Error Analysis for the Finite Element Approximation of Transmission Eigenvalues

In this paper we consider the transmission eigenvalue problem corresponding to acoustic scattering by a bounded isotropic inhomogeneous object in two dimensions. This is a non self-adjoint eigenvalue problem for a quadratic pencil of operators. In particular we are concerned with theoretical error analysis of a finite element method for computing the eigenvalues and corresponding eigenfunctions. Our analysis of convergence makes use of Osborn’s perturbation theory for eigenvalues of non self-adjoint compact operators. Some numerical examples are presented to confirm our theoretical error analysis.