An optimal control-based numerical method for scalar transmission problems with sign-changing coefficients

TL;DR

This paper introduces a novel numerical method based on optimal control for scalar transmission problems with sign-changing coefficients, common in electromagnetism, proving convergence without restrictive assumptions and demonstrating effectiveness through 2D experiments.

Contribution

The paper develops a new optimal control-based numerical approach for sign-changing coefficient problems, with proven convergence under minimal conditions, unlike previous methods.

Findings

Convergence of the method is proven without restrictive conditions.

The method effectively handles problems with sign-changing coefficients.

Numerical experiments validate the approach in 2D scenarios.

Abstract

In this work, we present a new numerical method for solving the scalar transmission problem with sign-changing coefficients. In electromagnetism, such a transmission problem can occur if the domain of interest is made of a classical dielectric material and a metal or a metamaterial, with for instance an electric permittivity that is strictly negative in the metal or metamaterial. The method is based on an optimal control reformulation of the problem. Contrary to other existing approaches, the convergence of this method is proved without any restrictive condition. In particular, no condition is imposed on the a priori regularity of the solution to the problem, and no condition is imposed on the meshes, other than that they fit with the interface between the two media. Our results are illustrated by some (2D) numerical experiments.