A reduced basis method for fractional diffusion operators I

Abstract

We propose and analyze new numerical methods to evaluate fractional norms and apply fractional powers of elliptic operators. By means of a reduced basis method, we project to a small dimensional subspace where explicit diagonalization via the eigensystem is feasible. The method relies on several independent evaluations of ({{,mathrm{I},}}-t_i^2Delta )^{-1}f, which can be computed in parallel. We prove exponential convergence rates for the optimal choice of sampling points t_i, provided by the so-called Zolotarëv points. Numerical experiments confirm the analysis and demonstrate the efficiency of our algorithm.