Abstract
The eigenvalue problem for the Laplace operator in two dimensions is classical in mathematics
and physics. Nevertheless, computational methods for estimating the eigenvalues are still of much current in-
terest, particularly in applications to acoustic and electromagnetic waveguides. Although our primary inter-
est is with the computational methods, there are a number of theoretical results on the behavior of the eigen-
values and eigenfunctions that are useful for understanding the methods and, in addition, are of interest in
themselves. These results are discussed first and then the various computational methods that have been used
to estimate the eigenvalues are reviewed with particular emphasis on methods that give error bounds. Some
of the more powerful techniques available are illustrated by applying them to a model problem.
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