THE INCLUSION OF DIRICHLET EIGENVALUES WITH SINGULARITY FUNCTIONS

We are concerned with the Dirichlet eigenvalue problem Delta u + lambd a u = 0 in G, u = 0 on Gamma, where G is a bounded, two dimensional do main with sufficiently smooth boundary Gamma. We deal with the ''Ansat z'' u(x, lambda) = Sigma(m=1)(M) a(m)Y(0)(root\x-y(m)\) to compute app roximate eigenpairs (u, lambda*) by the collocation method. Lower and upper eigenvalue bounds are estimated by an inclusion theorem due to Kuttler-Sigillito. In contrast to usual choices of trial functions, it is possible to control Me numerical stability by placing the source p oints in dependence of M. The effect arises from the logarithmic singu larity for x = y(m) and allows us to sharpen the eigenvalue bounds by increasing M. We give a strategy for placing the sources, present vari ous applications and give a comparison of results which indicates a hi gh efficiency of the method.