  Recently, a new eigenvalue problem, called the transmission eigenvalue
problem, has attracted many researchers. The problem arose in inverse
scattering theory for inhomogeneous media and has important applications in a
variety of inverse problems for target identification and nondestructive
testing. The problem is numerically challenging because it is non-selfadjoint
and nonlinear. In this paper, we propose a recursive integral method for
computing transmission eigenvalues from a finite element discretization of the
continuous problem. The method, which overcomes some difficulties of existing
methods, is based on eigenprojectors of compact operators. It is
self-correcting, can separate nearby eigenvalues, and does not require an
initial approximation based on some a priori spectral information. These
features make the method well suited for the transmission eigenvalue problem
whose spectrum is complicated. Numerical examples show that the method is
effective and robust.
