Eigenvalue problems for exponential-type kernels

Abstract

We study approximations of eigenvalue problems for integral operators associated with kernel functions of exponential type. We show convergence rate |λk − λk,h| ≤ Ckh2 in the case of lowest order approximation for both Galerkin and Nyström methods, where h is the mesh size, λk and λk,h are the exact and approximate kth largest eigenvalues, respectively. We prove that the two methods are numerically equivalent in the sense that |λ(kG,h) − λ(kN,h) | ≤ Ch2, where λ(kG,h) and λ(kN,h) denote the kth largest eigenvalues computed by Galerkin and Nyström methods, respectively, and C is a eigenvalue independent constant. The theoretical results are accompanied by a series of numerical experiments.