On Numerical Methods for Elliptic Transmission/boundary Value 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 differential equation subject to boundary and transmission conditions. The transmission conditions 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 discuss both a method to compute approximate eigenvalues based on the bisection method to find zeroes of a function, and a Finite Element Method to find the largest eigenvalue and associated eigenfunction. We also provide some numerical evidence as to which method is more efficient given the complexities of our problem.