Stable and convergent fully discrete interior-exterior coupling of Maxwell's equations

TL;DR

This paper introduces a stable, convergent fully discrete numerical method for Maxwell's equations with transparent boundary conditions, combining interior DG, boundary elements, and convolution quadrature.

Contribution

It develops a novel explicit-implicit coupling scheme using DG, boundary elements, and convolution quadrature, with proven stability and convergence.

Findings

Proven stability of the discretization scheme.

Demonstrated convergence of the method.

Implemented a simple stabilization term for full discretization.

Abstract

Maxwell's equations are considered with transparent boundary conditions, for initial conditions and inhomogeneity having support in a bounded, not necessarily convex three-dimensional domain or in a collection of such domains. The numerical method only involves the interior domain and its boundary. The transparent boundary conditions are imposed via a time-dependent boundary integral operator that is shown to satisfy a coercivity property. The stability of the numerical method relies on this coercivity and on an anti-symmetric structure of the discretized equations that is inherited from a weak first-order formulation of the continuous equations. The method proposed here uses a discontinuous Galerkin method and the leapfrog scheme in the interior and is coupled to boundary elements and convolution quadrature on the boundary. The method is explicit in the interior and implicit on the boundary. Stability and convergence of the spatial semidiscretization are proven, and with a computationally simple stabilization term, this is also shown for the full discretization.