The Eddy Current--LLG Equations: FEM-BEM Coupling and A Priori Error Estimates
This work provides a rigorous numerical analysis for a challenging nonlinear PDE system in electromagnetism, but the results are theoretical and domain-specific.
The authors develop a FEM-BEM coupling method for the eddy current-LLG equations in unbounded domains, requiring only two linear solves per time step, and prove unconditional weak convergence and a priori error estimates.
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.