te equivariant setting wic are equilibrium solutions to LLG. However, tere is no guarantee tat blow-up occurs, muc less tat it occurs in finite time. Proper numerical treatment of LLG is made difficult by te fact tat te nonlinearity occurs in te igest order derivative and te nonconvexity requirement u = 1. Explicit time integrators of ig order coupled wit occasional updates to ensure u = 1 are te most common strategies in engineering literature but suffer from nonreliable dynamics. On te oter and, implicit strategies to discretize LLG in time often introduce artificial damping wic prevents computed iterates from remaining on te spere, and wic also precludes a (discrete energy law. Recent remedies ave been made, partially addressing te dual requirements of efficiency and reliability: (i projection metods ave been constructed, independently dealing wit te nonconvex algebraic constraint; owever, no (discrete energy principle is available, and convergence to LLG is only known in te case of existing strong solutions to LLG; (ii explicit/implicit discretizations of Ginzburg-Landau penalizations tat involve an additional parameter ε > are used, wic allow for a discrete energy principle, possibly for restricted coices of spatio-temporal discretization parameters. We refer to [9] for a more detailed discussion in tis direction. 2

3 From tis perspective, it is callenging to construct efficient, convergent discretizations of LLG, a prerequisite to te reliable numerical study of singularity formation for weak solutions to LLG. To our knowledge only te work of Pistella and Valente [11] as made an attempt to seek singular solutions numerically, using a stable sceme wit a fourt order regularizing term, wose convergence beavior is not known so far. Teir study is eavily motivated by te work of Cang, Ding and Ye [4] on te blow-up of te armonic map eat flow. Tey specify equivariant data of degree greater tan one wic is known to blow-up for te armonic map eat flow wit fixed boundary data. Introducing a parameter β in front of te gyroscopic term in (1.2, tey fix λ = 1 and steadily increase te value of β and notice tat for β 1 4, blow-up still occurs. However, tey observe tat te singularity disappears for large β wic suggests te regularizing effect of te parameter β. We believe tat teir conclusion tat te gyroscopic term as a damping effect is a statement tat is valid only for te specific initial data tey coose. Tis study is designed to treat te LLG as a perturbation of te armonic map eat flow and ence gives little insigt into te more interesting question concerning te manner in wic te gyroscopic term contributes to blow-up. In tis paper, we report on our numerical findings of singularity formation for LLG in two dimensions, wit particular empasis on te regime of small λ. For tis purpose, we adopt a convergent finite elements plus projection sceme proposed by Alouges and Jaisson [1]. Te finite element formulation is well suited for our study of weak solutions. We improve results by supplying sufficient conditions for involved parameters yielding convergence and propose a modification to increase its efficiency. Tis yields te first practical, stable and convergent numerical sceme wic olds for arbitrarily small λ. For our study on singularity formation, we introduce a class of initial data wic is seen in our experiments to generate finite-time blow-up, even for small values of λ. 2. Approximation sceme and main result In tis section we describe te approximation sceme and state te main result of tis paper Preliminaries. Given a regular triangulation T of te polygonal or polyedral domain Ω R n into triangles or tetraedra for n = 2 or n = 3, respectively, we let := max{diam(k : K T } be te maximal mes-size of T. Te set of nodes in T is denoted by N and te function space S 1 (T W 1,2 (Ω consists of all continuous, T -elementwise affine functions. For eac z N te function ϕ z S 1 (T satisfies ϕ z (z = 1 and ϕ z (y = for all y N \ {z}. Trougout tis paper we set (v; w := Ω v w dx for v, w L2 (Ω. We write H 1 (A instead of H 1 (A; R l for l = 1, Approximation sceme. We follow ideas of Alouges and Jaisson in [1] to derive an approximation sceme for Landau-Lifsitz-Gilbert equation. Testing (1.5 wit a function φ and using (u t u (u φ = t u φ (owing to u = 1, (u t u φ = t u (u φ, and (u u φ = u (u φ we infer (u t u; u φ λ( t u; u φ = (1 + λ 2 ( u; u φ. We replace w = u φ and integrate by parts to verify λ( t u; w ( u t u; w = (1 + λ 2 ( u; w. Te fact tat w u = and u t u = almost everywere in Ω motivates an explicit numerical sceme in wic an approximation v of u t is computed in eac time step. Te updated approximation u + kv of u is ten projected in order to satisfy te constraint u = 1 in an appropriate way. Algoritm (A. Input: a time-step size k >, a positive integer J, a regular triangulation T of Ω, and u ( S1 (T 3 suc tat u ( (z = 1 for all z N. (a Set j :=. 3

4 (b Compute v (j+1 (c Set λ ( v (j+1 ( (j ; w u L (j := {w S 1 (T 3 : w (z u (j (z = for all z N } suc tat v(j+1 u (j+1 := z N and j := j + 1. (d Stop if j = J and go to (b oterwise. ; w = (1 + λ 2 ( u (j ; w u (j u (j (z + k v(j+1 (z (z + k v(j+1 (z ϕ z for all w L (j. Remarks 2.1. (i Te variational formulation in (b can be recast as a(v (j+1 ; w +b(v (j+1 ; w = l(w wit a continuous, elliptic, symmetric bilinear form a on L (j L (j, a continuous, skewsymmetric bilinear form b on L (j L (j, and a continuous linear form l on L (j. Ten, tere exists a unique solution v (j+1 L (j in (b. (ii Suppose tat u (j (z = 1 for some j and all z N. Since u(j (z v(j+1 (z = for all z N tere olds u (j (z + kv(j+1 (z 1 for all z N so tat (c in Algoritm (A is well defined and u (j+1 (z = 1 for all z N Approximation result. Convergence for te above sceme to a solution of te Landau- Lifsitz-Gilbert equation as been proved in [1] if k and subsequently. Here, we present a refined convergence result. Definition 2.1. Given u H 1 (Ω suc tat u = 1 almost everywere in Ω, a function u is called a weak solution of (1.3 if for all T > tere olds (i u H 1 ((, T Ω u(, = u in te sense of traces, (ii u = 1 almost everywere in (, T Ω, (iii for almost all T (, T tere olds λ 1 + λ 2 t u 2 dx dt + 1 u(t 2 dx 1 u 2 dx, (,T Ω 2 Ω 2 Ω and (iv for all φ C (Ω T wit Ω T = (, T Ω, tere olds ( t u φ dx dt + λ u t u φ dx dt = (1 + λ 2 u (u φ dx dt. Ω T Ω T Ω T Teorem A. Given t T Jk suc tat t [jk, (j + 1k] for some j J 1 and x Ω let û,k (t, x:= t jk u (j+1 (j + 1k t k (x + u (j k (x. Let u H 1 (Ω and suppose u ( u in H 1 (Ω for. If T is quasi-uniform and (k, suc tat k 1n/2 ten tere exists a subsequence of ( û,k wic weakly converges in H 1 ((, T Ω to a weak solution of (1.3. We refer to Section 3 for a proof of Teorem A and to Lemma 3.2 and Teorem 3.1 for more precise statements, in particular, a priori bounds wit explicit dependence on te possibly small parameter λ; tracing tis parameter is motivated from [2, Prop. 5.1], were solutions to te Caucy problem (1.1 are constructed as certain limits of weak solutions u λ (λ to (1.5. Trougout Section 3 we use several ideas from [1]. Section 4 discusses a modification of Algoritm (A wic leads to simpler linear systems in (b but still allows for weak subconvergence to a solution. 4

5 3. Proof of Teorem A Trougout tis section we assume tat T is quasi-uniform and make repeated use of te following inverse estimates: Tere exists an -independent constant c > suc tat for all 1 p and φ S 1 (T tere olds (3.1 φ L p (Ω c 1 φ L p (Ω and φ L 4 (Ω c n/4 φ L 2 (Ω. We let I denote te nodal interpolation operator on T. Given a sequence (a (j : j =, 1,.., J we set d t a (j+1 := k 1 (a (j+1 a (j for j =, 1,.., J 1. We abbreviate µ:= 1 + λ 2. Lemma 3.1. For eac j =, 1, 2,.., J let r (j+1 := k ( d t u (j+1 v (j+1. Tere exists an (, k, λ, µ, j- independent constant c 1 > suc tat for all j =, 1, 2,..., J 1 tere olds v (j+1 L 2 (Ω c (µ/λ 1 u (j L 2 (Ω, r (j+1 L 1 (Ω c 1 k 2 v (j+1 2 L 2 (Ω, r (j+1 L 2 (Ω c 1 k 2 n/2 v (j+1 2 L 2 (Ω, d t u (j+1 Proof. Coosing w = v (j+1 λ µ v(j+1 2 L 2 (Ω u(j For all z N tere olds r (j+1 Since u (j (z v(j k2 v (j+1 L 2 (Ω ( 1 + c 1 k n/2 v (j+1 L 2 (Ω (j+1 v L 2 (Ω. in (b of Algoritm (A and using (3.1 yields L 2 (Ω v (j+1 L 2 (Ω c 1 u (j L 2 (Ω v (j+1 L 2 (Ω. (z = u (j+1 (z u (j (z kv(j+1 (z u (j = (z + kv(j+1 (z u (j (z + kv(j+1 (z ( (j u (z + kv(j+1 (z = 1 u (j (z + kv(j+1 (z. (z = we find 1 u (j (z + kv(j+1 (z = 1 + k 2 v (j+1 (z (z 2 and ence r (j+1 (z 1 2 k2 v (j+1 (z 2 for all z N. Since tere exists c > suc tat for all 1 p and all φ S 1 (T tere olds c 1 φ p L p (Ω n z N φ (z p c φ p L p (Ω we verify te second assertion of te lemma and r (j+1 were we used (3.1. We ten verify d t u (j+1 L 2 (Ω v (j+1 L 2 (Ω ck 2 v (j+1 2 L 4 (Ω ck2 n/2 v (j+1 2 L 2 (Ω, L 2 (Ω + k 1 r (j+1 wic finises te proof of te lemma. L 2 (Ω ( 1 + ck n/2 v (j+1 L 2 (Ω 5 (j+1 v L 2 (Ω

6 Lemma 3.2. For all J J tere olds ( 1 Γ1 J 1 λ k d t u (j Γ 2 L 2 (Ω + µ 2 2 u(j j= J 1 λ(1 Γ 1 k j= v (j+1 2 L 2 (Ω 2 L 2 (Ω + µ 2 u(j 2 L 2 (Ω µ 2 u( 2 L 2 (Ω, were Γ 1 := c 2 ( c1 + (1 + C 2 (µ/λk 2 and Γ 2 := c c 1 (µ/λk 1n/2 C for C := u ( L 2 (Ω, and provided tat Γ 1 1 and c c 1 (µ/λk 1n/2 1. Te inductive argument used in te subsequent proof is borrowed from [3]. Proof. We coose w = v (j+1 u (j = u(j+1 λ v (j+1 Te identity implies λ v (j+1 kd t u (j+1 to verify 2 L 2 (Ω = µ( u (j µ ( u (j+1 = µ ( u (j+1 in (b of Algoritm (A and use v (j+1 ; d tu (j+1 ; d t u (j+1 ; d t u (j+1 µk = 2 L 2 (Ω + µ 2 d t u (j+1 2 L 2 (Ω = µ k + µk 1 ( u (j + µk dt u (j+1 2 d tu (j+1 ( u (j ; r(j+1 = d t u (j+1 2 L 2 (Ω + µk1( u (j 2 L 2 (Ω µ 2 d t u (j+1 2 L 2 (Ω ; r(j+1 k 1 r (j+1 ; r(j+1 + µk 2 d tu (j+1 2 L 2 (Ω. Hölder s inequality, (3.1, u (j L (Ω 1, and te second assertion of Lemma 3.1 lead to (3.2 λ v (j+1 2 L 2 (Ω + µ 2 d t u (j+1 2 L 2 (Ω c2 2 µ k r(j+1 L 1 (Ω + c 2 2 µk d t u (j+1 2 L 2 (Ω c 2 c 1 µk 2 v (j+1 2 L 2 (Ω + c2 µk 2 d t u (j+1 2 L 2 (Ω. Suppose tat u (j L 2 (Ω C (wic olds for j = and C = u ( L 2 (Ω. Since c 1 k c 1 (λ/µ1+n/2, te first assertion of Lemma 3.1 yields to (3.3 c 1 k n/2 v (j+1 L 2 (Ω c 1 (λ/µ v(j+1 L 2 (Ω C. A combination of tis bound wit te fourt estimate of Lemma 3.1 and (3.2 sows (3.4 λ v (j+1 2 L 2 (Ω + µ 2 d t u (j+1 2 L 2 (Ω c 2 c 1µk 2 v (j+1 2 L 2 (Ω + c2 µk2( 2 v (j C 2 L 2 (Ω = λγ 1 v (j+1 2 L 2 (Ω. Since Γ 1 1 tis implies u (j+1 L 2 (Ω C. Terefore, (3.4 olds for all j =, 1, 2,..., J 1 and multiplication of (3.4 wit k and summation over j =, 1, 2,..., J 1 prove J 1 λ(1 Γ 1 k j= v (j+1 2 L 2 (Ω + µ 2 u(j We combine te fourt estimate of Lemma 3.1 and (3.3 to verify L 2 (Ω ( 1 + c c 1 (µ/λk 1n/2 C v (j+1 d t u (j+1 A combination of te last two estimates proves te lemma. 6 2 L 2 (Ω µ 2 u( 2 L 2 (Ω. L 2 (Ω = (1 + Γ 2 v (j+1 L 2 (Ω.. and

7 Definition 3.1. For x Ω and t [jk, (j + 1k, j J 1, define û,k (t, x:= t jk k u (j+1 (x + (j + 1k t k u (j (x, u,k (t, x:= u(j (x, v+,k (t, x:= v(j+1 (x, r +,k (t, x:= r(j+1 (x. Lemma 3.3. Suppose tat Γ 1 1/2, assume tat T [, Jk], and define Ω T := (, T Ω. For all w L 2 (, T ; H 1 (Ω; R 3 suc tat w (t, S 1 (T 3 for almost all t (, T and w (t, z u (t, z = for all z N and almost all t (, T tere olds ( λ t û,k w dx dt u,k tû,k w dx dt + µ u,k w dx dt Ω T Ω T Ω T were Λ:= c c 1 C (1 + λ(µ/λk 1n/2. ( C (µ/λ 1/2 T 1/2 Λ w 2 L 2 (Ω dt Proof. Replacing v +,k = tû,k k 1 r +,k in (b of Algoritm (A we find for almost all t (, T tat LHS(t:= λ ( ( t û,k ; w (u,k µ ( tû,k ; w + u 2,k ; w = 1 ( λr + k,k w ( u,k r+,k ; w =: RHS(t. Hölder inequalities, te estimates of Lemma 3.1, and (3.3 prove for almost all t (jk, (j + 1k tat k RHS(t λ r (j+1 L 2 (Ω w L 2 (Ω + u (j L (Ω r (j+1 L 2 (Ω w L 2 (Ω (λ + 1c 1 k 2 n/2 v (j+1 2 L 2 (Ω w L 2 (Ω (λ + 1c c 1 (µ/λc k 2 1n/2 v (j+1 L 2 (Ω w L 2 (Ω. An integration over (, T sows T T ( T 1/2 ( LHS(t dt RHS(t dt Λ v + T,k 2 L 2 (Ω w 2 L 2 (Ω and te bound T v+,k 2 L 2 (Ω dt k J1 j= v(j+1 2 L 2 (Ω (µ/λc2 proof. 1/2 of Lemma 3.2 finises te Teorem 3.1. Suppose tat (k, suc tat k 1n/2 and u ( u in H 1 (Ω. Given any T [, Jk] and Ω T := (, T Ω tere exists u H 1 (, T ; L 2 (Ω L (, T ; H 1 (Ω suc tat (after extraction of a subsequence û,k u in H 1 (Ω T. Tere olds u(t, x = 1 for almost all (t, x Ω T, u(, = u in te sense of traces, (3.5 λ t u 2 dx dt + µ u(x, T 2 dx µ u (x 2 dx 2 2 (,T Ω for almost all T (, T, and for all φ C (Ω T tere olds (3.6 t u φ dx dt + λ (u t u φ dx dt = µ u (u φ dx dt Ω T Ω T Ω T Ω Proof. Lemma 3.2 and te estimate û,k u,k L 2 (Ω k t û,k L 2 (Ω yield te existence of some u H 1 (Ω T suc tat, after extraction of a subsequence, û,k u in H 1 (Ω T, u,k u in L2 (Ω T, û,k, u,k u in L (, T ; H 1 (Ω. 7 Ω

8 Notice tat u,k (z, t = 1 for all z N and almost all t (, T. Hence, for all K T tere olds u,k 2 1 L 2 (K c [ u,k 2 1] L 2 (K = c 2( u,k u,k L 2 (K 2c u,k L 2 (K. Tis implies tat u,k 1 in L2 (Ω T and ence u(x, t = 1 for almost all (x, t Ω T. Continuity of te trace operator and û,k (, u in H 1 (Ω imply u(, = u in Ω in te sense of traces. Passing to te limits in te estimate of Lemma 3.2 we verify (3.5. Given φ C (Ω T let w := u φ and w := I (u,k φ. An application of te triangle inequality sows w w L 2 (Ω I (u,k φ u,k φ L 2 (Ω + u,k φ u φ L 2 (Ω c (u,k φ L 2 (Ω + φ L (Ω u,k u L 2 (Ω and proves tat w w in L 2 (Ω T. Since t û,k t u in L 2 (Ω T we verify tat (3.7 t û,k w dx dt t u w dx dt. Ω T Ω T Te bound u,k 1 almost everywere in Ω T yields ( u,k ( tû,k w dx dt = u,k tû,k (w w dx dt Ω T Ω (3.8 T ( + u,k ( tû,k w dx dt u t u Ω T Ω w dx dt. T Tere olds u,k w dx dt = u,k I (u,k φ dx dt Ω T Ω T = u,k ( I (u,k φ u,k φ dx dt + u,k (u,k φ dx dt. Ω T Ω T Notice tat u,k is T -elementwise affine and φ C (Ω T so tat for eac K T we ave (3.9 ( I (u,k φ u,k φ L 2 (K c D 2 (u,k φ L 2 (K c ( φ W 2, (K u,k L 2 (K + 1 and ence tat ( I (u,k φ u,k φ in L 2 (Ω T. We use u,k ( u,k φ = u (,k u,k φ and u (u φ = u (u φ to verify u,k (u,k φ dx dt = u,k (u,k φ dx dt Ω T Ω T u (u φ dx dt = u (u φ dx dt Ω T Ω T A combination of te previous assertions sows (3.1 u,k w dx dt u w dx dt. Ω T Ω T Since w w in L 2 (Ω T we obtain a uniform bound for T w 2 L 2 (Ω dt. Using (3.7-(3.1 to pass to te limits in te estimate of Lemma 3.3 we verify tat ( λ t u (u φ dx dt u t u (u φ dx dt = µ u (u φ dx dt. Ω T Ω T Ω T 8

9 Tere olds t u (u φ = (u t u φ and (u t u (u φ = t u φ (since u = 1 wic implies (3.6 and finises te proof of te teorem. 4. Increased efficiency troug reduced integration In order to increase te efficiency of our approximation sceme we employ reduced integration, i.e., we use a modified Algoritm (A wic is obtained by replacing (b in Algoritm (A by te following: (b Compute v (j+1 λ ( v (j+1 ; w ( u (j v(j+1 ; w L (j := {w S 1 (T 3 : w (z u (j = (1 + λ2 ( u (j (z = for all z N } suc tat ; w for all w L (j. Here, given any η, ψ C(Ω; R l we set ( η; ψ := Ω I (η ψ dx. Since φ 2 L 2 (Ω (φ ; φ for all φ S 1 (T, Lemma 3.1 and Lemma 3.2 remain uncanged for Algoritm (A. A modified version of Lemma 3.3 olds: Using tat ( t û,k ; w ( t û,k ; w c2 t û,k L 2 (Ω w L 2 (Ω and (u,k tû,k ; w (u,k tû,k ; w = ( t û,k ; u,k w ( t û,k ; u,k w ( t û,k ; u,k w ( t û,k ; I [u,k w ] + ( t û,k ; I [u,k w ] ( t û,k ; I [u,k w ] c t û,k L 2 (Ω I [u,k w ] L 2 (Ω c 2 t û,k L (Ω( 2 w L 2 (Ω + w L (Ω u,k L 2 (Ω we verify wit te bounds of Lemma 3.2 tat ( λ t û,k w dx dt u,k tû,k w dx dt + µ u,k w dx dt Ω T Ω T Ω T ( C (µ/λ 1/2 T 1/2 ( Λ w 2 L 2 (Ω dt + c2 C (1 + λ(µ/λ 1/2 T 1/2 w 2 L 2 (Ω dt + c 2 C 2 (µ/λ 1/2 T 1/2 w L (Ω T for all w, as in Lemma 3.3. Te proof of Teorem 3.1 ten requires bounds for T w 2 L 2 (Ω dt and w L (Ω T wit w = I (u,k φ. Te first bound can be deduced from (3.9 and te second one follows immediately from u,k L (Ω T = Experiments Seeking Blow-Up We report te results of our experiments on singularity formation of (1.3 for Ω = (1/2, 1/2 2. Since we will work in te equivariant setting, a setting tat as been considered in work on singularity formation for te armonic map eat flow problem, e.g., [4, 5], we begin wit some notation. Let (α, r denote domain polar coordinates and (θ, φ sperical coordinates on S 2. A point (θ, φ corresponds to te point (cos θ sin φ, sin θ sin φ, cos φ. An equivariant map is given by (α, r (θ, φ(r, were θ = lα + θ(r, l Z. Our coice of initial data is influenced by strong evidence tat static solutions play a crucial role in singularity formation of LLG, as tey do in te armonic map eat flow. From te formulation 9

10 of LLG given in (1.6, we see tat te static solutions of LLG are exactly tose of te armonic map eat flow, namely, armonic maps wic are solutions to u = u 2 u. Tere are plenty of nontrivial equivariant armonic maps. A family of solutions φ : D S 2 results from te observation tat φ(r = 2 tan 1 r is a solution. By scaling r r/ρ, ρ >, te maps φ ρ (r = 2 tan 1 (r/ρ are also solutions. In te construction of te Struwe solution carried out in [6], as a singular time is approaced, energy concentration occurs and after appropriate rescaling, a armonic map separates. Using te energy bound E(u E(u and te observation tat for u armonic and conformal, tat E(u = 1 u 2 = Area of Image(u, 2 Ω an immediate consequence of te Struwe solution construction in te equivariant setting is tat if E[u ] < 4π, no singularities can form. Recently [7] reports orbital stability of LLG about armonic maps before blow-up. Even toug te blow-up time as not analytically been sown to be finite, our candidate initial data supports tis picture tat near te time of blow-up, a armonic map is approaced. Before specifying te class of initial data, we preemptively remark tat te essential property of our initial data is tat it mimic a armonic map about to be bubbled off, rater tan te property tat it is of degree one. In [4], a necessary and sufficient condition for blow-up is obtained for te equivariant armonic map eat flow wit fixed boundary data; in tat setting, initial data of degree one does not blow-up. Not only are we considering te LLG but we are not fixing boundary data Candidate Initial Data. Te initial data u is prescribed as follows: { 2 tan θ = α, φ(r = 1 ρ(r, r 1/2 π, r 1/2, ρ(r = r A(r A(r is cosen wit te following properties: (i A(1/2 = (ii Te energy density E(u = φ 2 r + r 2 sin φ 2 as a decreasing profile, peaked at r = and at r = 1/2. A function tat satisfies tese conditions is A = (1 2r 4 /s, were increasing s sarpens te concentration of te energy density about te origin. Elementary calculations sow tat u (r = (cos θ sin φ(r, sin θ sin φ(r, cos φ(r is ten given by u (r = (2xA, 2yA, A 2 r 2 /(A 2 + r 2 Notice tat u (r = (,, 1 for r 1/2 so tat for Ω = (1/2, 1/2 2, tis initial data ten wraps around te spere once. 6. Numerical Experiments In tis section we report on te practical performance of Algoritms (A and (A in some numerical experiments and study finite time blow-up of weak solutions. Moreover, we investigate te dependence of numerical approximations upon te parameter λ. Te implementation of te algoritm was performed in MATLAB wit an assemblation of te stiffness matrices in C. Te constraints included in te subspace L (j were directly incorporated in te linear systems wic were solved using te backslas operator in MATLAB. 1

11 Example 6.1. Let Ω:= (1/2, 1/2 2 and let u : Ω R 3 be defined by { (,, 1 for x 1/2, u (x:= ( 2xA, A 2 x 2 / ( A 2 + x 2 for x 1/2, were A := (1 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+1 alved squares wit edge lengt := 2 l. Motivated by Lemma 3.2 we use k = (µ/λ 5/2 /1 (unless oterwise stated, were te additional power 1/2 guarantees tat k 2, k 1n/2 (for n = 2. As discrete initial data we employed te nodal interpolant of u, i.e., we set u ( := I T l u in all experiments. We ran Algoritm (A in Example 6.1 wit s = 1, l = 4, and λ = 1, 1/4. Figure 1 sows snapsots of te numerical solution for λ = 1 at various times. Te plots in Figure 1 display te ortogonal projection of te vector field û,k (t, onto te plane {(x, y, : x, y R}. We observe tat for t.586 te vector û,k (t, canges its direction from (1,, to (1,,. Figure 2 magnifies tis cange of direction for λ = 1. Similar snapsots for λ = 1/4 are sown in Figures 3 and 4. For te smaller λ te vector at te origin canges its direction at a significantly later time Instability of te numerical sceme for k = O( 2 and stabilizing effect of reduced integration. Our first numerical experiment reveals tat te relation k 2 is not sufficient to guarantee stability and convergence of our approximation sceme. We ran Algoritms (A and (A in Example 6.1 wit λ = 1, s = 1 and using te triangulations T l for l = 4, 5. For bot Algoritms we tried te time step sizes k 1 = (µ/λ 5/2 /1 and k 2 = (µ/λ 2 /1. Figure 5 displays te energy E(û,k (t = 1 û,k (t 2 dx 2 as a function of time in te interval (, 1. Te energy is not decreasing for k 2 in Algoritm (A wic indicates instability of Algoritm (A if te time-step size violates te conditions of Lemma 3.2. Te results also sow tat reduced integration stabilizes te sceme as no instability is observable wen Algoritm (A is used wit te large time-step size k 2. We remark tat reduced integration significantly increased te efficiency of our sceme, e.g., in te above experiments te CPU time for Algoritm (A was about 1% of tat of Algoritm (A Beavior for. For fixed λ = 1 and s = 4 we tried l = 4, 5, 6 in Example 6.1. In Figure 6 we displayed te energy E(û,k (t and te W 1, semi-norm û,k (t L (Ω as functions of t for t (, 6/1 for l = 4, 5, 6. For eac l = 4, 5, 6 û,k (t L (Ω assumes te maximum value (among functions v S 1 (T l 3 wit v (z = 1 for all z N. We observe tat for decreasing mes-size te blow-up time (te time at wic û,k (t L (Ω assumes its maximum approaces t Dependence of blow-up time on λ. In order to study te dependence of blow-up beaviour on te parameter λ we ran Algoritm (A in Example 6.1 for fixed l = 5 and s = 1 and for λ = 1/16, 1/4, 1, 4, 16. Te plot in Figure 7 sows tat te blow-up time approaces zero for increasing λ and seems to tend to a time larger tan.8 for decreasing λ Increasing te parameter s. Our final numerical experiment analyses te dependence of (approximate solutions on te parameter s wic defines te initial data u. We approximated solutions in Example 6.1 for λ = 1 and s = 1, 4, 16. Te parameter l = 4 defined all triangulations in tis experiment. Figure 8 displays te energies E(û,k (t and te W 1, seminorms for te solutions obtained for te different parameters s. We observe tat te blow-up time decreases for larger values of s. 11 Ω

12 Figure 1. Numerical approximation û,k (t, in Example 6.1 wit s = 1, l = 4, and λ = 1 for t =,.98,.195,.293,.391,.488,.586,.684, Figure 2. Nodal values û,k (t, z for nodes z N close to te origin in Example 6.1 wit s = 1, l = 4, and λ = 1 for t =.98,.488,

13 Figure 3. Numerical approximation û,k (t, in Example 6.1 wit s = 1, l = 4, and λ = 1/4 for t =,.1,.2,.3,.4,.5,.6,.199, Figure 4. Nodal values û,k (t, z for nodes z N close to te origin in Example 6.1 wit s = 1, l = 4, and λ = 1/4 for t =.3,.6,

14 E[ u,k (t ] (reduced integration, k 2.5, = 1/16 E[ u,k (t ] (reduced integration, k 2, = 1/16 E[ u,k (t ] (reduced integration, k 2.5, = 1/32 E[ u,k (t ] (reduced integration, k 2, = 1/32 E[ u,k (t ] (exact integration, k 2.5, = 1/16 E[ u,k (t ] (exact integration, k 2, = 1/16 E[ u,k (t ] (exact integration, k 2.5, = 1/32 E[ u,k (t ] (exact integration, k 2, = 1/ t Figure 5. Energy for different discretization parameters in Algoritms (A (exact integration and (A (reduced integration E[ u (t ],k (=1/16 u,k (t 1, (=1/16 E[ u,k (t ] (=1/32 u,k (t 1, (=1/32 E[ u,k (t ] (=1/64 u,k (t 1, (=1/ t Figure 6. Energy and W 1, semi-norm for decreasing mes-sizes in Example 6.1 wit λ = 1 and s = 4. 14

15 E[ u (t ],k (λ = 1/16 u (t,k 1, (λ = 1/16 E[ u (t ],k (λ = 1/4 u (t,k 1, (λ = 1/4 E[ u,k (t ] (λ = 1 u (t,k 1, (λ = 1 E[ u,k (t ] (λ = 4 u,k (t 1, (λ = 4 E[ u,k (t ] (λ = 16 u,k (t 1, (λ = t Figure 7. Energy and W 1, semi-norm for different values of λ in Example E[ u (t ],k (s=1 u (t,k 1, (s=1 E[ u (t ],k (s=4 u (t,k 1, (s=4 E[ u (t ],k (s=16 u (t,k 1, (s= t Figure 8. Energy and W 1, semi-norm for various coices of te parameter s in te definition of u in Example

16 Acknowledgment: Part of te work was written wen S.B. visited Forscungsinstitut für Matematik (ETH Züric in January 25 and Brown University in Marc 25. S.B. gratefully acknowledges ospitality by te Department of Matematics of te University of Maryland at College Park. References [1] François Alouges and Pascal Jaisson. Convergence of a finite elements discretization for te Landau-Lifsitz equations. preprint, 23. [2] François Alouges and Alain Soyeur. On global weak solutions for Landau-Lifsitz equations: existence and nonuniqueness. Nonlinear Anal., 18(11: , [3] Jon W. Barrett, Sören Bartels, Xiaobing Feng, and Andreas Prol. A convergent and constraint-preserving finite finite element metod for te p-armonic flow into speres. (manuscript. [4] Kung-Cing Cang, Wei Yue Ding, and Rugang Ye. Finite-time blow-up of te eat flow of armonic maps from surfaces. J. Differential Geom., 36(2:57 515, [5] Jean-Micel Coron. Nonuniqueness for te eat flow of armonic maps. Ann. Inst. H. Poincaré Anal. Non Linéaire, 7(4: , 199. [6] Boling Guo and Min-Cun Hong. Te Landau-Lifsitz equation of te ferromagnetic spin cain and armonic maps. Calc. Var. Partial Differential Equations, 1(1: , [7] Stepen Gustafson, Kyungkeun Kang, and Tai-Peng Tsai. Scrödinger flow near armonic maps. in preparation, 25. [8] Joy Ko. Te construction of a partially regular solution to te landau-lifsitz-gilbert equation in R 2. preprint, 25. [9] Martin Kružík and Andreas Prol. Recent developments in modeling, analysis, and numerics of ferromagnetism. SIAM Review (accepted, also downloadable at: ttp:// [1] Cristof Melcer. Existence of partially regular solutions for Landau-Lifsitz equations in R 3. to appear in Comm. PDE. [11] Francesca Pistella and Vanda Valente. Numerical study of te appearance of singularities in ferromagnets. Adv. Mat. Sci. Appl., 12(2:83 816, 22. [12] Jalal Sata and Congcun Zeng. preprint. 23. [13] Micael Struwe. On te evolution of armonic mappings of Riemannian surfaces. Comment. Mat. Helv., 6(4: , Department of Matematics, Humboldt-Universität zu Berlin, Unter den Linden 6, D-199 Berlin, Germany address: sba@mat.u-berlin.de Department of Matematics, Brown University, Providence, RI 2912, USA. address: joyko@mat.brown.edu Department of Matematics, ETH, CH-892 Züric, Switzerland. address: apr@mat.etz.c 16