Finite Element Approximation of Transmission Eigenvalues for Anisotropic Media

TL;DR

This paper develops a finite element method for approximating transmission eigenvalues in anisotropic media, proving convergence and introducing a spectral indicator method for computation, with numerical validation.

Contribution

It formulates the anisotropic transmission eigenvalue problem as a holomorphic Fredholm operator eigenvalue problem and proves convergence of the finite element approximation.

Findings

Convergence of the finite element method is rigorously established.

A spectral indicator method effectively computes eigenvalues.

Numerical examples validate the theoretical results.

Abstract

The transmission eigenvalue problem arises from the inverse scattering theory for inhomogeneous media and has important applications in many qualitative methods. The problem is posted as a system of two second order partial differential equations and is essentially nonlinear, non-selfadjoint, and of higher order. It is nontrivial to develop effective numerical methods and the proof of convergence is challenging. In this paper, we formulate the transmission eigenvalue problem for anisotropic media as an eigenvalue problem of a holomorphic Fredholm operator function of index zero. The Lagrange finite elements are used for discretization and the convergence is proved using the abstract approximation theory for holomorphic operator functions. A spectral indicator method is developed to compute the eigenvalues. Numerical examples are presented for validation.