Computational homogenization of time-harmonic Maxwell's equations

TL;DR

This paper develops a practical numerical homogenization method for time-harmonic Maxwell's equations using the Localized Orthogonal Decomposition, extending previous theoretical results to natural boundary conditions and demonstrating effectiveness through numerical experiments.

Contribution

It adapts the LOD-based homogenization approach to natural boundary conditions and provides a computable method with numerical validation.

Findings

The method is effective in 2D and 3D simulations.

Boundary source values influence computational complexity and accuracy.

The approach extends previous theoretical results to practical numerical implementation.

Abstract

In this paper we consider a numerical homogenization technique for curl-curl-problems that is based on the framework of the Localized Orthogonal Decomposition and which was proposed in [D. Gallistl, P. Henning, B. Verf\"urth. SIAM J. Numer. Anal. 56-3:1570-1596, 2018] for problems with essential boundary conditions. The findings of the aforementioned work establish quantitative homogenization results for the time-harmonic Maxwell's equations that hold beyond assumptions of periodicity, however, a practical realization of the approach was left open. In this paper, we transfer the findings from essential boundary conditions to natural boundary conditions and we demonstrate that the approach yields a computable numerical method. We also investigate how boundary values of the source term can effect the computational complexity and accuracy. Our findings will be supported by various numerical experiments, both in $2D$ and $3D$.