The Eddy Current--LLG Equations: FEM-BEM Coupling and A Priori Error Estimates

TL;DR

This paper develops and analyzes a FEM-BEM coupled numerical method for the nonlinear eddy current and LLG equations, proving convergence and error estimates, with numerical validation.

Contribution

It introduces a coupled FEM-BEM approach for the nonlinear eddy current-LLG system and provides rigorous convergence and error analysis.

Findings

Unconditional weak convergence of finite-element solutions.

A priori error estimates under smoothness assumptions.

Numerical experiments confirm theoretical results.

Abstract

We analyze 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 discretized 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. We prove unconditional weak convergence (of a subsequence) of the finite-element solutions towards a weak solution. We establish a priori error estimates if a sufficiently smooth strong solution exists. Numerical experiments underlining the theoretical results are presented.