A finite element approximation for the stochastic Landau--Lifshitz--Gilbert equation with multi-dimensional noise
aa r X i v : . [ m a t h . NA ] M a r A FINITE ELEMENT APPROXIMATION FOR THE STOCHASTICLANDAU–LIFSHITZ–GILBERT EQUATION WITHMULTI-DIMENSIONAL NOISE
BENIAMIN GOLDYS, JOSEPH GROTOWSKI, AND KIM-NGAN LE
Abstract.
We propose an unconditionally convergent linear finite element scheme forthe stochastic Landau–Lifshitz–Gilbert (LLG) equation with multi-dimensional noise. Byusing the Doss-Sussmann technique, we first transform the stochastic LLG equation intoa partial differential equation that depends on the solution of the auxiliary equation forthe diffusion part. The resulting equation has solutions absolutely continuous with respectto time. We then propose a convergent θ -linear scheme for the numerical solution ofthe reformulated equation. As a consequence, we are able to show the existence of weakmartingale solutions to the stochastic LLG equation. Contents
1. Introduction 12. Definition of a weak solution and the main result 33. The auxiliary equation for the diffusion part 44. Equivalence of weak solutions 135. The finite element scheme 166. The main result 197. Appendix 28Acknowledgements 28References 281.
Introduction
The deterministic Landau-Lifschitz-Gilbert (LLG) equation provides a basis for the the-ory and applications of ferromagnetic materials and fabrication of magnetic memories inparticular, see for example [15, 9, 12, 17]. Let us recall, that in this theory we considera ferromagnetic material filling the domain D and a function u ∈ H , (cid:0) D, S (cid:1) , where S stands for the unit sphere in R , represents a configuration of magnetic moments acrossthe domain D , that is u ( x ) is the magnetisation vector at the point x ∈ D . Accordingto the Landau and Lifschitz theory of ferrormagnetizm [17], modified later by Gilbert [12], Date : January 25, 2018.2000
Mathematics Subject Classification.
Primary 35Q40, 35K55, 35R60, 60H15, 65L60, 65L20, 65C30;Secondary 82D45.
Key words and phrases. stochastic partial differential equation, Landau–Lifshitz–Gilbert equation, finiteelement, ferromagnetism. the time evolution of magnetic moments M ( t, x ) is described, in the simplest case, by theLandau-Lifschitz-Gilbert (LLG) equation(1.1) ∂ M ∂t = λ M × ∆ M − λ M × ( M × ∆ M ) in (0 , T ) × D, ∂ M ∂ n = 0 in (0 , T ) × ∂D, M (0 , · ) = M ( · ) in D, where λ = 0 and λ > n stands for the outward normal vector on ∂D ; see e.g. [9]. We assume that M ∈ H , (cid:0) D, S (cid:1) , and then one can show that(1.2) | M ( t, x ) | = 1 , t ∈ [0 , T ] , x ∈ D In this paper we are concerned with a stochastic version of the LLG equation. Randomlyfluctuating fields were originally introduced in physics by N´eel in [ ? ] as formal quantitiesresponsible for magnetization fluctuations. The necessity of being able to describe deviationsfrom the average magnetization trajectory in an ensemble of noninteracting nanoparticleswas later emphasised by Brown in [6, 7]. According to a non-rigorous arguments of Brownthe magnetisation M evolves randomly according to a stochastic version of (1.1) that takesthe form, (see [8] for more details about the physical background and derivation of thisequation) d M = (cid:0) λ M × ∆ M − λ M × ( M × ∆ M ) (cid:1) dt + P qi =1 ( M × g i ) ◦ dW i ( t ) ,∂ M ∂ n = 0 on (0 , T ) × ∂D, M (0 , · ) = M in D, (1.3)where g i ∈ W , ∞ ( D ), i = 1 , · · · , q , satisfy the homogeneous Neumann boundary conditionsand ( W i ) qi =1 is a q -dimensional Wiener process. In view of the property (1.2) for thedeterministic system, we require that M also satisfies (1.2). To this end we are forced touse the Stratonovich differential ◦ dW i ( t ) in equation (1.3). Mathematical theory of equation(1.3) has been initiated only recently, in [8], where the existence of weak martingale solutionsto (1.3) was proved for the case q = 1 using the Galerkin-Faedo approximations. Let us note,that usually the Galerkin-Faedo approximations do not provided a useful computational toolfor solving an equation.The aim of this paper is two-fold. We will prove the existence of solutions to the stochasticLLG equation (1.3) and at the same time will provide an efficient and flexible algorithmfor solving numerically this equation. To this end we will use the finite element methodand a new transformation of the Stratonovich type equation (1.3) to a deterministic PDE(4.2) with coefficients determined by a stochastic ODE (3.6) that can be solved separately.The deterministic PDE we obtain, has solutions absolutely continuous with respect to time,hence convenient for the construction of a convergent finite elemetn scheme. Our approachis based on the Doss-Sussmann technique [11, 18]. This transformation was introduced in[13] to study the stochastic LLG equation with a single Wiener process ( q = 1), in which EM FOR STOCHASTIC LANDAU–LIFSHITZ–GILBERT EQUATION 3 case the auxiliary ODE is deterministic. Since the vector fields u × g i are non-commuting,the case of q > q >
1. We note here that under appropriate assumptions even the caseof infinite-dimensional noise ( q = ∞ ) can be handled in exactly the same way.Let us recall that the first convergent finite element scheme for the stochastic LLG equa-tion was studied in [5] and is based on a Crank–Nicolson type time-marching evolution,relying on a nonlinear iteration solved by a fixed point method. On the other hand, therehas been an intensive development of a new class of numerical methods for the LLG equa-tion (1.1) based on a linear iterations, yielding unconditional convergence and stability [1, 3].The ideas developed there are extended and generalized in [13, 2] in order to take into ac-count the stochastic term. A fully linear discrete scheme for (1.3) is studied in [13] butwith one-dimensional noise. The method is based on the so–called Doss-Sussmann tech-nique [11, 18], which allows one to replace the stochastic partial differential equation (PDE)by an equivalent PDE with random coefficients. In contrast, [2] considers, for a more generalnoise, a projection scheme applied directly to the original stochastic equation (1.3). How-ever, this approach requires a quite specific and complicated treatment of the stochasticterm. In this paper, we propose a convergent θ -linear scheme for the numerical solution ofthe tranformed equation and prove unconditional stability and convergence for the schemewhen θ > /
2. To the best of our knowledge this is a new result for this problem.The paper is organised as follows. In Section 2 we define the notion of weak martin-gale solutions to (1.3) and state our main result. In Section 3, we introduce an auxiliarystochastic ODE and prove some properties of solution necessary for the transformation ofequation (1.3) to a deterministic PDE with random coefficients. Details of this transforma-tion are presented in Section 4. We also show in this section how a weak solution to (1.3)can be obtained from a weak solution of the reformulated form. In Section 5 we introduceour finite element scheme and present a proof for the stability of approximate solutions.Section 6 is devoted to the proof of the main theorem, namely the convergence of finiteelement solutions to a weak solution of the reformulated equation. Finally, in the Appendixwe collect, for the reader’s convenience, a number of facts that are used in the course of theproof.Throughout this paper, c denotes a generic constant that may take different values atdifferent occurrences. In what follows we will also use the notation D T = (0 , T ) × D .2. Definition of a weak solution and the main result
In this section we state the definition of a weak solution to (1.3) and present our mainresult. Before doing so, we introduce some suitable Sobolev spaces, and fix some notation.The standing assumption for the rest of the paper is that D is a bounded open domain in R with a smooth boundary.For any U ⊂ R d , d ≥
1, we denote by L ( U ) the space of Lebesgue square-integrablefunctions defined on U and taking values in R . The function space H ( U ) is defined as: H ( U ) = (cid:26) u ∈ L ( U ) : ∂ u ∂x i ∈ L ( U ) for i ≤ d. (cid:27) . BENIAMIN GOLDYS, JOSEPH GROTOWSKI, AND KIM-NGAN LE
Remark 2.1.
For u , v ∈ H ( D ) we denote u × ∇ v := (cid:18) u × ∂ v ∂x , u × ∂ v ∂x , u × ∂ v ∂x (cid:19) ∇ u × ∇ v := X i =1 ∂ u ∂x i × ∂ v ∂x i h w × ∇ v , ∇ u i L ( D ) := X i =1 (cid:28) w × ∂ v ∂x i , ∂ u ∂x i (cid:29) L ( D ) ∀ w ∈ L ∞ ( D ) . Definition 2.2.
Given T ∈ (0 , ∞ ) and a family of functions { g i : i = 1 , . . . , q } ⊂ L ∞ ( D ) ,a weak martingale solution (Ω , F , ( F t ) t ∈ [0 ,T ] , P , W, M ) to (1.3) , for the time interval [0 , T ] ,consists of (a) a filtered probability space (Ω , F , ( F t ) t ∈ [0 ,T ] , P ) with the filtration satisfying the usualconditions, (b) a q -dimensional ( F t ) -adapted Wiener process W = ( W t ) t ∈ [0 ,T ] , (c) a progressively measurable process M : [0 , T ] × Ω → L ( D ) such that (1) M ( · , ω ) ∈ C ([0 , T ]; H − ( D )) for P -a.e. ω ∈ Ω ; (2) E (cid:16) ess sup t ∈ [0 ,T ] k∇ M ( t ) k L ( D ) (cid:17) < ∞ ; (3) | M ( t, x ) | = 1 for each t ∈ [0 , T ] , a.e. x ∈ D , and P -a.s.; (4) for every t ∈ [0 , T ] , for all ψ ∈ C ∞ ( D ) , P -a.s.: h M ( t ) , ψ i L ( D ) − h M , ψ i L ( D ) = − λ Z t h M × ∇ M , ∇ ψ i L ( D ) ds − λ Z t h M × ∇ M , ∇ ( M × ψ ) i L ( D ) ds + q X i =1 Z t h M × g i , ψ i L ( D ) ◦ dW i ( s ) . (2.1)As the main result of this paper, we will establish a finite element scheme defined viaa sequence of functions which are piecewise linear in both the space and time variables.We also prove that this sequence contains a subsequence converging to a weak martingalesolution in the sense of Definition 2.2. A precise statement will be given in Theorem 6.9.3. The auxiliary equation for the diffusion part
In this section we introduce the auxiliary equation (3.12) that will be used in the nextsection to define a new variable from M , and establish some properties of its solution.Let g , . . . , g q ∈ C (cid:0) D, R (cid:1) , be fixed. For i = 1 , . . . , q , and x ∈ D we define linear operators G i ( x ) : R → R by G i ( x ) u = u × g i ( x ). In what follows we suppress the argument x . Itis easy to check that G ⋆i = − G i , (3.1) and (cid:0) G i (cid:1) ⋆ = G i . (3.2) EM FOR STOCHASTIC LANDAU–LIFSHITZ–GILBERT EQUATION 5
We will consider a stochastic Stratonovitch equation on the algebra L (cid:0) R (cid:1) of linear oper-ators in R :(3.3) Z t = I + q X i =1 Z t G i Z s ◦ dW i ( s ) , t ≥ . Lemma 3.1.
Let g , . . . , g q ∈ C (cid:0) D, R (cid:1) . Then the following holds.(a) For every x ∈ D equation (3.3) has a unique strong solution, which has a t -continuousversion in L (cid:0) R (cid:1) .(b) For every t ≥ and x ∈ D (3.4) | Z t u | = | u | P − a.s for every u ∈ R . In particular, for every t ≥ the operator Z t is invertible and Z − t = Z ⋆t .(c) If moreover g , . . . , g q ∈ C α (cid:0) D, R (cid:1) for a certain α ∈ (0 , then the mapping ( t, x ) → Z t ( x ) has a continuous version in L (cid:0) R (cid:1) .Proof. Equation (3.3) can be equivalently written as an Itˆo equation(3.5) Z t = I + 12 q X i =1 Z t G i Z s ds + q X i =1 Z t G i Z s dW i ( s ) , t ≥ . Since the coefficients of equation (3.5) are Lipschitz, the existence and uniqueness of strongsolutions to equation (3.5), and the existence of its continuous version is standard, see forexample Theorem 18.3 in [16]. Hence, the same result holds for (3.3).To prove (b) we fix x ∈ D , t ≥ u ∈ R and put Z u = Z u . Then equation (3.5) yields(3.6) Z u t = u + 12 q X i =1 t Z G i Z u s ds + q X i =1 t Z G i Z u s dW i ( s ) . Applying the Itˆo formula to the process | Z u t | and invoking (3.1) we obtain d | Z u t | = 2 h Z u t , dZ u t i + q X i =1 | G i Z u t | dt = q X i =1 (cid:10) Z u t , G i Z u t (cid:11) dt + 2 q X i =1 h Z u t , G i Z u t i dW i ( t ) + q X i =1 | G i Z u t | dt = 0 , or equivalently | Z u t | = | u | , for all t ≥ , P − a.s.. To prove (c), we begin by letting 0 ≤ s < t ≤ T and x, y ∈ D . For any p ≥ E | Z t ( y ) − Z s ( x ) | p ≤ p − E | Z t ( y ) − Z s ( y ) | p + 2 p − E | Z s ( y ) − Z s ( x ) | p . It is well known that there exists C > E | Z t ( y ) − Z s ( y ) | p ≤ C | t − s | p . If there exists α ∈ (0 ,
1] such that | g i ( x ) − g i ( y ) | ≤ c i | x − y | α , x, y ∈ D, i = 1 , . . . , q
BENIAMIN GOLDYS, JOSEPH GROTOWSKI, AND KIM-NGAN LE then for a certain
C >
0, for any h ∈ R there holds(3.9) | G i ( x ) h | ≤ C | h | , | ( G i ( x ) − G i ( y )) h | ≤ C | x − y | α | h | , (cid:12)(cid:12)(cid:0) G i ( x ) − G i ( y ) (cid:1) h (cid:12)(cid:12) ≤ C | x − y | α | h | . Then Z s ( y ) − Z s ( x ) = 12 q X i =1 Z s G i ( y ) Z r ( y ) dr + q X i =1 Z s G i ( y ) Z r ( y ) dW i ( r ) − q X i =1 Z s G i ( x ) Z r ( x ) dr − q X i =1 Z s G i ( x ) Z r ( x ) dW i ( r )= 12 q X i =1 Z s (cid:0) G i ( y ) − G i ( x ) (cid:1) Z r ( y ) dr + 12 q X i =1 Z s G i ( x ) ( Z r ( y ) − Z r ( x )) dr + q X i =1 Z s ( G i ( y ) − G i ( x )) Z r ( y ) dW i ( r ) + q X i =1 Z s G i ( x ) ( Z r ( y ) − Z r ( x )) dW i ( r ) . Using (3.9) we obtain E | Z s ( y ) − Z s ( x ) | p ≤ e C | x − y | αp + e C Z s E | Z r ( y ) − Z r ( x ) | p dr. Therefore, invoking the Gronwall Lemma we obtain(3.10) E | Z s ( y ) − Z s ( x ) | p ≤ e Ce e CT | x − y | αp . Combining (3.7), (3.8) and (3.10) we obtain(3.11) E | Z t ( y ) − Z s ( x ) | p ≤ c | t − s | p + c | x − y | αp . Let β > r = d + 1 + β . Let p be chosen in such a way that p ≥ r and pα ≥ r . The set [0 , T ] × D can be covered by a finite number of open sets B k with the property | t − s | r + | x − y | r < B k . In each B k , (3.11) then yields E | Z t ( y ) − Z s ( x ) | p ≤ c ( | t − s | r + | x − y | r ) , and the result then follows by the Kolmogorv-Chentsov theorem, see p. 57 of [16]. (cid:3) Lemma 3.2.
Assume that g i ∈ C αb ( D, R ) . Then the following holds.(a) For every t ≥ we have Z t ∈ C b ( D, L ( R )) P -a.s.(b) For every x ∈ D the process ξ t ( x ) = ∇ Z t ( x ) is the unique solution of the linear Itˆoequation dξ t ( x ) = 12 q X i =1 (cid:0) G i ξ t ( x ) + H i Z t ( x ) (cid:1) dt + q X i =1 (cid:0) G i ξ t ( x ) + I i Z t ( x ) (cid:1) dW i ( t ) , with ξ ( x ) = 0 and the operators H i , I i ∈ L (cid:0) R (cid:1) defined as I i u = u × ∇ g i and H i u = G i u × ∇ g i + G i ( u × ∇ g i ) . EM FOR STOCHASTIC LANDAU–LIFSHITZ–GILBERT EQUATION 7 (c) For every γ < min (cid:0) α, (cid:1) the mapping ( t, x ) → ∇ Z t ( x ) is γ -H¨older continuous.(d) We have E sup t ≤ T sup x ∈ D |∇ Z t | < ∞ . Proof. (a) Let E denote the Banach space of continuous and adapted processes Z takingvalues in the space of linear operators L (cid:0) R (cid:1) and endowed with the norm k Z k E = E sup t ≤ T | Z t | ! / . For every x ∈ D we define a mapping K : D × E → D × E, K ( x, Z )( t ) = I + q X i =1 Z t G i ( x ) Z s ◦ dW i ( s ) . It is easy to check that the assumptions of Lemma 9.2, p. 238 in [10] are satisfied andtherefore (a) holds.(b) The proof is completely analogous to the proof of Theorem 9.8 in [10], and is henceomitted.(c) The proof is analogous to the proof of part (c) of Lemma 3.1.(d) The estimate follows easily from (c). (cid:3)
For every u ∈ L ( D ) we will consider the L ( D )-valued process Z t ( u ) defined by[ Z t ( u )]( x ) = Z t ( x ) u ( x ) x − a.e. Clearly,(3.12) Z t ( u ) = u + q X i =1 Z t Z s ( u ) × g i ◦ dW i ( s ) , t ≥ , where the equality holds in L ( D ). The process Z t is now an operator-valued process takingvalues L (cid:0) L ( D ) (cid:1) and it will still be denoted by Z t . The next lemmas follow immediatelyfrom the properties of the matrix-valued process considered above. Lemma 3.3.
Assume that { g i : i = 1 , . . . , q } ⊂ C αb ( D ) . Then for every u ∈ L ( D ) thestochastic differential equation (3.12) has a unique strong continuous solution in L ( D ) .Moreover, there exists Ω ⊂ Ω such that P (Ω ) = 1 and for every ω ∈ Ω the followingholds.(a) For all t ≥ and every u ∈ L ( D ) , | Z t ( ω, u ) | = | u | . (b) For every t ≥ the mapping u → Z t ( ω, u ) defines a linear bounded operator Z t ( ω ) on L ( D ) . In particular, (3.13) Z t ( ω, u + v ) = Z t ( ω, u ) + Z t ( ω, v ) . Moreover, for every
T > there exists a constant C T > such that (3.14) E sup t ≤ T | Z t ( u ) | L ( D ) ≤ C T | u | L ( D ) . BENIAMIN GOLDYS, JOSEPH GROTOWSKI, AND KIM-NGAN LE (c) For every t ≥ the operator Z t ( ω ) is invertible and the inverse operator is the uniquesolution of the stochastic differential equation on L ( D ) : (3.15) Z − t ( u ) = u − q X i =1 Z t Z − s G i ( u ) ◦ dW i ( s ) , u ∈ L ( D ) . Finally, (3.16) Z − t ( ω ) = Z ⋆t ( ω ) . Lemma 3.4.
Assume that g i ∈ C αb (cid:0) D, R (cid:1) for i = 1 , · · · , q . Then, for every u ∈ H ( D ) Z t ( u ) ∈ H P -a.s. Furthermore, the process ξ t ( u ) := ∇ Z t ( u ) , is the unique solution of thelinear equation dξ t ( u ) = 12 q X i =1 (cid:0) G i ξ t ( u ) + H i Z t ( u ) (cid:1) dt + q X i =1 (cid:0) G i ξ t ( u ) + I i Z t ( u ) (cid:1) dW i ( t ) , with ξ ( u ) = ∇ u . Lemma 3.5.
For any u , v ∈ L ( D ) , there holds for all t ≥ and P -a.s.: Z t ( u × v ) = Z t ( u ) × Z t ( v ) , (3.17) Proof.
Let Z u t := Z t ( u ) and Z v t := Z t ( v ) for all t ≥
0. We now prove (3.17); the prop-erty (3.13) can be obtained in the same manner. Using the Itˆo formula for Z u t × Z v t and (3.6),we obtain d ( Z u t × Z v t ) = dZ u t × Z v t + Z u t × dZ v t + q X i =1 ( Z u t × g i ) × ( Z v t × g i ) dt = q X i =1 (cid:0) Z u t × ( Z v t × g i ) − Z v t × ( Z u t × g i ) (cid:1) dW i ( t )+ 12 q X i =1 (cid:0) Z u t × (cid:0) ( Z v t × g i ) × g i (cid:1) − Z v t × (cid:0) Z u t × g i ) × g i (cid:1)(cid:1) dt (3.18) + q X i =1 ( Z u t × g i ) × ( Z v t × g i ) dt. Using an elementary identity(3.19) a × ( b × c ) = h a , c i b − h a , b i c , a , b , c ∈ R , we find that(3.20) Z u t × ( Z v t × g i ) − Z v t × ( Z u t × g i ) = ( Z u t × Z v t ) × g i and Z u t × (cid:0) ( Z v t × g i ) × g i (cid:1) − Z v t × (cid:0) Z u t × g i ) × g i (cid:1) = h Z u t , g i i ( Z v t × g i ) − h Z v t , g i i ( Z u t × g i ) − Z u t × g i ) × ( Z v t × g i )= (cid:0) ( Z u t × Z v t ) × g i (cid:1) × g i − Z u t × g i ) × ( Z v t × g i ) . (3.21) EM FOR STOCHASTIC LANDAU–LIFSHITZ–GILBERT EQUATION 9
Invoking (3.20) and (3.21), equation (3.18) we obtain d ( Z u t × Z v t ) = 12 q X i =1 (cid:0) ( Z u t × Z v t ) × g i (cid:1) × g i dt + q X i =1 (cid:0) ( Z u t × Z v t ) × g i (cid:1) dW i ( t ) . Therefore, the process V t := Z u t × Z v t is a solution of the following stochastic differentialequation: ( dV t = P qi =1 (cid:0) V t × g i (cid:1) × g i dt + P qi =1 (cid:0) V t × g i (cid:1) dW i ( t ) V = u × v . On the other hand, it follows from (3.6) that the process Z t ( u × v ) satisfies the sameequation. Hence, (3.17) follows from the uniqueness of solutions to (3.18). (cid:3) Lemma 3.6.
For any u , v ∈ H ( D ) , there holds for all t ≥ and P -a.s.: h∇ Z t ( u ) , ∇ Z t ( v ) i L ( D ) = h∇ u , ∇ v i L ( D ) + F ( t, u , v ) , (3.22) with F ( t, u , v ) := q X i =1 Z t F ,i ( s, u , v ) ds + q X i =1 Z t F ,i ( s, u , v ) dW i ( s ) where F ,i ( t, u , v ) := (cid:10) ∇ Z t ( u ) , ( H i − G i I i ) Z t ( v ) (cid:11) L ( D ) − (cid:10) ∇ ( H i − G i I i ) Z t ( u ) , Z t ( v ) (cid:11) L ( D ) + h I i Z t ( u ) , I i Z t ( v ) i L ( D ) ; and F ,i ( t, u , v ) := h∇ Z t ( u ) , I i Z t ( v ) i L ( D ) − h∇ ( I i Z t ( u )) , Z t ( v ) i L ( D ) Proof.
Let ξ u t := ξ t ( u ) and ξ v t := ξ t ( v ) for all t ≥
0. In addition, we consider a C ∞ function φ : (cid:0) L ( D ) (cid:1) → R defined by φ ( x , y ) = h x , y i L ( D ) . By using the Itˆo Lemma we obtain d h ξ u t , ξ v t i L ( D ) = h dξ u t , ξ v t i L ( D ) + h ξ u t , dξ v t i L ( D ) + h dξ u t , dξ v t i L ( D ) = q X i =1 (cid:18) (cid:10) G i ξ u t + H i Z u t , ξ v t (cid:11) L ( D ) + (cid:10) ξ u t , G i ξ v t + H i Z v t (cid:11) L ( D ) + h G i ξ u t + I i Z u t , G i ξ v t + I i Z v t i L ( D ) (cid:19) dt + q X i =1 (cid:18) h G i ξ u t + I i Z u t , ξ v t i L ( D ) + h ξ u t , G i ξ v t + I i Z v t i L ( D ) (cid:19) dW i ( t ) . (3.23) Using (3.1) and (3.2), we deduce from (3.23) that d h ξ u t , ξ v t i L ( D ) = q X i =1 (cid:18) h H i Z u t , ξ v t i L ( D ) + h ξ u t , H i Z v t i L ( D ) + h G i ξ u t , I i Z v t i L ( D ) + h I i Z u t , G i ξ v t i L ( D ) + h I i Z u t , I i Z v t i L ( D ) (cid:19) dt + q X i =1 (cid:18) h I i Z u t , ξ v t i L ( D ) + h ξ u t , I i Z v t i L ( D ) (cid:19) dW i ( t )= q X i =1 (cid:18) (cid:10) ( H i − G i I i ) Z u t , ξ v t (cid:11) L ( D ) + (cid:10) ξ u t , ( H i − G i I i ) Z v t (cid:11) L ( D ) + h I i Z u t , I i Z v t i L ( D ) (cid:19) dt + q X i =1 (cid:18) h ξ u t , I i Z v t i L ( D ) + h I i Z u t , ξ v t i L ( D ) (cid:19) dW i ( t ) . Integrating by parts for the first and the last term in the right hand side of the aboveequation and noting the homogeneous Neumann boundary condition of g i , we obtain d h ξ u t , ξ v t i L ( D ) = q X i =1 (cid:18) − (cid:10) ∇ ( H i − G i I i ) Z u t , Z v t (cid:11) L ( D ) + (cid:10) ξ u t , ( H i − G i I i ) Z v t (cid:11) L ( D ) + h I i Z u t , I i Z v t i L ( D ) (cid:19) dt + q X i =1 (cid:18) h ξ u t , I i Z v t i L ( D ) − h∇ ( I i Z u t ) , Z v t i L ( D ) (cid:19) dW i ( t ) . (3.24)Hence, the resutl follows from replacing t by s and intergrating (3.24) over [0 , t ]. (cid:3) Remark 3.7.
By using integration by parts and the homogeneous Neumann boundary con-ditions of g i for i = 1 , · · · , q we obtain some symmetry properties of functions F ,i , F ,i and F : for any u , v , v , v ∈ H ( D ) , F ,i ( t, u , v ) = F ,i ( t, v , u ); F ,i ( t, u , v ) = F ,i ( t, v , u ); and hence, F ( t, u , v ) = F ( t, v , u ) . Furthermore, it follows from (3.13) that F ( t, u , v + v ) = F ( t, u , v ) + F ( t, u , v ) . The following lemmas state some important properties of F used throughout this paper. Lemma 3.8.
Assume that g i ∈ W , ∞ ( D ) for i = 1 , · · · , q . Then for any u , v ∈ L (Ω; H ( D )) there exists a constant c depending on T and { g i } i =1 , ··· ,q such that E sup t ∈ [0 ,T ] k∇ Z t ( u ) k L ( D ) ≤ c E k u k H ( D ) , (3.25) and for any ǫ > , E sup s ∈ [0 ,t ] (cid:12)(cid:12) F ( s, u , v ) (cid:12)(cid:12) ≤ cǫ E k u k H ( D ) + cǫ − E k v k L ( D ) . (3.26) EM FOR STOCHASTIC LANDAU–LIFSHITZ–GILBERT EQUATION 11
Proof.
It follows from (3.22) that E k∇ Z t ( u ) k L ( D ) = E k∇ u k L ( D ) + E [ F ( t, u , u )](3.27) ≤ E k∇ u k L ( D ) + q X i =1 E Z t (cid:12)(cid:12) F ,i ( τ, u , u ) (cid:12)(cid:12) dτ + E (cid:12)(cid:12) q X i =1 Z t F ,i ( τ, u , u ) dW i ( τ ) (cid:12)(cid:12) . (3.28)For convenience, we next estimate | F ( τ, u , v ) | , which is a slightly more general versionof | F ( τ, u , u ) | . By using the elementary inequality(3.29) ab ≤ ǫa + ǫ − b , the assumption g i ∈ W , ∞ ( D ) and (3.3), there holds (cid:12)(cid:12) F ,i ( τ, u , v ) (cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:10) ∇ Z τ ( u ) , ( H i − G i I i ) Z τ ( v ) (cid:11) L ( D ) (cid:12)(cid:12) + (cid:12)(cid:12)(cid:10) ∇ ( H i − G i I i ) Z τ ( u ) , Z τ ( v ) (cid:11) L ( D ) (cid:12)(cid:12) + (cid:12)(cid:12) h I i Z τ ( u ) , I i Z τ ( v ) i L ( D ) (cid:12)(cid:12) ≤ c (cid:0) ǫ k∇ Z τ ( u ) k L ( D ) + ǫ k Z τ ( u ) k L ( D ) + ǫ − k Z τ ( v ) k L ( D ) (cid:1) ≤ c (cid:0) ǫ k∇ Z τ ( u ) k L ( D ) + ǫ k u k L ( D ) + ǫ − k v k L ( D ) (cid:1) . This implies that E Z t (cid:12)(cid:12) F ,i ( τ, u , v ) (cid:12)(cid:12) dτ ≤ cǫt E k u k L ( D ) + cǫ − t E k v k L ( D ) + cǫ E Z t k∇ Z τ ( u ) k L ( D ) dτ. (3.30)Then, by using the Burkholder-Davis-Gundy inequality, H¨older inequality, (3.3) and (3.29),we estimate E sup s ∈ [0 ,t ] (cid:12)(cid:12) q X i =1 Z s F ,i ( τ, u , v ) dW i ( τ ) (cid:12)(cid:12) ≤ c E (cid:12)(cid:12) q X i =1 Z t (cid:0) F ,i ( τ, u , v ) (cid:1) dτ (cid:12)(cid:12) / ≤ c E (cid:12)(cid:12)Z t (cid:0) k∇ Z τ ( u ) k L ( D ) k Z τ ( v ) k L ( D ) + k Z τ ( u ) k L ( D ) k Z τ ( v ) k L ( D ) (cid:1) dτ (cid:12)(cid:12) / ≤ c E (cid:12)(cid:12)Z t (cid:0) k∇ Z τ ( u ) k L ( D ) k v k L ( D ) + k u k L ( D ) k v k L ( D ) (cid:1) dτ (cid:12)(cid:12) / ≤ c E (cid:20) k v k L ( D ) (cid:0)Z t k∇ Z τ ( u ) k L ( D ) dτ (cid:1) / (cid:21) + ct / E (cid:2) k u k L ( D ) k v k L ( D ) (cid:3) (3.31) ≤ cǫt / E k u k L ( D ) + cǫ − ( t / + 1) E k v k L ( D ) + cǫ E Z t k∇ Z τ ( u ) k L ( D ) dτ. (3.32)We use (3.30) and (3.32) with v = u and ǫ = 1 together with (3.27) to deduce E k∇ Z t ( u ) k L ( D ) ≤ c E k u k H ( D ) + c E Z t k∇ Z τ ( u ) k L ( D ) dτ. Hence, the result (3.25) follows immediately by using Gronwall’s inequality.
To prove (3.26) we note that E sup s ∈ [0 ,t ] (cid:12)(cid:12) F ( s, u , v ) (cid:12)(cid:12) ≤ q X i =1 E Z t (cid:12)(cid:12) F ,i ( τ, u , v ) (cid:12)(cid:12) dτ + E sup s ∈ [0 ,t ] (cid:12)(cid:12) q X i =1 Z s F ,i ( τ, u , v ) dW i ( τ ) (cid:12)(cid:12) . (3.33)Hence, it follows from (3.30), (3.32) and (3.25) that E sup s ∈ [0 ,t ] (cid:12)(cid:12) q X i =1 Z s F ,i ( τ, u , v ) dW i ( τ ) (cid:12)(cid:12) ≤ cǫ E k u k L ( D ) + cǫ − E k v k L ( D ) + cǫ E Z t k∇ Z τ ( u ) k L ( D ) dτ ≤ c (cid:0) ǫ E k u k H ( D ) + ǫ − E k v k L ( D ) (cid:1) , which completed the proof of the lemma. (cid:3) Lemma 3.9.
For any u ∈ L (cid:0) Ω; L ( D ) (cid:1) , v ∈ L (cid:0) Ω; H ( D ) (cid:1) and ≤ s ≤ T , there exists aconstant c depending on { g i } i =1 , ··· ,q such that E | F ( s, u , v ) | ≤ cs (cid:0) E [ k u k L ( D ) ] (cid:1) / (cid:0) E [ k∇ v k L ( D ) ] (cid:1) / + c ( s / + s ) (cid:0) E [ k u k L ( D ) ] (cid:1) / (cid:0) E [ k v k L ( D ) ] (cid:1) / . Proof.
From the definition of function F in Lemma 3.6 and the triangle inequality, thereholds E | F ( s, u , v ) | ≤ q X i =1 E Z s (cid:12)(cid:12) F ,i ( τ, u , v ) (cid:12)(cid:12) dτ + E (cid:12)(cid:12) q X i =1 Z s F ,i ( τ, u , v ) dW i ( τ ) (cid:12)(cid:12) . (3.34)From Remark 3.7, we note that F ,i ( τ, u , v ) = F ,i ( τ, v , u ) , and therefore, by using (3.31), the last term of (3.34) can be estimated as follows: E (cid:12)(cid:12) q X i =1 Z s F ,i ( τ, u , v ) dW i ( τ ) (cid:12)(cid:12) = E (cid:12)(cid:12) q X i =1 Z s F ,i ( τ, v , u ) dW i ( τ ) (cid:12)(cid:12) ≤ c E (cid:20) k u k L ( D ) (cid:0)Z s k∇ Z τ ( v ) k L ( D ) dτ (cid:1) / (cid:21) + cs / E (cid:2) k u k L ( D ) k v k L ( D ) (cid:3) . (3.35)We now estimate (cid:12)(cid:12) F ,i ( τ, u , v ) (cid:12)(cid:12) by integrating by parts and then using H¨older’s inequality,the assumption g i ∈ W , ∞ ( D ) and (3.3) as follows: (cid:12)(cid:12) F ,i ( τ, u , v ) (cid:12)(cid:12) = (cid:12)(cid:12) − (cid:10) Z τ ( u ) , ∇ (cid:0) ( H i − G i I i ) Z τ ( v ) (cid:1)(cid:11) L ( D ) + (cid:10) ( H i − G i I i ) Z τ ( u ) , ∇ Z τ ( v ) (cid:11) L ( D ) + h I i Z τ ( u ) , I i Z τ ( v ) i L ( D ) (cid:12)(cid:12) ≤ c k u k L ( D ) (cid:0) k∇ Z τ v k L ( D ) + k v k L ( D ) (cid:1) , EM FOR STOCHASTIC LANDAU–LIFSHITZ–GILBERT EQUATION 13 and therefore, E Z s (cid:12)(cid:12) F ,i ( τ, u , v ) (cid:12)(cid:12) dτ ≤ c E (cid:20) k u k L ( D ) (cid:0)Z s k∇ Z τ v k L ( D ) dτ (cid:1)(cid:21) + cs E (cid:2) k u k L ( D ) k v k L ( D ) (cid:3) . (3.36)Hence, by using H¨older inequality we obtain from (3.34)–(3.36) that there holds: E | F ( s, u , v ) | ≤ c E (cid:20) k u k L ( D ) (cid:0)Z s k∇ Z τ v k L ( D ) dτ (cid:1)(cid:21) + c ( s / + s ) E (cid:2) k u k L ( D ) k v k L ( D ) (cid:3) ≤ c (cid:0) E [ k u k L ( D ) ] (cid:1) / (cid:0) E (cid:2) ( Z s k∇ Z τ v k L ( D ) dτ ) (cid:3)(cid:1) / + c ( s / + s ) (cid:0) E [ k u k L ( D ) ] (cid:1) / (cid:0) E [ k v k L ( D ) ] (cid:1) / . (3.37)Via the Minkowski inequality and (3.25), we observe that (cid:0) E (cid:2) ( Z s k∇ Z τ v k L ( D ) dτ ) (cid:3)(cid:1) / ≤ Z s (cid:0) E [ k∇ Z τ v k L ( D ) ] (cid:1) / dτ ≤ cs (cid:0) E [ k∇ v k L ( D ) ] (cid:1) / . (3.38)The required result follows from (3.37) and (3.38), which completes the proof of this lemma. (cid:3) Equivalence of weak solutions
In this section we use the process ( Z t ) t ≥ defined in the preceding section to define a newprocess m from M . Let(4.1) m ( t, x ) = Z − t M ( t, x ) ∀ t ≥ , a.e. x ∈ D. We will show that this new variable m is differentiable with respect to t .In the next lemma, we introduce the equation satisfied by m so that M is a solutionto (1.3) in the sense of (2.1). Lemma 4.1. If m ( · , ω ) ∈ H (0 , T ; L ( D )) ∩ L (0 , T ; H ( D )) , for P -a.s. ω ∈ Ω , satisfies | m ( t, x ) | = 1 ∀ t ≥ , a.e. x ∈ D, P − a.s. , and for any ψ ∈ L (0 , T ; W , ∞ ( D )) h ∂ t m , ψ i L ( D T ) + λ Z T h Z s m × ∇ Z s m , ∇ Z s ψ i L ( D ) ds + λ Z T h Z s m × ∇ Z s m , ∇ Z s ( m × ψ ) i L ( D ) ds = 0 , P -a.s. . (4.2) Then M = Z t m satisfies (2.1) P -a.s..Proof. Using Itˆo’s formula for M = Z t m , we deduce M ( t ) = M (0) + q X i =1 Z t Z m × g i ◦ dW i ( s ) + Z t Z ( ∂ t m ) ds = M (0) + q X i =1 Z t M × g i ◦ dW i ( s ) + Z t Z s ( ∂ t m ) ds. Multiplying both sides by a test function ψ ∈ C ∞ ( D ) and integrating over D we obtain h M ( t ) , ψ i L ( D ) = h M (0) , ψ i L ( D ) + q X i =1 Z t h M × g i , ψ i L ( D ) ◦ dW i ( s )+ Z t h Z s ( ∂ t m ) , ψ i L ( D ) ds = h M (0) , ψ i L ( D ) + q X i =1 Z t h M × g i , ψ i L ( D ) ◦ dW i ( s )+ Z t (cid:10) ∂ t m , Z − s ψ (cid:11) L ( D ) ds, (4.3)where in the last step we used (3.16). On the other hand, it follows from (4.2) that, for all ξ ∈ L (0 , t ; W , ∞ ( D )), there holds: Z t h ∂ t m , ξ i L ( D ) ds = − λ Z t h Z s m × ∇ Z s m , ∇ Z s ξ i L ( D ) ds − λ Z t h Z s m × ∇ Z s m , ∇ Z s ( m × ξ ) i L ( D ) ds. (4.4)Using (4.4) with ξ = Z − s ψ for the last term on the right hand side of (4.3) we deduce h M ( t ) , ψ i L ( D ) = h M (0) , ψ i L ( D ) + q X i =1 Z t h M × g i , ψ i L ( D ) ◦ dW i ( s ) − λ Z t (cid:10) M × ∇ M , ∇ ψ (cid:1)(cid:11) L ( D ) ds − λ Z t (cid:10) M × ∇ M , ∇ Z s ( m × Z − s ψ ) (cid:11) L ( D ) ds. It follows from (3.17) that h M ( t ) , ψ i L ( D ) = h M (0) , ψ i L ( D ) + q X i =1 Z t h M × g i , ψ i L ( D ) ◦ dW i ( s ) − λ Z t (cid:10) M × ∇ M , ∇ ψ (cid:1)(cid:11) L ( D ) ds − λ Z t h M × ∇ M , ∇ ( M × ψ ) i L ( D ) ds, which complete the proof. (cid:3) The following lemma shows that the constraint on | m | is inherited by | M | . Lemma 4.2.
The process M satisfies | M ( t, x ) | = 1 ∀ t ≥ , a.e. x ∈ D, P − a.s.if and only if m defined in (4.1) satisfies | m ( t, x ) | = 1 ∀ t ≥ , a.e. x ∈ D, P − a.s. . EM FOR STOCHASTIC LANDAU–LIFSHITZ–GILBERT EQUATION 15
Proof.
The proof follows by using (3.16): | m | = h m , m i = (cid:10) Z − t M , Z − t M (cid:11) = (cid:10) M , Z t Z − t M (cid:11) = h M , M i = | M | . (cid:3) In the next lemma we provide a relationship between equation (4.2) and its Gilbert form.
Lemma 4.3.
Let m ( · , ω ) ∈ H (0 , T ; L ( D )) ∩ L (0 , T ; H ( D )) for P -a.s. ω ∈ Ω satisfy (4.5) | m ( t, x ) | = 1 , t ∈ (0 , T ) , x ∈ D, and λ h ∂ t m , ϕ i L ( D T ) + λ h m × ∂ t m , ϕ i L ( D T ) = µ Z T h∇ Z s m , ∇ Z s ( m × ϕ ) i L ( D ) ds, (4.6) where µ = λ + λ . Then m satisfies (4.2) .Proof. For each ψ ∈ L (0 , T ; W , ∞ ( D )), using Lemma 7.1 in the Appendix, there exists ϕ ∈ L (0 , T ; H ( D )) such that(4.7) λ ϕ + λ ϕ × m = ψ . We can write (4.6) as h ∂ t m , λ ϕ + λ ϕ × m i L ( D T ) + λ Z T h Z s m × ∇ Z s m , ∇ Z s ( λ ϕ ) i L ( D ) ds + λ Z T h∇ Z s m , ∇ Z s ( λ ϕ × m ) i L ( D ) ds = 0 . (4.8)From (4.5) and (3.3), we obtain that(4.9) | Z t m ( t, x ) | = 1 , ∀ t ∈ (0 , T ) and x ∈ D. On the other hand, by using (4.9), (3.19) and a standard identity(4.10) h a , b × c i = h b , c × a i = h c , a × b i , for all a , b , c ∈ R , we obtain λ Z T h Z s m × ∇ Z s m , ∇ Z s ( λ ϕ × m ) i L ( D ) ds + λ Z T h∇ Z s m , ∇ Z s ( λ ϕ ) i L ( D ) ds − λ Z T (cid:10) |∇ Z s m | Z s m , Z s ( λ ϕ ) (cid:11) L ( D ) ds = 0 . (4.11)Moreover, we have(4.12) − λ Z T (cid:10) |∇ Z s m | Z s m , λ Z s ϕ × Z s m (cid:11) L ( D ) ds = 0 . Summing (4.8), (4.11) and (4.12) gives h ∂ t m , λ ϕ + λ ϕ × m i L ( D T ) + λ Z T h Z s m × ∇ Z s m , ∇ Z s ( λ ϕ + λ ϕ × m ) i L ( D ) ds + λ Z T h∇ Z s m , ∇ Z s ( λ ϕ + λ ϕ × m ) i L ( D ) ds − λ Z T (cid:10) |∇ Z s m | Z s m , Z s ( λ ϕ + λ ϕ × m ) (cid:11) L ( D ) ds = 0 The desired equation (4.2) follows by noting (4.7) and using (3.19), (4.10) and (4.9). (cid:3)
Remark 4.4.
By using (4.10) and (4.5) we can rewrite (4.6) as λ h m × ∂ t m , w i L ( D T ) − λ h ∂ t m , w i L ( D T ) = µ Z T h∇ Z s m , ∇ Z s w i L ( D ) ds, (4.13) or equivalently, thanks to Lemma 3.6, λ h m × ∂ t m , w i L ( D T ) − λ h ∂ t m , w i L ( D T ) = µ h∇ m , ∇ w i L ( D T ) + µ Z T F ( t, m ( t, · ) , w ( t, · )) dt, (4.14) where w = m × ϕ for ϕ ∈ L (0 , T ; H ( D )) . We note in particular that w · m = 0 . Thisproperty will be exploited later in the design of the finite element scheme. We state the following lemma as a consequence of Lemmas 4.3, 4.2 and 4.1.
Lemma 4.5.
Let m ( · , ω ) ∈ H (0 , T ; L ( D )) ∩ L (0 , T ; H ( D )) for P -a.s. ω ∈ Ω . If m is asolution of (4.5) – (4.6) , then M = Z t m is a weak martingale solution of (1.3) in the senseof Definition 2.2.Proof. By using Lemmas 4.1, 4.2 and 4.3 together with the imbedding H (0 , T ; L ( D )) ֒ → C (0 , T ; H − ( D ), we deduce that M satisfies (1), (2), (3), (4) in Definition 2.2, which com-pletes the proof. (cid:3) Thanks to the above lemma, we now can now restrict our attention to solving equa-tion (4.6) rather than (2.1). 5.
The finite element scheme
In this section we design a finite element scheme to find approximate solutions to (4.6).In the next section, we prove that the finite element solutions converge to a solution of (4.6).Then, thanks to Lemma 4.5, we obtain a weak solution of (2.1).Let T h be a regular tetrahedrization of the domain D into tetrahedra of maximal mesh-size h . We denote by N h := { x , . . . , x N } the set of vertices and introduce the finite-elementspace V h ⊂ H ( D ), which is the space of all continuous piecewise linear functions on T h .A basis for V h can be chosen to be { φ n ξ , φ n ξ , φ n ξ } ≤ n ≤ N , where { ξ i } i =1 , ··· , is thecanonical basis for R and φ n ( x m ) = δ n,m . Here δ n,m denotes the Kronecker delta symbol.The interpolation operator from C ( D ) onto V h , denoted by I V h , is defined by I V h ( v ) = N X n =1 v ( x n ) φ n ( x ) ∀ v ∈ C ( D, R ) . Before introducing the finite element scheme, we state the following result proved byBartels [4], which will be used in the subsequent analysis.
Lemma 5.1.
Assume that (5.1) Z D ∇ φ i · ∇ φ j d x ≤ for all i, j ∈ { , , · · · , J } and i = j. Then for all u ∈ V h satisfying | u ( x l ) | ≥ , l = 1 , , · · · , J , there holds (5.2) Z D (cid:12)(cid:12)(cid:12)(cid:12) ∇ I V h (cid:18) u | u | (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) d x ≤ Z D |∇ u | d x . EM FOR STOCHASTIC LANDAU–LIFSHITZ–GILBERT EQUATION 17
When d = 2, we note that condition (5.1) holds for Delaunay triangulation. Roughly speak-ing, a Delaunay triangulation is one in which no vertex is contained inside the perimeter ofany triangle. When d = 3, condition (5.1) holds if all dihedral angles of the tetrahedra in T h | L ( D ) are less than or equal to π/
2; see [4]. In what follows we assume that (5.1) holds.To discretize the equation (4.6), we fix a positive integer J , choose the time step k tobe k = T /J and define t j = jk , j = 0 , · · · , J . For j = 1 , , . . . , J , the solution m ( t j , · ) isapproximated by m ( j ) h ∈ V h , which is computed as follows.Since m t ( t j , · ) ≈ m ( t j +1 , · ) − m ( t j , · ) k ≈ m ( j +1) h − m ( j ) h k , we can define m ( j +1) h from m ( j ) h by(5.3) m ( j +1) h = m ( j ) h + k v ( j ) h , where v ( j ) h is an approximation of m t ( t j , · ). Hence, it suffices to propose a scheme tocompute v ( j ) h .Motivated by the property ∂ t m · m = 0, we will find v ( j ) h in the space W ( j ) h defined by(5.4) W ( j ) h := n w ∈ V h | w ( x n ) · m ( j ) h ( x n ) = 0 , n = 1 , . . . , N o . Given m ( j ) h ∈ V h , we use (4.14) to define v ( j ) h instead of using (4.6) so that the same testand trial functions can be used (see Remark 4.4). Hence, we define by v ( j ) h ∈ W ( j ) h satisfyingthe following equation λ D m ( j ) h × v ( j ) h , w ( j ) h E L ( D ) − λ D v ( j ) h , w ( j ) h E L ( D ) = µ D ∇ ( m ( j ) h + kθ v ( j ) h ) , ∇ w ( j ) h E L ( D ) + µF ( t j , m ( j ) h , w ( j ) h ) P -a.s. . (5.5)We summarise the algorithm as follows. Algorithm 5.1.Step 1:
Set j = 0 . Choose m (0) h = I V h m . Step 2:
Find v ( j ) h ∈ W ( j ) h satisfying (5.5) . Step 3:
Define m ( j +1) h ( x ) := N X n =1 m ( j ) h ( x n ) + k v ( j ) h ( x n ) (cid:12)(cid:12)(cid:12) m ( j ) h ( x n ) + k v ( j ) h ( x n ) (cid:12)(cid:12)(cid:12) φ n ( x ) . Step 4:
Set j = j + 1 , and return to Step 2 if j < J . Stop if j = J . Since (cid:12)(cid:12)(cid:12) m (0) h ( x n ) (cid:12)(cid:12)(cid:12) = 1 and v ( j ) h ( x n ) · m ( j ) h ( x n ) = 0 for all n = 1 , . . . , N and j = 0 , . . . , J ,we obtain (by induction)(5.6) (cid:12)(cid:12)(cid:12) m ( j ) h ( x n ) + k v ( j ) h ( x n ) (cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12) m ( j ) h ( x n ) (cid:12)(cid:12)(cid:12) = 1 , j = 0 , . . . , J. In particular, (5.6) shows that the algorithm is well defined.We finish this section by proving the following three lemmas concerning some propertiesof m ( j ) h and R h,k . Lemma 5.2.
For any j = 0 , . . . , J , k m ( j ) h k L ∞ ( D ) ≤ and k m ( j ) h k L ( D ) ≤ | D | , where | D | denotes the measure of D .Proof. The first inequality follows from (5.6) and the second can be obtained by integrating m ( j ) h ( x ) over D . (cid:3) Lemma 5.3.
There exist a deterministic constant c depending on m , { g i } qi =1 , λ and λ such that for any θ ∈ [1 / , and for j = 1 , . . . , J there holds E (cid:13)(cid:13)(cid:13) ∇ m ( j ) h (cid:13)(cid:13)(cid:13) L ( D ) + k j − X i =1 µ − λ E (cid:13)(cid:13)(cid:13) v ( i ) h (cid:13)(cid:13)(cid:13) L ( D ) + k (2 θ − j − X i =1 E (cid:13)(cid:13)(cid:13) ∇ v ( i ) h (cid:13)(cid:13)(cid:13) L ( D ) ≤ c. Proof.
Taking w ( j ) h = v ( j ) h in equation (5.5) yields to the following identity: − λ (cid:13)(cid:13)(cid:13) v ( j ) h (cid:13)(cid:13)(cid:13) L ( D ) = µ D ∇ m ( j ) h , ∇ v ( j ) h E L ( D ) + µkθ (cid:13)(cid:13)(cid:13) ∇ v ( j ) h (cid:13)(cid:13)(cid:13) L ( D ) + µF (cid:0) t j , m ( j ) h , v ( j ) h (cid:1) , or equivalently µ D ∇ m ( j ) h , ∇ v ( j ) h E L ( D ) = − λ (cid:13)(cid:13)(cid:13) v ( j ) h (cid:13)(cid:13)(cid:13) L ( D ) − µkθ (cid:13)(cid:13)(cid:13) ∇ v ( j ) h (cid:13)(cid:13)(cid:13) L ( D ) − µF (cid:0) t j , m ( j ) h , v ( j ) h (cid:1) . (5.7)From Lemma 5.1 it follows that (cid:13)(cid:13)(cid:13) ∇ m ( j +1) h (cid:13)(cid:13)(cid:13) L ( D ) ≤ (cid:13)(cid:13)(cid:13) ∇ ( m ( j ) h + k v ( j ) h ) (cid:13)(cid:13)(cid:13) L ( D ) , or equivalently, (cid:13)(cid:13)(cid:13) ∇ m ( j +1) h (cid:13)(cid:13)(cid:13) L ( D ) ≤ (cid:13)(cid:13)(cid:13) ∇ m ( j ) h (cid:13)(cid:13)(cid:13) L ( D ) + k (cid:13)(cid:13)(cid:13) ∇ v ( j ) h (cid:13)(cid:13)(cid:13) L ( D ) + 2 k D ∇ m ( j ) h , ∇ v ( j ) h E L ( D ) . Therefore, together with (5.7), we deduce (cid:13)(cid:13)(cid:13) ∇ m ( j +1) h (cid:13)(cid:13)(cid:13) L ( D ) + 2 kµ − λ (cid:13)(cid:13)(cid:13) v ( j ) h (cid:13)(cid:13)(cid:13) L ( D ) + k (2 θ − (cid:13)(cid:13)(cid:13) ∇ v ( j ) h (cid:13)(cid:13)(cid:13) L ( D ) ≤ (cid:13)(cid:13)(cid:13) ∇ m ( j ) h (cid:13)(cid:13)(cid:13) L ( D ) − kF (cid:0) t j , m ( j ) h , v ( j ) h (cid:1) . Thus, it follows from (3.26) that E (cid:13)(cid:13)(cid:13) ∇ m ( j +1) h (cid:13)(cid:13)(cid:13) L ( D ) + 2 kµ − λ E (cid:13)(cid:13)(cid:13) v ( j ) h (cid:13)(cid:13)(cid:13) L ( D ) + k (2 θ − E (cid:13)(cid:13)(cid:13) ∇ v ( j ) h (cid:13)(cid:13)(cid:13) L ( D ) ≤ E (cid:13)(cid:13)(cid:13) ∇ m ( j ) h (cid:13)(cid:13)(cid:13) L ( D ) + 2 k E h(cid:12)(cid:12) F (cid:0) t j , m ( j ) h , v ( j ) h (cid:1)(cid:12)(cid:12)i ≤ (1 + kcǫT ) E (cid:13)(cid:13)(cid:13) ∇ m ( j ) h (cid:13)(cid:13)(cid:13) L ( D ) + ckǫ ( T + T / ) E (cid:13)(cid:13)(cid:13) m ( j ) h (cid:13)(cid:13)(cid:13) L ( D ) + ckǫ − ( T + T / + 1) E (cid:13)(cid:13)(cid:13) v ( j ) h (cid:13)(cid:13)(cid:13) L ( D ) . EM FOR STOCHASTIC LANDAU–LIFSHITZ–GILBERT EQUATION 19
By choosing ǫ = µ − λ c ( T + T / +1) in the right hand side of this inequality and using Lemma 5.2we deduce E (cid:13)(cid:13)(cid:13) ∇ m ( j +1) h (cid:13)(cid:13)(cid:13) L ( D ) + kµ − λ E (cid:13)(cid:13)(cid:13) v ( j ) h (cid:13)(cid:13)(cid:13) L ( D ) + k (2 θ − E (cid:13)(cid:13)(cid:13) ∇ v ( j ) h (cid:13)(cid:13)(cid:13) L ( D ) ≤ ck + (1 + kc ) E (cid:13)(cid:13)(cid:13) ∇ m ( j ) h (cid:13)(cid:13)(cid:13) L ( D ) . Replacing j by i in the above inequality and summing for i from 0 to j − E (cid:13)(cid:13)(cid:13) ∇ m ( j ) h (cid:13)(cid:13)(cid:13) L ( D ) + k j − X i =1 µ − λ E (cid:13)(cid:13)(cid:13) v ( i ) h (cid:13)(cid:13)(cid:13) L ( D ) + k (2 θ − j − X i =1 E (cid:13)(cid:13)(cid:13) ∇ v ( i ) h (cid:13)(cid:13)(cid:13) L ( D ) ≤ ckj + c k∇ m (0) h k L ( D ) + kc j − X i =1 E (cid:13)(cid:13)(cid:13) ∇ m ( i ) h (cid:13)(cid:13)(cid:13) L ( D ) . (5.8)Since m ∈ H ( D ) it can be shown that there exists a deterministic constant c dependingonly on m such that(5.9) k∇ m (0) h k L ( D ) ≤ c. Hence, inequality (5.8) implies E (cid:13)(cid:13)(cid:13) ∇ m ( j ) h (cid:13)(cid:13)(cid:13) L ( D ) + k j − X i =1 µ − λ E (cid:13)(cid:13)(cid:13) v ( i ) h (cid:13)(cid:13)(cid:13) L ( D ) + k (2 θ − j − X i =1 E (cid:13)(cid:13)(cid:13) ∇ v ( i ) h (cid:13)(cid:13)(cid:13) L ( D ) ≤ c + kc j − X i =1 E (cid:13)(cid:13)(cid:13) ∇ m ( i ) h (cid:13)(cid:13)(cid:13) L ( D ) . (5.10)By using induction and (5.9) we can show that E k∇ m ih k L ( D ) ≤ c (1 + ck ) i . Summing over i from 0 to j − x ≤ e x we obtain k j − X i =0 E (cid:13)(cid:13) ∇ m ih (cid:13)(cid:13) L ( D ) ≤ ck (1 + ck ) j − ck ≤ e ckJ = c. This together with (5.10) gives the desired result. (cid:3) The main result
In this section, we will use the finite element function m ( j ) h to construct a sequence offunctions that converges in an appropriate sense to a function that is a weak martingalesolution of (1.3) in the sense of Definition 2.2.The discrete solutions m ( j ) h and v ( j ) h constructed via Algorithm 5.1 are interpolated intime in the following definition. Definition 6.1.
For all x ∈ D , u , v ∈ V h and all t ∈ [0 , T ] , let j ∈ { , ..., J } be such that t ∈ [ t j , t j +1 ) . We then define m h,k ( t, x ) := t − t j k m ( j +1) h ( x ) + t j +1 − tk m ( j ) h ( x ) , m − h,k ( t, x ) := m ( j ) h ( x ) , v h,k ( t, x ) := v ( j ) h ( x ) ,F k ( t, u , v ) := F ( t j , u , v ) P − a.s. . We note that m h,k ( t ) is an F t j adapted process for t ∈ [ t j , t j +1 ). The above sequenceshave the following obvious bounds. Lemma 6.2.
There exist a deterministic constant c depending on m , g , µ , µ and T such that for all θ ∈ [1 / , , E k m ∗ h,k k L ( D T ) + E (cid:13)(cid:13) ∇ m ∗ h,k (cid:13)(cid:13) L ( D T ) + E k v h,k k L ( D T ) + k (2 θ − E k∇ v h,k k L ( D T ) ≤ c, where m ∗ h,k = m h,k or m − h,k . In particular, when θ ∈ [0 , ) , E k m ∗ h,k k L ( D T ) + E (cid:13)(cid:13) ∇ m ∗ h,k (cid:13)(cid:13) L ( D T ) + (cid:0) θ − kh − (cid:1) E k v h,k k L ( D T ) ≤ c. Proof.
It is easy to see that k m − h,k k L ( D T ) = k J − X i =0 k m ( i ) h k L ( D ) and k v h,k k L ( D T ) = k J − X i =0 k v ( i ) h k L ( D ) . Both inequalities are direct consequences of Definition 6.1, Lemmas 5.2, and 5.3, notingthat the second inequality requires the use of the inverse estimate (see e.g. [14]) k∇ v ( i ) h k L ( D ) ≤ ch − k v ( i ) h k L ( D ) . (cid:3) The next lemma provides a bound of m h,k in the H -norm and establishes relationshipsbetween m − h,k , m h,k and v h,k . Lemma 6.3.
Assume that h and k approach , with the further condition k = o ( h ) when θ ∈ [0 , ) . The sequences { m h,k } , { m − h,k } , and { v h,k } defined in Definition 6.1 satisfy thefollowing properties: E k m h,k k H ( D T ) ≤ c, (6.1) E k m h,k − m − h,k k L ( D T ) ≤ ck , (6.2) E k v h,k − ∂ t m h,k k L ( D T ) ≤ ck, (6.3) E k| m h,k | − k L ( D T ) ≤ c ( h + k ) . (6.4) Proof.
The results can be obtained by using Lemma 6.2 and the arguments in the proofof [13, Lemma 6.3]. (cid:3)
We now prove some properties of F and F k , which will be used in the next two lemmas. EM FOR STOCHASTIC LANDAU–LIFSHITZ–GILBERT EQUATION 21
Lemma 6.4.
For any u , v ∈ L (cid:0) Ω; L (0 , T ; H ( D )) (cid:1) , there exists a constant c dependingon T and { g i } i =1 , ··· ,q such that (6.5) E [ Z T | F ∗ ( t, u ( t, · ) , v ( t, · )) | dt ] ≤ c (cid:0) E [ k u k L ( D T ) ] (cid:1) / (cid:18)(cid:0) E [ k∇ v k L ( D T ) ] (cid:1) / + (cid:0) E [ k v k L ( D T ) ] (cid:1) / (cid:19) , here, F ∗ = F or F k . Furthermore, E [ Z T | F ( t, u ( t, · ) , v ( t, · )) − F k ( t, u ( t, · ) , v ( t, · )) | dt ] ≤ ck / (cid:0) E [ k u k L ( D T ) ] (cid:1) / (cid:18)(cid:0) E [ k∇ v k L ( D T ) ] (cid:1) / + (cid:0) E [ k v k L ( D T ) ] (cid:1) / (cid:19) . (6.6) Proof.
Proof of (6.5): The first result of the lemma for F ∗ = F can be deduced fromLemma 3.9 by replacing s ≡ t , u ≡ u ( t, · ), v ≡ v ( t, · ) and using H¨older’s inequality asfollows: E [ Z T | F ( t, u ( t, · ) , v ( t, · )) | dt ] = Z T E [ | F ( t, u ( t, · ) , v ( t, · )) | ] dt ≤ c Z T (cid:0) E [ k u ( t, · ) k L ( D ) ] (cid:1) / (cid:0) E [ k∇ v ( t, · ) k L ( D ) ] (cid:1) / dt + c Z T (cid:0) E [ k u ( t, · ) k L ( D ) ] (cid:1) / (cid:0) E [ k v ( t, · ) k L ( D ) ] (cid:1) / dt (6.7) ≤ c (cid:0) E [ k u k L ( D T ) ] (cid:1) / (cid:18)(cid:0) E [ k∇ v k L ( D T ) ] (cid:1) / + (cid:0) E [ k v k L ( D T ) ] (cid:1) / (cid:19) . We first note that E [ Z T | F k ( t, u ( t, · ) , v ( t, · )) | dt ] = Z T E [ | F k ( t, u ( t, · ) , v ( t, · )) | ] dt = J − X j =0 Z t j +1 t j E [ | F ( t j , u ( t, · ) , v ( t, · )) | ] dt, then apply Lemma 3.9 for s ≡ t j , u ≡ u ( t, · ) and v ≡ v ( t, · ) to deduce E [ Z T | F k ( t, u ( t, · ) , v ( t, · )) | dt ] ≤ c J − X j =0 t j Z t j +1 t j (cid:0) E [ k u ( t, · ) k L ( D ) ] (cid:1) / (cid:0) E [ k∇ v ( t, · ) k L ( D ) ] (cid:1) / dt + c J − X j =0 ( t / j + t j ) Z t j +1 t j (cid:0) E [ k u ( t, · ) k L ( D ) ] (cid:1) / (cid:0) E [ k v ( t, · ) k L ( D ) ] (cid:1) / dt ≤ cT Z T (cid:0) E [ k u ( t, · ) k L ( D ) ] (cid:1) / (cid:0) E [ k∇ v ( t, · ) k L ( D ) ] (cid:1) / dt + c ( T + T / ) Z T (cid:0) E [ k u ( t, · ) k L ( D ) ] (cid:1) / (cid:0) E [ k v ( t, · ) k L ( D ) ] (cid:1) / dt. Hence, (6.5) with function F ∗ = F k follows by using H¨older’s inequality. Proof of (6.6): Noting that E [ Z T | F ( t, u ( t, · ) , v ( t, · )) − F k ( t, u ( t, · ) , v ( t, · )) | dt ]= J X j =0 Z t j +1 t j E [ | F ( t, u ( t, · ) , v ( t, · )) − F ( t j , u ( t, · ) , v ( t, · )) | ] dt := J X j =0 Z t j +1 t j E [ | ˜ F j ( t − t j , u ( t, · ) , v ( t, · )) | ] dt. (6.8)Here ˜ F j ( t, x , y ) := q X i =1 Z t ˜ F j ,i ( s, x , y ) ds + q X i =1 Z t ˜ F j ,i ( s, x , y ) d ˜ W i ( s ) , in which ˜ F j ,i ( s, x , y ) = F ,i ( s + t j , x , y ), ˜ F j ,i ( s, x , y ) = F ,i ( s + t j , x , y ) and ˜ W i ( s ) = W i ( s + t j ) − W i ( t j ). By using the same arguments as in the proof of Lemma 3.9 we obtainthe same result for the upper bound of ˜ F j , namely E | ˜ F j ( s, u , v ) | ≤ cs (cid:0) E [ k u k L ( D ) ] (cid:1) / (cid:0) E [ k∇ v k L ( D ) ] (cid:1) / + c ( s / + s ) (cid:0) E [ k u k L ( D ) ] (cid:1) / (cid:0) E [ k v k L ( D ) ] (cid:1) / . Hence, there holds Z t j +1 t j E | ˜ F j ( s, u , v ) | ds ≤ c Z t j +1 t j s (cid:0) E [ k u k L ( D ) ] (cid:1) / (cid:0) E [ k∇ v k L ( D ) ] (cid:1) / ds + c Z t j +1 t j ( s / + s ) (cid:0) E [ k u k L ( D ) ] (cid:1) / (cid:0) E [ k v k L ( D ) ] (cid:1) / ds, ≤ ck Z t j +1 t j (cid:0) E [ k u k L ( D ) ] (cid:1) / (cid:0) E [ k∇ v k L ( D ) ] (cid:1) / ds + c ( k / + k ) Z t j +1 t j (cid:0) E [ k u k L ( D ) ] (cid:1) / (cid:0) E [ k v k L ( D ) ] (cid:1) / ds. (6.9)Therefore, it follows from (6.8) and (6.9) that E [ Z T | F ( t, u ( t, · ) , v ( t, · )) − F k ( t, u ( t, · ) , v ( t, · )) | dt ] ≤ ck Z T (cid:0) E [ k u k L ( D ) ] (cid:1) / (cid:0) E [ k∇ v k L ( D ) ] (cid:1) / ds + c ( k / + k ) Z T (cid:0) E [ k u k L ( D ) ] (cid:1) / (cid:0) E [ k v k L ( D ) ] (cid:1) / ds. The result follows immediately by using H¨older’s inequality, which completes the proof ofthe lemma. (cid:3)
The following two Lemmas 6.5 and 6.6 show that m − h,k and m h,k , respectively, satisfy adiscrete form of (4.6). EM FOR STOCHASTIC LANDAU–LIFSHITZ–GILBERT EQUATION 23
Lemma 6.5.
Assume that h and k approach with the following conditions (6.10) k = o ( h ) when ≤ θ < / ,k = o ( h ) when θ = 1 / , no condition when / < θ ≤ . Then for any ψ ∈ C ∞ (cid:0) (0 , T ); C ∞ ( D ) (cid:1) , there holds P -a.s. − λ D m − h,k × v h,k , m − h,k × ψ E L ( D T ) + λ D v h,k , m − h,k × ψ E L ( D T ) + µ D ∇ ( m − h,k + kθ v h,k ) , ∇ ( m − h,k × ψ ) E L ( D T ) + µ Z T F k ( t, m − h,k , m − h,k × ψ ) dt = X j =1 I j , where I := D − λ m − h,k × v h,k + λ v h,k , m − h,k × ψ − I V h ( m − h,k × ψ ) E L ( D T ) ,I := µ D ∇ ( m − h,k + kθ v h,k ) , ∇ ( m − h,k × ψ − I V h ( m − h,k × ψ )) E L ( D T ) ,I := µ Z T F k ( t, m − h,k , m − h,k × ψ ) − F k ( t, m − h,k , I V h ( m − h,k × ψ )) dt. Furthermore, E | I i | = O ( h ) for i = 1 , , .Proof. For t ∈ [ t j , t j +1 ), we use equation (5.5) with w ( j ) h = I V h (cid:0) m − h,k ( t, · ) × ψ ( t, · ) (cid:1) to see − λ D m − h,k ( t, · ) × v h,k ( t, · ) , I V h (cid:0) m − h,k ( t, · ) × ψ ( t, · ) (cid:1)E L ( D ) + λ D v h,k ( t, · ) , I V h (cid:0) m − h,k ( t, · ) × ψ ( t, · ) (cid:1)E L ( D ) + µ D ∇ ( m − h,k ( t, · ) + kθ v h,k ( t, · )) , ∇ I V h (cid:0) m − h,k ( t, · ) × ψ ( t, · ) (cid:1)E L ( D ) + µF k ( t, m − h,k ( t, · ) , I V h (cid:0) m − h,k ( t, · ) × ψ ( t, · ) (cid:1) ) = 0 . Integrating both sides of the above equation over ( t j , t j +1 ) and summing over j = 0 , . . . , J − − λ D m − h,k × v h,k , I V h (cid:0) m − h,k × ψ (cid:1)E L ( D T ) + λ D v h,k , I V h (cid:0) m − h,k × ψ (cid:1)E L ( D T ) + µ D ∇ ( m − h,k + kθ v h,k ) , ∇ I V h (cid:0) m − h,k × ψ (cid:1)E L ( D T ) + µ Z T F k ( t, m − h,k , I V h ( m − h,k × ψ )) dt = 0 . This implies − λ D m − h,k × v h,k , m − h,k × ψ E L ( D T ) + λ D v h,k , m − h,k × ψ E L ( D T ) + µ D ∇ ( m − h,k + kθ v h,k ) , ∇ ( m − h,k × ψ ) E L ( D T ) + µ Z T F k ( t, m − h,k , m − h,k × ψ ) dt = I + I + I . Hence it suffices to prove that E | I i | = O ( h ) for i = 1 , ,
3. First, by using Lemma 5.2 weobtain(6.11) k m − h,k k L ∞ ( D T ) ≤ sup ≤ j ≤ J k m ( j ) h k L ∞ ( D ) ≤ , and(6.12) k m h,k k L ∞ ( D T ) ≤ ≤ j ≤ J k m ( j ) h k L ∞ ( D ) ≤ . Lemma 6.2 and (6.11) together with H¨older’s inequality and Lemma 7.2 yield E | I | ≤ c E h(cid:16) k m − h,k k L ∞ ( D T ) + 1 (cid:17) k v h,k k L ( D T ) k m − h,k × ψ − I V h ( m − h,k × ψ ) k L ( D T ) i ≤ c E h k v h,k k L ( D T ) k m − h,k × ψ − I V h ( m − h,k × ψ ) k L ( D T ) i ≤ c (cid:0) E [ k v h,k k L ( D T ) ] (cid:1) / (cid:0) E [ k m − h,k × ψ − I V h ( m − h,k × ψ ) k L ( D T ) ] (cid:1) / ≤ ch. The bound for E | I | can be obtained similarly, using Lemma 6.2 and noting that when θ ∈ [0 , ], a suitable bound on k k∇ v h,k k L ( D T ) can be deduced from the inverse estimateas follows: k k∇ v h,k k L ( D T ) ≤ ckh − k v h,k k L ( D T ) ≤ ckh − . The bound for E | I | can be obtained by noting the linearity of F in Remark 3.7 and usingLemmas 6.4 and 7.2. Indeed, E | I | = µ E (cid:12)(cid:12)Z T F k ( t, m − h,k , m − h,k × ψ − I V h ( m − h,k × ψ )) dt (cid:12)(cid:12) ≤ µ E Z T (cid:12)(cid:12) F k ( t, m − h,k , m − h,k × ψ − I V h ( m − h,k × ψ )) (cid:12)(cid:12) dt ≤ c (cid:0) E [ k m − h,k k L ( D T ) ] (cid:1) / (cid:18)(cid:0) E [ k∇ (cid:0) m − h,k × ψ − I V h ( m − h,k × ψ ) (cid:1) k L ( D T ) ] (cid:1) / + (cid:0) E [ k m − h,k × ψ − I V h ( m − h,k × ψ ) k L ( D T ) ] (cid:1) / (cid:19) ≤ ch. This completes the proof of the lemma. (cid:3)
Lemma 6.6.
Assume that h and k approach 0 satisfying (6.10) . Then for any ψ ∈ C ∞ (cid:0) (0 , T ); C ∞ ( D ) (cid:1) , there holds P -a.s. − λ h m h,k × ∂ t m h,k , m h,k × ψ i L ( D T ) + λ h ∂ t m h,k , m h,k × ψ i L ( D T ) + µ h∇ ( m h,k ) , ∇ ( m h,k × ψ ) i L ( D T ) + µ Z T F ( t, m h,k , m h,k × ψ ) dt = X j =1 I j , (6.13) EM FOR STOCHASTIC LANDAU–LIFSHITZ–GILBERT EQUATION 25 where I = − λ D m − h,k × v h,k , m − h,k × ψ E L ( D T ) + λ h m h,k × ∂ t m h,k , m h,k × ψ i L ( D T ) ,I = λ D v h,k , m − h,k × ψ E L ( D T ) − λ h ∂ t m h,k , m h,k × ψ i L ( D T ) ,I = µ D ∇ ( m − h,k + kθ v h,k ) , ∇ ( m − h,k × ψ ) E L ( D T ) − µ h∇ ( m h,k ) , ∇ ( m h,k × ψ ) i L ( D T ) ,I = µ Z T (cid:0) F ( t, m h,k , m h,k × ψ ) − F k ( t, m − h,k , m − h,k × ψ ) (cid:1) dt. Furthermore, E | I i | = O ( h ) for i = 1 , · · · , and E | I | = O ( h + k / ) .Proof. From Lemma 6.5 it follows that − λ h m h,k × ∂ t m h,k , m h,k × ψ i L ( D T ) + λ h ∂ t m h,k , m h,k × ψ i L ( D T ) + µ h∇ ( m h,k ) , ∇ ( m h,k × ψ ) i L ( D T ) + µ Z T F k ( t, m h,k , m h,k × ψ ) dt = I + · · · + I . Hence it suffices to prove that E | I i | = O ( h ) for i = 4 , · · · ,
6. First, by using the triangleinequality and H¨older’s inequality, we obtain λ − | I | ≤ (cid:12)(cid:12)(cid:12)(cid:12)D ( m − h,k − m h,k ) × v h,k , m − h,k × ψ E L ( D T ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)D m h,k × v h,k , ( m − h,k − m h,k ) × ψ E L ( D T ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) h m h,k × ( v h,k − ∂ t m h,k ) , m h,k × ψ i L ( D T ) (cid:12)(cid:12)(cid:12) , ≤ k m − h,k − m h,k k L ( D T ) k v h,k k L ( D T ) (cid:0) k m − h,k k L ∞ ( D T ) + k m h,k k L ∞ ( D T ) (cid:1) k ψ k L ∞ ( D T ) + k v h,k − ∂ t m h,k k L ( D T ) k m h,k k L ∞ ( D T ) k ψ k L ∞ ( D T ) . Therefore, the required bound on E | I | can be obtained by using (6.11), (6.12) and Lem-mas 6.2, 6.3. The bounds on E | I | and E | I | can be obtaineded similarly.In order to prove the bound for E | I | , we first use the triangle inequality then Remark 3.7and Lemma 6.4 to obtain E | I | ≤ E Z T | F ( t, m h,k , m h,k × ψ ) − F k ( t, m h,k , m h,k × ψ ) | dt + E Z T (cid:12)(cid:12)(cid:12) F k ( t, m h,k − m − h,k , m h,k × ψ ) (cid:12)(cid:12)(cid:12) dt + E Z T (cid:12)(cid:12)(cid:12) F k ( t, m − h,k , ( m h,k − m − h,k ) × ψ ) (cid:12)(cid:12)(cid:12) dt ≤ ck / (cid:0) E [ k m h,k k L ( D T ) ] (cid:1) / (cid:18)(cid:0) E [ k∇ m h,k k L ( D T ) ] (cid:1) / + (cid:0) E [ k m h,k k L ( D T ) ] (cid:1) / (cid:19) + c (cid:0) E [ k m h,k − m − h,k k L ( D T ) ] (cid:1) / (cid:18)(cid:0) E [ k∇ m h,k k L ( D T ) ] (cid:1) / + (cid:0) E [ k m h,k k L ( D T ) ] (cid:1) / (cid:19) ≤ c ( h + k / ) , in which (6.1) and (6.2) are used to obtain the last inequality. This completes the proof ofthe lemma. (cid:3) In order to prove the convergence of random variables m h,k , we first state a result oftightness for the family L ( m h,k ). We then use the Skorohod theorem to define anotherprobability space and an almost surely convergent sequence defined in this space whoselimit is a weak martingale solution of equation (4.14). The proof of the following resultsare omitted since they are relatively simple modification of the proof of the correspondingresults from [13]. Lemma 6.7.
Assume that h and k approach , and further that (6.10) holds. Then the setof laws {L ( m h,k ) } on the Banach space C (cid:0) [0 , T ]; H − ( D ) (cid:1) ∩ L (0 , T ; L ( D )) is tight. Proposition 6.8.
Assume that h and k approach , and further that (6.10) holds. Thenthere exist: (a) a probability space (Ω ′ , F ′ , P ′ ) ; (b) a sequence { m ′ h,k } of random variables defined on (Ω ′ , F ′ , P ′ ) and taking values in C (cid:0) [0 , T ]; H − ( D ) (cid:1) ∩ L (0 , T ; L ( D )) ; and (c) a random variable m ′ defined on (Ω ′ , F ′ , P ′ ) and taking values in C (cid:0) [0 , T ]; H − ( D ) (cid:1) ∩ L (0 , T ; L ( D )) ,satisfying (1) L ( m h,k ) = L ( m ′ h,k ) , (2) m ′ h,k → m ′ in C (cid:0) [0 , T ]; H − ( D ) (cid:1) ∩ L (0 , T ; L ( D )) strongly, P ′ -a.s..Moreover, the sequence { m ′ h,k } satisfies E [ k m ′ h,k k H ( D T ) ] ≤ c, (6.14) E [ k| m ′ h,k | − k L ( D T ) ] ≤ c ( h + k ) , (6.15) k m ′ h,k k L ∞ ( D T ) ≤ c P ′ -a.s., (6.16) here, c is a positive constant only depending on { g i } i =1 , ··· ,q . We are now ready to state and prove our main theorem.
Theorem 6.9.
Assume that
T > , M ∈ H ( D ) satisfies ( ?? ) and g i ∈ W , ∞ ( D ) for i = 1 , · · · , q satisfy the homogeneous Neumann boundary condition. Then m ′ , the sequence { m ′ h,k } and the probability space (Ω ′ , F ′ , P ′ ) given by Proposition 6.8 satisfy: (1) the sequence of { m ′ h,k } converges to m ′ weakly in L (Ω ′ ; H ( D T )) ; and (2) (cid:0) Ω ′ , F ′ , ( F ′ t ) t ∈ [0 ,T ] , P ′ , M ′ (cid:1) is a weak martingale solution of (1.3) , where M ′ ( t ) := Z t m ′ ( t ) ∀ t ∈ [0 , T ] , a.e. x ∈ D. Proof.
From (6.16) and property (2) of Proposition 6.8, there exists a set V ⊂ Ω ′ such that P ′ ( V ) = 1 and for all ω ′ ∈ V there hold k m ′ h,k ( ω ′ ) k L ( D T ) ≤ c and m ′ h,k ( ω ′ ) → m ′ ( ω ′ ) in L ( D T ) strongly . Hence, by using Lebesgue’s dominated convergence theorem, we deduce(6.17) m ′ h,k → m ′ in L (Ω ′ ; L ( D T )) strongly , which implies from (6.14) that(6.18) m ′ h,k → m ′ in L (Ω ′ ; H ( D T )) weakly . EM FOR STOCHASTIC LANDAU–LIFSHITZ–GILBERT EQUATION 27
In order to prove Part (2), by noting Lemma 4.5 and Remark 4.4 we only need to provethat m ′ satisfies (4.5) and (4.14), namely(6.19) | m ′ ( t, x ) | = 1 , t ∈ (0 , T ) , x ∈ D, P ′ -a.s.and(6.20) I ( m ′ , ϕ ) = 0 P ′ -a.s. ∀ ϕ ∈ L (0 , T ; H ( D )) , where I ( m ′ , ϕ ) := λ (cid:10) m ′ × ∂ t m ′ , m ′ × ϕ (cid:11) L ( D T ) − λ (cid:10) ∂ t m ′ , m ′ × ϕ (cid:11) L ( D T ) − µ (cid:10) ∇ m ′ , ∇ ( m ′ × ϕ ) (cid:11) L ( D T ) − µ Z T F ( t, m ′ ( t, · ) , m ′ ( t, · ) × ϕ ( t, · )) dt. By using (6.15) and (6.17), we obtain (6.19) immediately.In order to prove (6.20), we first find the equation satisfied by m ′ h,k and then pass to thelimit when h and k approach 0.By using Lemmas 6.6 and property (1) of Proposition 6.8, it follows that for any ψ ∈ C ∞ (0 , T ; C ∞ ( D )) that there holds(6.21) E |I ( m ′ h,k , ψ ) | = O ( h + k / ) . To pass to the limit in (6.21), we first using (6.17)–(6.19) and the same arguments as in [13,Theorem 6.8] to obtain that as h and k tend to 0, (cid:10) m ′ h,k × ∂ t m ′ h,k , m ′ h,k × ϕ (cid:11) L (Ω ′ ; L ( D T )) → (cid:10) m ′ × ∂ t m ′ , m ′ × ϕ (cid:11) L (Ω ′ ; L ( D T )) , (6.22) (cid:10) ∂ t m ′ h,k , m ′ h,k × ϕ (cid:11) L (Ω ′ ; L ( D T )) → (cid:10) ∂ t m ′ , m ′ × ϕ (cid:11) L (Ω ′ ; L ( D T )) , (6.23) (cid:10) ∇ m ′ h,k , ∇ ( m ′ h,k × ϕ ) (cid:11) L (Ω ′ ; L ( D T )) → (cid:10) ∇ m ′ , ∇ ( m ′ × ϕ ) (cid:11) L (Ω ′ ; L ( D T )) . (6.24)Then, by using Remark 3.7 and (6.5) with F ∗ = F , we estimate E Z T (cid:12)(cid:12) F ( t, m ′ h,k ( t, · ) , m ′ h,k ( t, · ) × ϕ ( t, · )) − F ( t, m ′ ( t, · ) , m ′ ( t, · ) × ϕ ( t, · )) (cid:12)(cid:12) dt ≤ E Z T (cid:12)(cid:12) F ( t, m ′ h,k ( t, · ) − m ′ ( t, · ) , m ′ h,k ( t, · ) × ϕ ( t, · )) (cid:12)(cid:12) dt + E Z T (cid:12)(cid:12) F ( t, (cid:0) m ′ h,k ( t, · ) − m ′ ( t, · ) (cid:1) × ϕ ( t, · ) , m ′ ( t, · )) (cid:12)(cid:12) dt ≤ c k m ′ h,k − m ′ k L (Ω ′ ; L ( D T )) (cid:0) k∇ m ′ h,k k L (Ω ′ ; L ( D T )) + k m ′ h,k k L (Ω ′ ; L ( D T )) + k∇ m ′ k L (Ω ′ ; L ( D T )) + k m ′ k L (Ω ′ ; L ( D T )) (cid:1) . Since m ′ ∈ L (Ω ′ ; H ( D T )), it follows from (6.14) and (6.17) that(6.25) E (cid:2)Z T F ( t, m ′ h,k ( t, · ) , m ′ h,k ( t, · ) × ϕ ( t, · )) dt (cid:3) → E (cid:2)Z T F ( t, m ′ ( t, · ) , m ′ ( t, · ) × ϕ ( t, · )) dt (cid:3) , as h and k tend to 0. From (6.22)–(6.25) we deduce that E |I ( m ′ h,k , ψ ) − I ( m ′ , ψ ) | → , and hence, together with (6.21) E |I ( m ′ , ψ ) | = 0. This implies (6.20) which completes theproof of our main theorem. (cid:3) Appendix
For the reader’s convenience we will recall the following results, which are proved in [13].
Lemma 7.1.
For any real constants λ and λ with λ = 0 , if ψ , ζ ∈ R satisfy | ζ | = 1 ,then there exists ϕ ∈ R satisfying (7.1) λ ϕ + λ ϕ × ζ = ψ . As a consequence, if ζ ∈ H ( D T ) with | ζ ( t, x ) | = 1 a.e. in D T and ψ ∈ L (0 , T ; W , ∞ ( D )) ,then ϕ ∈ L (0 , T ; H ( D )) . Lemma 7.2.
For any v ∈ C ( D ) , v h ∈ V h and ψ ∈ C ∞ ( D T ) , k I V h v k L ∞ ( D ) ≤ k v k L ∞ ( D ) , k m − h,k × ψ − I V h ( m − h,k × ψ ) k L ([0 ,T ] , H ( D )) ≤ ch k m − h,k k L ([0 ,T ] , H ( D )) k ψ k W , ∞ ( D T ) , where m − h,k is defined in Defintion 6.1 The next lemma defines a discrete L p -norm in V h , equivalent to the usual L p -norm. Lemma 7.3.
There exist h -independent positive constants C and C such that for all p ∈ [1 , ∞ ] and u ∈ V h , C k u k p L p (Ω) ≤ h d N X n =1 | u ( x n ) | p ≤ C k u k p L p (Ω) , where Ω ⊂ R d , d=1,2,3. Acknowledgements
The authors acknowledge financial support through the ARC Discovery projects DP140101193and DP120101886. They are grateful to Vivien Challis for a number of helpful conversations.
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