On Numerical Methods for Elliptic Transmission Eigenvalue Problems

The use of numerical tools to solve challenging problems in mathematics has exploded in the past several decades. The purpose of this paper is to compare the results of two different types of numerical methods in finding solutions to the eigenvalue problem for a second order elliptic partial differential equations (PDE) with boundary and transmission conditions. Transmission properties result from jumps in the coefficients of the equation and require more complex numerical methods to solve the eigenvalue problem than when the coefficients are continuous. We present the setup of both the bisection method to solve the exact equation satisfied by the eigenvalues and an application of the power method on a Finite Element Method discretization to find the largest eigenvalues and eigenfunction. We also provide some numerical evidence as to which method is more efficient given the complexities of our problem.