Numerical analysis for electromagnetic scattering from nonlinear boundary conditions

TL;DR

This paper develops a numerical method for electromagnetic scattering problems involving nonlinear boundary conditions, deriving and analyzing a nonlinear boundary integral equation system with proven error bounds and demonstrating its effectiveness through numerical experiments.

Contribution

It introduces a novel boundary integral equation approach for nonlinear electromagnetic scattering with rigorous error analysis and practical numerical implementation.

Findings

The method achieves optimal convergence rates.

Error bounds are explicitly derived.

Numerical experiments confirm theoretical results.

Abstract

This work studies time-dependent electromagnetic scattering from obstacles whose interaction with the wave is fully determined by a nonlinear boundary condition. In particular, the boundary condition studied in this work enforces a power law type relation between the electric and magnetic field along the boundary. Based on time-dependent jump conditions of classical boundary operators, we derive a nonlinear system of time-dependent boundary integral equations that determines the tangential traces of the scattered electric and magnetic fields. These fields can subsequently be computed at arbitrary points in the exterior domain by evaluating a time-dependent representation formula. Fully discrete schemes are obtained by discretising the nonlinear system of boundary integral equations with Runge--Kutta based convolution quadrature in time and Raviart--Thomas boundary elements in space. Error bounds with explicitly stated convergence rates are proven, under the assumption of sufficient regularity of the exact solution. The error analysis is conducted through novel techniques based on time-discrete transmission problems and the use of a new discrete partial integration inequality. Numerical experiments illustrate the use of the proposed method and provide empirical convergence rates.