This talk summarizes joint works by the speaker and Anne-Sophie Bonnet-Ben
Dhia, Lucas Chesnel, Camille Carvalho and Juan-Pablo Borthagaray on how to solve problems with discontinuous, sign-changing coefficients.
In electromagnetic theory, the effective response of specifically designed materials is modeled by strictly negative coefficients: these are the so-called negative materials. Transmission problems with discontinuous, sign-changing coefficients then occur in the presence of negative materials surrounded by classical materials. For general geometries, establishing Fredholmness of these transmission problems is well-understood thanks to the T-coercivity approach.
Let σ be a parameter that is strictly positive in some part of the computational domain, and strictly negative elsewhere. We focus on the scalar source problem: find u such that divσ∇u − ω^2 u = f plus boundary condition, where f is some data and ω is the pulsation. Denoting by σ+ the strictly positive value, and by σ− the strictly negative value, one can prove that there exists a critical interval, such that the scalar source problem is well-posed in the Fredholm sense if, and only, if, the ratio σ−/σ+ lies outside the critical interval. One may derive similar results for the related eigenvalue problem.
When the ratio σ−/σ+ lies outside the critical interval, the shape of the interface separating the two materials must be taken into account to solve the problems numerically. We propose a treatment which allows to design meshing rules for an arbitrary polygonal interface and then recover standard error estimates.
This treatment relies on the use of simple geometrical transforms to define the meshes.
In a last part, we discuss the case where the ratio σ−/σ+ lies inside the critical interval.