Iterative, finite difference solution of interior eigenvalues and eigenfunctions of Laplace's operator

Abstract

This paper discusses and compares two iterative methods for solutions of the Helmholtz equation. Ordinary successive overrelaxation (SOR) fails to converge to eigenvalues other than the first. One convergent iterative approach investigated is the accelerated method of Kaczmarz. The other method, proposed in this paper, obtains the solution by redefining the problem such that the new system matrix is symmetric and positive semidefinite for all eigenvalues. SOR is then guaranteed to converge. The resulting matrix is the one defined by premultiplication of the original system matrix by its transpose. This cannot be done directly as excessive computer store would be required for high accuracy. An algorithm is described which produces only one row at a time of the new matrix equation, using linear combinations of a few easily generated rows of the original system, thus reserving almost all store for the eigenvector. SOR on the positive definite system (PDSOR) is then applied. Numerical results obtained indicate that PDSOR converges far more rapidly than accelerated Kaczmarz.