Temporal convergence of extrapolated BDF-2 scheme for the Maxwell-Landau-Lifshitz equations

Abstract

The Maxwell-Landau-Lifshitz equations are used to describe certain electromagnetic phenomena and are a strongly nonlinear parabolic system. This paper focuses on the optimal error estimates of a linearized second-order BDF semi-discrete scheme for the numerical approximation of the solution to the Maxwell-Landau-Lifshitz equations. The proposed algorithm is a semi-implicit scheme by using the extrapolation technique and the implicit-explicit method to linearize the nonlinear terms. Furthermore, the second-order temporal convergence rate $\mathcal{O}({\tau }^2)$ is derived, where τ is the time step size. Finally, the numerical experiment is presented to confirm the theoretical result.

Introduction

The Landau-Lifshitz (LL) equation describes the evolution of magnetization in continuum ferromagnets and plays a fundamental role in the understanding of non-equilibrium magnetism. Let $\mathrm{\Omega }\subset {\mathbf{\text{R}}}^3$ be a bounded and convex domain with smooth boundary ∂Ω. The unknown magnetization m satisfies the following nonlinear parabolic problem in ${\mathbf{\text{R}}}^{+}\times \mathrm{\Omega }$:

$${\partial }_t\mathbf{\text{m}}-\mathbf{\text{m}}\times \mathrm{\Delta }\mathbf{\text{m}}+\lambda \mathbf{\text{m}}\times (\mathbf{\text{m}}\times \mathrm{\Delta }\mathbf{\text{m}})=0,$$

where $\lambda >0$ represents the damping constant. Multiplying (1.1) by m, it is easy to see that

$$\frac{d}{dt}|\mathbf{\text{m}}(t,x){|}^2=0,\text{for any}t>0,x\in \mathrm{\Omega },$$

which means that $|\mathbf{\text{m}}|$ remains unchanged in time. Usually, one assumes that the initial value $\mathbf{\text{m}}(0)={\mathbf{\text{m}}}_0\in {\mathbb{S}}^2$, where ${\mathbb{S}}^2$ is the unit sphere in ${\mathbf{\text{R}}}^3$. Then the magnetization m always satisfies

$$|\mathbf{\text{m}}(t,x)|=1\text{for any}t>0,x\in \mathrm{\Omega }.$$

In designing numerical algorithms for the LL equation, a crucial issue is how to preserve the point-wise constraint (1.2) of numerical solutions in fully discrete level. There are two different strategies to deal with it: one implementing the unit length constraint approximately and another preserving the unit length constraint exactly. For the first strategy, the point-wise constraint (1.2) can be relaxed by introducing some penalty terms [26], [27]. On the other hand, we note that the point-wise constraint (1.2) can be deduced from the equation (1.1), and some linearized semi-implicit schemes were proposed and studied in [5], [13], [20], in which only the solution of a linear system is required at each time step. The last technique in the first strategy is the projection free method. For example, the projection free method was applied to the decoupled integration of the coupling of the Landau–Lifshitz–Gilbert (LLG) equation with a spin diffusion equation for the spin accumulation in [1]. The error estimate of the projection free method for the LLG equation was presented in [18]. Akrivis et al. presented an orthogonal projection-free method based on $L^2$-averaged orthogonality constraints and higher order nonconforming finite element approximations [3].

For the second strategy, one is to design a suitable scheme embedding the constraint implicitly as the LL equation does. Most of these schemes are nonlinear and fully implicit and an inner iteration is required for solving the nonlinear discrete system [11], [15]. Another way preserving the point-wise constraint (1.2) of numerical solutions is the projection method. An orthogonal sphere projection method was firstly introduced by Alouges and Jaisson in [4] for the LLG equation, where one has to build a new finite element space which is orthogonal point-wisely to the finite element solution at the previous time step. A simple and natural projection method is to update the intermediate magnetization by projecting it onto the unit sphere ${\mathbb{S}}^2$ to get the end-of-step magnetization [17]. Recently, optimal error estimates for such projection method are proved in [6] for the finite difference methods of Euler and Crank-Nicolson schemes and in [7] for the k-th order finite element method of Euler scheme, where $k\ge 2$.

In this paper, we consider the Maxwell-Landau-Lifshitz (MLL) equations which are used to describe certain electromagnetic phenomena in a ferromagnet occupying the domain Ω and are governed by the following coupled system in $(0,T]\times \mathrm{\Omega }$:

$${\partial }_t\mathbf{\text{m}}-\mathbf{\text{m}}\times (\mathrm{\Delta }\mathbf{\text{m}}+\mathbf{\text{H}})+\lambda \mathbf{\text{m}}\times (\mathbf{\text{m}}\times (\mathrm{\Delta }\mathbf{\text{m}}+\mathbf{\text{H}}))=0,$$

$${\partial }_t\mathbf{\text{H}}+\text{curl}(\text{curl}\mathbf{\text{H}})=-\beta {\partial }_t\mathbf{\text{m}},$$

where $\beta \ge 0$ represents the magnetic permeability. The unknown functions m and H denote the magnetization and the magnetic field, respectively. The system (1.3)-(1.4) is supplemented with initial conditions

$$\mathbf{\text{m}}(0,x)={\mathbf{\text{m}}}_0(x)\in {\mathbb{S}}^2,\mathbf{\text{H}}(0,x)={\mathbf{\text{H}}}_0(x)\text{for}x\in \mathrm{\Omega }$$

and boundary conditions

$$\mathrm{\nabla }\mathbf{\text{m}}\cdot \mathbf{\text{n}}=0,\mathbf{\text{H}}\times \mathbf{\text{n}}=0,\text{on}(0,T]\times \partial \mathrm{\Omega },$$

where n is the unit outward normal vector to ∂Ω.

It is clear that if ${\mathbf{\text{m}}}_0\in {\mathbb{S}}^2$, the solution m to (1.3) still satisfies the point-wise constraint

$$|\mathbf{\text{m}}(t,x)|=1,\forall x\in \mathrm{\Omega },0\le t\le T.$$

By (1.2) and the following vector formula:

$$\mathbf{\text{a}}\times (\mathbf{\text{b}}\times \mathbf{\text{c}})=(\mathbf{\text{a}}\cdot \mathbf{\text{c}})\mathbf{\text{b}}-(\mathbf{\text{a}}\cdot \mathbf{\text{b}})\mathbf{\text{c}},\mathbf{\text{a}},\mathbf{\text{b}},\mathbf{\text{c}}\in {\mathbf{\text{R}}}^3,$$

the system (1.3)-(1.4) are equivalent to

$${\partial }_t\mathbf{\text{m}}-\lambda \mathrm{\Delta }\mathbf{\text{m}}-\mathbf{\text{m}}\times \mathrm{\Delta }\mathbf{\text{m}}-\lambda |\mathrm{\nabla }\mathbf{\text{m}}{|}^2\mathbf{\text{m}}=\mathbf{\text{m}}\times \mathbf{\text{H}}-\lambda \mathbf{\text{m}}\times (\mathbf{\text{m}}\times \mathbf{\text{H}}),$$

$${\partial }_t\mathbf{\text{H}}+\text{curl}(\text{curl}\mathbf{\text{H}})=-\beta {\partial }_t\mathbf{\text{m}}$$

by noting the fact that the point-wise constraint (1.7) can be deduced from (1.5) and (1.8). Multiplying (1.8) by 2m and setting $z=|\mathbf{\text{m}}{|}^2$ leads to

$$z_t-\lambda \mathrm{\Delta }z=2\lambda |\mathrm{\nabla }\mathbf{\text{m}}{|}^2(z-1),$$

where we use

$$z_t=2({\mathbf{\text{m}}}_t\cdot \mathbf{\text{m}}),\mathrm{\Delta }z=2(\mathrm{\Delta }\mathbf{\text{m}}\cdot \mathbf{\text{m}})+2|\mathrm{\nabla }\mathbf{\text{m}}{|}^2.$$

Denote $w=z-1$. Then (1.10) reduces to

$$w_t-\lambda \mathrm{\Delta }w=2\lambda |\mathrm{\nabla }\mathbf{\text{m}}{|}^{2}w.$$

Testing (1.11) by 2w and using the Gronwall's inequality, we obtain

$${\Vert w(t)\Vert }_{L^2}^2+2\lambda \underset{0}{\overset{t}{\int }}{\Vert \mathrm{\nabla }w(s)\Vert }_{L^2}^{2}ds\le {\Vert w(0)\Vert }_{L^2}^2\exp \left(4\lambda \underset{0}{\overset{t}{\int }}{\Vert \mathrm{\nabla }\mathbf{\text{m}}(s)\Vert }_{L^2}^{2}ds\right)=0$$

for any $t>0$. In the above inequality, we noted $w(0)=|{\mathbf{\text{m}}}_0{|}^2-1=0$ from (1.5). Thus, $w\equiv 0$ which implies that the constraint (1.7) holds.

The construction of efficiently numerical schemes for the MLL equations is difficult due to the nonlinear character of (1.8) and the point-wise constraint (1.2). The nonlinear implicit finite element schemes were proposed for solving the Maxwell-Landau-Lifshitz-Gilbert (MLLG) equations numerically in [8], [9], where the unit-length constraint of numerical solutions can be preserved at all nodes and the unconditional convergence of the algorithm was proved in [8]. Based on the orthogonal sphere projection method proposed in [4], two linear finite element schemes for the MLLG equations were studied in [10] and the authors proved that the numerical solutions converge towards weak solutions of the MLLG equations. We noted that there have some works on numerical schemes for the coupling of LLG equation with the quasi-static eddy-current formulation of the Maxwell equations, such as the projection-free methods [18], the orthogonal projection methods [19], [22], [23].

In this paper, we propose and study the second-order backward differentiation formula (BDF-2) time-discrete scheme for the approximation of the MLL equations (1.8)-(1.9) with initial and boundary conditions (1.5)-(1.6). Since the point-wise constraint (1.2) can be deduced from (1.8), the proposed BDF-2 scheme is a semi-implicit scheme by using the implicit-explicit and extrapolation methods to linearize the nonlinear terms, and the unit-length of numerical solutions is satisfied approximately. The first-order Euler semi-implicit time-discrete scheme has been studied in [14], where the author proved the optimal convergence rate $\mathcal{O}(\tau )$ for the magnetization in $L^2$-norm and the sub-optimal convergence rate $\mathcal{O}({\tau }^{1/2})$ for the magnetization and the magnetic field in $H^1$-norm and $L^2$-norm, respectively. For the proposed BDF-2 scheme in this paper, the optimal temporal convergence rate $\mathcal{O}({\tau }^2)$ is derived for the magnetization and the magnetic field in $H^1$-norm and $\mathbf{\text{H}}(\text{curl})$-norm, respectively, by using the method of mathematical induction, where $\mathbf{\text{H}}(\text{curl})$-norm is defined in next section.

This paper is organized as follows. In next section, we present the BDF-2 time-discrete scheme for (1.8)-(1.9) with (1.5)-(1.6), and state the main result on the temporal convergence rate under some regularity assumptions. In Section 3, we recall some known inequalities used in the proof of main result, and give the proof by the method of mathematical induction. In Section 4, we provide numerical results to confirm the theoretical result on the temporal convergence rate $\mathcal{O}({\tau }^2)$. Finally, a conclusion section is given.