A postprocessing technique for a discontinuous Galerkin discretization of time-dependent Maxwell's equations

TL;DR

This paper introduces a local postprocessing method for DG discretization of Maxwell's equations that achieves superconvergence, is computationally efficient, and is inspired by HDG techniques, improving solution accuracy in electromagnetic simulations.

Contribution

The paper develops a novel, local postprocessing technique for DG discretizations of Maxwell's equations that enhances convergence rates and is computationally inexpensive.

Findings

Postprocessed field converges one order faster in H(curl)-norm.

Method is local and independent per cell and time step.

Numerical experiments confirm superconvergence.

Abstract

We present a novel postprocessing technique for a discontinuous Galerkin (DG) discretization of time-dependent Maxwell's equations that we couple with an explicit Runge-Kutta time-marching scheme. The postprocessed electromagnetic field converges one order faster than the unprocessed solution in the H(curl)-norm. The proposed approach is local, in the sense that the enhanced solution is computed independently in each cell of the computational mesh, and at each time step of interest. As a result, it is inexpensive to compute, especially if the region of interest is localized, either in time or space. The key ideas behind this postprocessing technique stem from hybridizable discontinuous Galerkin (HDG) methods, which are equivalent to the analyzed DG scheme for specific choices of penalization parameters. We present several numerical experiments that highlight the superconvergence properties of the postprocessed electromagnetic field approximation.