Computation of Transmission Eigenvalues for Elastic Waves

TL;DR

This paper develops and analyzes numerical methods for computing the smallest elasticity transmission eigenvalues, which are crucial in inverse scattering, using finite element approaches and a secant method.

Contribution

It introduces a new nonlinear function approach and proves convergence for computing elasticity transmission eigenvalues, with verification through finite element methods.

Findings

Effective computation of transmission eigenvalues demonstrated

Convergence of the secant method proved

Numerical examples verify theoretical results

Abstract

The goal of this paper is to develop numerical methods computing a few smallest elasticity transmission eigenvalues, which are of practical importance in inverse scattering theory. The problem is challenging since it is nonlinear, non-self-adjoint, and of fourth order. We construct a nonlinear function whose values are generalized eigenvalues of a series of self-adjoint fourth order problems. The roots of the function are the transmission eigenvalues. Using an $H^2$-conforming finite element for the self-adjoint fourth order eigenvalue problems, we employ a secant method to compute the roots of the nonlinear function. The convergence of the proposed method is proved. In addition, a mixed finite element method is developed for the purpose of verification. Numerical examples are presented to verify the theory and demonstrate the effectiveness of the two methods.