A virtual element approximation for the modified transmission eigenvalues for natural materials

TL;DR

This paper introduces a virtual element method for approximating modified transmission eigenvalues in inverse scattering problems involving natural materials, addressing non-coercivity issues and providing error estimates and numerical validation.

Contribution

It presents a novel virtual element approximation approach for the modified transmission eigenvalue problem, including well-posedness analysis, error estimates, and numerical demonstrations.

Findings

Method effectively approximates eigenvalues in complex materials.

Error estimates validate the accuracy of the virtual element approach.

Numerical examples confirm the method's effectiveness.

Abstract

In this paper, we discuss a virtual element approximation for the modified transmission eigenvalue problem in inverse scattering for natural materials. In this case, due to the positive artificial diffusivity parameter in the considered problem, the sesquilinear form at the left end of the variational form is not coercive. We first demonstrate the well-posedness of the discrete source problem using the $\mathds{T}$-coercivity property, then provide the a priori error estimates for the approximate eigenspaces and eigenvalues, and finally reports several numerical examples. The numerical experiments show that the proposed method is effective