We propose a θ-linear scheme for the numerical solution of the quasi-static Maxwell-Landau-Lifshitz-Gilbert (MLLG) equations. Despite the strong nonlinearity of the Landau-Lifshitz-Gilbert equation, the proposed method results in a linear system at each time step. We prove that as the time and space steps tend to zero (with no further conditions when θ ∈ ( 1 2 , 1]), the finite element solutions converge weakly to a weak solution of the MLLG equations. Numerical results are presented to show the applicability of the method.

We propose a numerical integrator for the coupled system of the eddy-current equation with the nonlinear Landau-Lifshitz-Gilbert equation. The considered effective field contains a general field contribution, and we particularly cover exchange, anisotropy, applied field and magnetic field (stemming from the eddycurrent equation). Even though the considered problem is nonlinear, our scheme requires only the solution of two linear systems per time-step. Moreover, our algorithm decouples both equations so that in each time-step, one linear system is solved for the magnetization, and afterwards one linear system is solved for the magnetic field. Unconditional convergence -at least of a subsequencetowards a weak solution is proved, and our analysis even provides existence of such weak solutions. Numerical experiments with micromagnetic benchmark problems underline the performance and the stability of the proposed algorithm.

A time-fractional Fokker-Planck initial-boundary value problem is considered, with differential operator ut − ∇ • (∂ 1−α t κα∇u − F∂ 1−α t u), where 0 < α < 1. The forcing function F = F(t, x), which is more difficult to analyse than the case F = F(x) investigated previously by other authors. The spatial domain Ω ⊂ R d , where d ≥ 1, has a smooth boundary. Existence, uniqueness and regularity of a mild solution u is proved under the hypothesis that the initial data u 0 lies in L 2 (Ω). For 1/2 < α < 1 and u 0 ∈ H 2 (Ω) ∩ H 1 0 (Ω), it is shown that u becomes a classical solution of the problem. Estimates of time derivatives of the classical solution are derived-these are known to be needed in numerical analyses of this problem. 2000 Mathematics Subject Classification. 35R11.

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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.