Numerical homogenization for indefinite H(curl)-problems

TL;DR

This paper develops a numerical homogenization method for indefinite H(curl)-problems, specifically for time-harmonic Maxwell's equations with rough coefficients, achieving optimal error estimates through a stable, quasi-local operator.

Contribution

It extends previous homogenization techniques to indefinite problems, providing a stable, quasi-local operator for error correction in Maxwell's equations.

Findings

Achieves order optimal error estimates w.r.t. mesh size.

Extends previous H(curl) homogenization methods to indefinite problems.

Provides a stable, quasi-local correction operator for Maxwell's equations.

Abstract

In this paper, we present a numerical homogenization scheme for indefinite, time-harmonic Maxwell's equations involving potentially rough (rapidly oscillating) coefficients. The method involves an $\mathbf{H}(\mathrm{curl})$-stable, quasi-local operator, which allows for a correction of coarse finite element functions such that order optimal (w.r.t. the mesh size) error estimates are obtained. To that end, we extend the procedure of [D. Gallistl, P. Henning, B. Verf\"urth, Numerical homogenization of H(curl)-problems, arXiv:1706.02966, 2017] to the case of indefinite problems. In particular, this requires a careful analysis of the well-posedness of the corrector problems as well as the numerical homogenization scheme.