Spectral element approximation of Fredholm integral eigenvalue problems

The Karhunen-Loève expansion of a Gaussian process, a common tool on finite element methods for differential equations with stochastic coefficients, is based on the spectral decomposition of its covariance function. The eigenpairs of the covariance are expressed as a Fredholm integral equation of second kind, which can be readily approximated with piecewise-constant finite elements. In this work, the spectral element method with GaussLobatto-Legendre (GLL) collocation points is employed to approximate this eigenvalue problem. Similarly to piecewise-constant finite elements, this approach is simple to implement and does not lead to generalized discrete eigenvalue problems (considering that the numerical integration is also performed with GLL points), with the additional advantage of providing high-order approximations of the eigenfunctions. Numerical experiments involving covariance functions in oneand two-dimensional domains illustrate the effectiveness of this approach.