Eigenvalue Problems for Exponential-Type Kernels

Abstract

Abstract We study approximations of eigenvalue problems for integral operators associated with kernel functions of exponential type. We show convergence rate | λ k - λ k , h | ≤ C k ⁢ h 2 {\lvert\lambda_{k}-\lambda_{k,h}\rvert\leq C_{k}h^{2}} in the case of lowest order approximation for both Galerkin and Nyström methods, where h is the mesh size, λ k {\lambda_{k}} and λ k , h {\lambda_{k,h}} are the exact and approximate kth largest eigenvalues, respectively. We prove that the two methods are numerically equivalent in the sense that | λ k , h ( G ) - λ k , h ( N ) | ≤ C ⁢ h 2 {|\lambda^{(G)}_{k,h}-\lambda^{(N)}_{k,h}|\leq Ch^{2}} , where λ k , h ( G ) {\lambda^{(G)}_{k,h}} and λ k , h ( N ) {\lambda^{(N)}_{k,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.