A rational preconditioner for multi-dimensional Riesz fractional diffusion equations

Abstract

We propose a rational preconditioner for an efficient numerical solution of linear systems arising from the discretization of multi-dimensional Riesz fractional diffusion equations. In particular, the discrete problem is obtained by employing finite difference or finite element methods to approximate the fractional derivatives of order α with α∈(1,2]. The proposed preconditioner is then defined as a rational approximation of the Riesz operator expressed as the integral of the standard heat diffusion semigroup. We show that, being the sum of k inverses of shifted Laplacian matrices, the resulting preconditioner belongs to the generalized locally Toeplitz class. As a consequence, we are able to provide the asymptotic description of the spectrum of the preconditioned matrices and we show that, despite the lack of clustering just as for the Laplacian, our preconditioner for α close to 1 and k≠1 reasonably small, provides better results than the Laplacian itself, while sharing the same computational complexity.