Spectral Analysis and Multigrid Methods for Finite Volume Approximations of Space-Fractional Diffusion Equations

Abstract

We consider a boundary value problem in weak form of a steady-state Riesz space-fractional diffusion equation (FDE) of order $2-\alpha$ with $0<\alpha<1$. By using a finite volume approximation technique on uniform grids, we obtain a large linear system, whose coefficient matrix can be viewed as the sum of diagonal matrices times dense Toeplitz matrices. We study in detail the hidden nature of the resulting sequence of coefficient matrices, and we show that they fall in the class of generalized locally Toeplitz (GLT) sequences. The associated GLT symbol is obtained as the sum of products of functions, involving the Wiener generating functions of the Toeplitz components and the diffusion coefficients of the considered FDE. By exploiting a few analytical features of the GLT symbol, we obtain spectral information used for designing efficient preconditioners and multigrid methods. Several numerical experiments, both in the 1D and 2D cases, are reported and discussed, in order to show the optimality of the proposed algorithms.