A boundary integral equation formulation for transient electromagnetic transmission problems on Lipschitz domains

TL;DR

This paper develops a boundary integral equation approach for transient electromagnetic transmission problems on Lipschitz domains, providing stability, solvability, and error estimates for numerical discretizations.

Contribution

It introduces a coupled boundary integral formulation for electromagnetic transmission, proves stability and unique solvability in Laplace and time domains, and derives error estimates for numerical schemes.

Findings

Stable and uniquely solvable boundary integral system in Laplace domain

Proven stability and solvability in the time domain

Derived error estimates for Galerkin and Convolution Quadrature discretizations

Abstract

We propose a boundary integral formulation for the dynamic problem of electromagnetic scattering and transmission by homogeneous dielectric obstacles. In the spirit of Costabel and Stephan, we use the transmission conditions to reduce the number of unknown densities and to formulate a system of coupled boundary integral equations describing the scattered and transmitted waves. The system is transformed into the Laplace domain where it is proven to be stable and uniquely solvable. The Laplace domain stability estimates are then used to establish the stability and unique solvability of the original time domain problem. Finally, we show how the bounds obtained in both Laplace and time domains can be used to derive error estimates for semi discrete Galerkin discretizations in space and for fully discrete numerical schemes that use Convolution Quadrature for time discretization and a conforming Galerkin method for discretization of the space variables.