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

TL;DR

This paper introduces a linear finite element scheme for solving the nonlinear quasi-static Maxwell-Landau-Lifshitz-Gilbert equations, proving convergence and demonstrating practical effectiveness through numerical experiments.

Contribution

A novel $ heta$-linear finite element method for MLLG equations that simplifies computation while ensuring convergence to weak solutions.

Findings

The method produces linear systems at each step despite nonlinearity.

Convergence of solutions is proven as discretization parameters tend to zero.

Numerical results confirm the method's applicability and accuracy.

Abstract

We propose a $\theta$-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 $\theta\in(1/2,1]$), 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.