Eigenvalue Problems for Exponential-Type Kernels

We study approximations of eigenvalue problems for integral operators associated with kernel functions of exponential type. We show convergence rate |λk − λk,h| ≤ Ckh 2 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 |λk,h(G)-λk,h(N)| ≤ Ch2, where λk,h(G) and λk,h(N) 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 experiment