This research proposes an effective numerical method for solving transmission problems with changing coefficients in specific geometries, such as those with corners. In electromagnetism, such transmission problems can arise when the domain of interest consists of a conventional dielectric material and a metal or metamaterial, where, for example, the electric permittivity is strictly negative in the metal or metamaterial.
We propose a new numerical method based on reformulating the problem as an optimal control problem. Unlike other existing approaches, the convergence of this method is demonstrated without imposing any restrictive conditions. Specifically, no preconditions are required regarding the a priori regularity of the solution, and no conditions are needed for the meshes, except that they must align with the interface between the two media. Our results are illustrated by several two-dimensional numerical experiments.
We propose a new numerical method based on reformulating the problem as an optimal control problem. Unlike other existing approaches, the convergence of this method is demonstrated without imposing any restrictive conditions. Specifically, no preconditions are required regarding the a priori regularity of the solution, and no conditions are needed for the meshes, except that they must align with the interface between the two media. Our results are illustrated by several two-dimensional numerical experiments.
[Picture]
A mesh of a geometry with a corner interface (on the left). Behavior of relative errors in $L^{2}$ and $H^{1}_{0}$ norm with respect to the meshsize $h$, with the optimal convergence rates observed for two different contrasts (on the center and right).