A Reduced Basis Method For Fractional Diffusion Operators I

2019/04/11 by Danczul, Tobias, Schöberl, Joachim
#26C15 (Secondary) #35J15 (Primary) 65N25 #35P10 #46B70 #65N12 #65N15 #65N30 #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.1904.05599

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 (\textrmI-t_i²Δ)^-1f, 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.

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