A discontinuous least squares finite element method for time-harmonic Maxwell equations

TL;DR

This paper introduces a novel discontinuous least squares finite element method for solving time-harmonic Maxwell equations, demonstrating stability and convergence without mesh constraints through theoretical analysis and numerical experiments.

Contribution

It develops a new discontinuous least squares finite element scheme for Maxwell equations with proven stability and convergence, expanding the numerical methods available for these problems.

Findings

Method is stable without mesh size restrictions

Convergence orders are established in energy and L2 norms

Numerical results confirm theoretical error estimates

Abstract

We propose and analyze a discontinuous least squares finite element method for solving the indefinite time-harmonic Maxwell equations. The scheme is based on the $L^2$ norm least squares functional with the weak imposition of the continuity across the interior faces. We minimize the functional over the piecewise polynomial spaces to seek numerical solutions. The method is shown to be stable without any constraint on the mesh size. We prove the convergence orders under both the energy norm and the $L^2$ norm. Numerical results in two and three dimensions are presented to verify the error estimates.