Divergence error based $p$-adaptive discontinuous Galerkin solution of time-domain Maxwell's equations

TL;DR

This paper introduces a $p$-adaptive discontinuous Galerkin method for time-domain Maxwell's equations, utilizing divergence error as an efficient indicator to adaptively refine the solution, reducing computational costs while maintaining high accuracy.

Contribution

The novel use of divergence error as a proxy for truncation error to drive $p$-adaptivity in DG time-domain methods for electromagnetic problems.

Findings

Divergence error correlates with truncation error and can guide adaptive refinement.

The method achieves high-order solutions with reduced computational cost.

Adaptive $p$-refinement improves efficiency over uniform polynomial degree approaches.

Abstract

A $p$-adaptive discontinuous Galerkin time-domain method is developed to obtain high-order solutions to electromagnetic scattering problems. A novel feature of the proposed method is the use of divergence error to drive the $p$-adaptive method. The nature of divergence error is explored and that it is a direct consequence of the act of discretization is established. Its relation with relative truncation error is formed which enables the use of divergence error as an inexpensive proxy to truncation error. Divergence error is used as an indicator to dynamically identify and assign spatial operators of varying accuracy to substantial regions in the computational domain. This results in a reduced computational cost than a comparable discontinuous Galerkin time-domain solution using uniform degree piecewise polynomial bases throughout.