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

Tonatiuh Sánchez-Vizuet

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.

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

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.
Paper Structure (5 sections, 9 theorems, 90 equations)

This paper contains 5 sections, 9 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. A Laplace domain system of BIEs
  3. Well posedness in the Laplace domain
  4. Time domain results
  5. Discrete and semi discrete error estimates.

Key Result

proposition thmcounterproposition

Let $({\bf j}_h,{\bf m}_h)$ be a solution pair to eq:WeakBIE. Then ${\bf E}_h^{scat}$ and ${\bf E}_h$ are defined using these densities through the integral representations eq:IntegralRepresentations if and only if they satisfy the following transmission problem

Theorems & Definitions (13)

  • proposition thmcounterproposition
  • proof
  • proposition thmcounterproposition
  • proof
  • theorem 1
  • proof
  • corollary thmcountercorollary
  • theorem 2: Sayas2016errata
  • theorem 3: Time-domain stability
  • proposition thmcounterproposition: Laplace domain error estimates
  • ...and 3 more