Homotopy Method for the Eigenvalue Problem for Partial Diierential Equations

Given a linear self-adjoint partial diierential operator L, the smallest few eigenvalues and eigenfunctions of L are computed by the homotopy (continuation) method. The idea of the method is very simple. From some initial operator L0 with known eigenvalues and eigenfunctions, deene the homotopy H (t) = (1 ? t)L0 + tL; 0 t 1. If the eigenfunctions of H (t0) are known, then they are used to determine the eigenpairs of H (t0 + dt) via the Rayleigh quotient iteration, for some value of dt. This is repeated until t becomes 1, when the solution to the original problem is found. A fundamental problem is the selection of the stepsize dt. A simple criterion to select dt is given. It is shown that the iterative solver to nd the eigenvector at each step can be stabilized by applying a low-rank perturbation to the relevant matrix. By carrying out a small part of the calculation in higher precision, it is demonstrated that eigenvectors corresponding to clustered eigenvalues can be computed to high accuracy. Some numerical results for the Helmholtz eigenvalue problem are given.