A mixed element scheme of Helmholtz transmission eigenvalue problem for anisotropic media

TL;DR

This paper introduces a mixed finite element scheme for the Helmholtz transmission eigenvalue problem in anisotropic media, achieving optimal convergence and addressing computational challenges with large sparse eigenvalue problems.

Contribution

The paper develops a novel mixed formulation and discretization for anisotropic Helmholtz transmission eigenvalues, improving computational efficiency and eigenvalue approximation accuracy.

Findings

Achieves optimal convergence rates on convex and nonconvex domains.

Successfully reduces eigenvalue problem size through deflation techniques.

Demonstrates effectiveness with extensive numerical examples.

Abstract

In this paper, we study the Helmholtz transmission eigenvalue problem for inhomogeneous anisotropic media with the index of refraction $n(x)\equiv 1$ in two and three dimension. Starting with a nonlinear fourth order formulation established by Cakoni, Colton and Haddar [2009], by introducing some auxiliary variables, we present an equivalent mixed formulation for this problem, followed up with the finite element discretization. Using the proposed scheme, we rigorously show that the optimal convergence rate for the transmission eigenvalues both on convex and nonconvex domains can be expected. Moreover, by this scheme, we will obtain a sparse generalized eigenvalue problem whose size is so demanding even with a coarse mesh that its smallest few real eigenvalues fail to be solved by the shift and invert method. We partially overcome this critical issue by deflating the almost all of the $\infty$ eigenvalue of huge multiplicity, resulting in a drastic reduction of the matrix size without deteriorating the sparsity. Extensive numerical examples are reported to demonstrate the effectiveness and efficiency of the proposed scheme.