A minimax method for the spectral fractional Laplacian and related evolution problems

Abstract

We present a numerical method for the approximation of the inverse of the fractional Laplacian $(-\Delta)^{s}$, based on its spectral definition, using rational functions to approximate the fractional power $A^{-s}$ of a matrix $A$, for $0<s<1$. The proposed numerical method is fast and accurate, benefiting from the fact that the matrix $A$ arises from a finite element approximation of the Laplacian $-\Delta$, which makes it applicable to a wide range of domains with potentially irregular shapes. We make use of state-of-the-art software to compute the best rational approximation of a fractional power. We analyze the convergence rate of our method and validate our findings through a series of numerical experiments with a range of exponents $s \in (0,1)$. Additionally, we apply the proposed numerical method to different evolution problems that involve the fractional Laplacian through an interaction potential: the fractional porous medium equation and the fractional Keller-Segel equation. We then investigate the accuracy of the resulting numerical method, focusing in particular on the accurate reproduction of qualitative properties of the associated analytical solutions to these partial differential equations.