A finite element approximation for the quasi-static Maxwell--Landau--Lifshitz--Gilbert equations

Abstract

The quasi-static Maxwell-Landau-Lifshitz-Gilbert equations which describe the electromagnetic behaviour of a ferromagnetic material are highly nonlinear. Sophisticated numerical schemes are required to solve the equations, given their nonlinearity and the constraint that the solution stays on a sphere. We propose an implicit finite element solution to the problem. The resulting system of algebraic equations is linear which facilitates the solution process compared to nonlinear methods. We present numerical results to show the efficacy of the proposed method.

A convergent finite element approximation for the quasi-static Maxwell–Landau–Lifshitz–Gilbert equations

“…Gilbert introduces a different approach for description of damped precession in [9]: 5) in which µ = λ 2 1 + λ 2 2 . A proof of the equivalence between (1.5) and (1.1) can be found in [13]. It is easier to numerically solve (1.5) than (1.1) because the latter has a double cross term, namely m × (m × H eff ).…”

“…We choose the time step k = 10 −3 and the parameter θ in Algorithm 2.1 to be 0.7. The construction of the basis functions for W (j) h and Y h in this algorithm is discussed in [13]. At each iteration we need to solve a linear system of size (2N + M) × (2N + M), recalling that N is the number of vertices and M is the number of edges in the triangulation.…”

“…Since equations (1.1) and (1.5) are equivalent (a proof of which can be found in [13]), instead of solving (1.1)-(1.4) we solve (1.2)- (1.5). A weak solution of the problem is defined in the following definition.…”

“…It is required that k = o(h 2 ) when θ ∈ [0, 1 2 ), and k = o(h) when θ = 1 2 . The implementation aspect of the algorithm is reported in [13] where no convergence analysis is carried out.…”

A finite-difference scheme for a model of magnetization dynamics with inertial effects

“…The general framework (although without inertial effects, i.e., the case ζ = 0) was established in earlier papers by finite-difference/element methods; see, for instance, [7][8][9][10][11]. The following results concern systems coupling the LLG equation with the Maxwell system [4,[12][13][14][15][16]. In the case of magnetoelastic interactions, a finite-difference scheme is proposed and its stability discussed; see [17].…”