We propose an implicit, fully discrete scheme for the numerical solution of the Maxwell-Landau-Lifshitz-Gilbert equation which is based on linear finite elements and satisfies a discrete sphere constraint as well as a discrete energy law. As numerical parameters tend to zero, solutions weakly accumulate at weak solutions of the Maxwell-Landau-Lifshitz-Gilbert equation. A practical linearization of the nonlinear scheme is proposed and shown to converge for certain scalings of numerical parameters. Computational studies are presented to indicate finite-time blowup behavior and to study combined electromagnetic phenomena in ferromagnets for benchmark problems.

We propose a convergent finite element based discretization of the stochastic Landau-Lifshitz-Gilbert equation. Solutions of the discretization satisfy the sphereconstraint at nodal points of the spatial triangulation, and have finite energies. Computational studies are included.

We consider a system of nonlinear PDEs modeling nematic electrolytes, and construct a dissipative solution with the help of its implementable, structure-inheriting and space–time discretization. Computational studies are performed to study the mutual effects of electric, elastic and viscous effects onto the molecules in a nematic electrolyte.

We consider a finite element approximation of a phase field model for the evolution of voids by surface diffusion in an electrically conducting solid. The phase field equations are given by the nonlinear degenerate parabolic system, 1] on u and flux boundary conditions on all three equations. Here γ ∈ R >0 , α ∈ R ≥0 , is a non-smooth double well potential, and c(u) := 1 + u, b(u) := 1 − u 2 are degenerate coefficients. On extending existing results for the simplified two dimensional phase field model, we show stability bounds for our approximation and prove convergence, and hence existence of a solution to this nonlinear degenerate parabolic system in three space dimensions. Furthermore, a new iterative scheme for solving the resulting nonlinear discrete system is introduced and some numerical experiments are presented.Keywords Void electromigration · Surface diffusion · Phase field model · Degenerate Cahn-Hilliard equation · Fourth order degenerate parabolic system · Finite elements · Convergence analysis · Multigrid methods IntroductionIn the recent paper [9], abbreviated to BNS throughout this paper, the authors proposed and analysed a fully practical finite element approximation for a phase field model describing