Accurate discretization of a non-linear micromagnetic problem

In this paper we propose a finite element discretization of the Maxwell-Lan dau-Lifchitz-Gilbert equations governing the electromagnetic field in a fer romagnetic material. Our point of view is that it is desirable for the disc rete problem to possess conservation properties similar to the continuous s ystem. We first prove the existence of a new class of Liapunov functions fo r the continuous problem, and then for a variational formulation of the con tinuous problem. We also show a special continuous dependence result. Then we propose a family of mass-lumped finite element schemes for the problem. For the resulting semi-discrete problem we show that magnetization is conse rved and that semi-discrete Liapunov functions exist. Finally we show the r esults of some computations that show the behavior of the fully discrete Li apunov functions. (C) 2001 Elsevier Science B.V. All rights reserved.