The Eddy Current-LLG Equations-Part I: FEM-BEM Coupling

TL;DR

This paper presents a finite-element/boundary-element coupling method for simulating the nonlinear eddy current and Landau-Lifshitz-Gilbert equations, proving convergence of the numerical solutions in an unbounded domain.

Contribution

It introduces a coupled FEM-BEM numerical scheme for the nonlinear eddy current-LLG system and proves its unconditional weak convergence.

Findings

Proved weak convergence of the finite-element solutions.

Developed a numerical approach requiring only two linear systems per time step.

Analyzed a strongly nonlinear coupled system in unbounded domains.

Abstract

We analyse a numerical method for the coupled system of the eddy current equations in $\mathbb{R}^3$ with the Landau-Lifshitz-Gilbert equation in a bounded domain. The unbounded domain is discretised by means of finite-element/boundary-element coupling. Even though the considered problem is strongly nonlinear, the numerical approach is constructed such that only two linear systems per time step have to be solved. In this first part of the paper, we prove unconditional weak convergence (of a subsequence) of the finite-element solutions towards a weak solution. A priori error estimates will be presented in the second part.