# Spectral analysis and multigrid preconditioners for two-dimensional space-fractional diffusion equations

## TL;DR

This paper develops spectral analysis and multigrid preconditioners for 2D space-fractional diffusion equations, enabling efficient iterative solutions by exploiting Toeplitz structure and proving linear convergence.

## Contribution

It introduces a spectral analysis framework for 2D space-FDEs and designs multigrid preconditioners that ensure fast, robust iterative solutions with proven convergence rates.

## Key findings

- Spectral analysis of coefficient matrices guides preconditioner design.
- Multigrid methods achieve linear convergence rates.
- Preconditioned Krylov methods maintain efficiency with new strategies.

## Abstract

Fractional diffusion equations (FDEs) are a mathematical tool used for describing some special diffusion phenomena arising in many different applications like porous media and computational finance. In this paper, we focus on a two-dimensional space-FDE problem discretized by means of a second order finite difference scheme obtained as combination of the Crank-Nicolson scheme and the so-called weighted and shifted Gr\"unwald formula. By fully exploiting the Toeplitz-like structure of the resulting linear system, we provide a detailed spectral analysis of the coefficient matrix at each time step, both in the case of constant and variable diffusion coefficients. Such a spectral analysis has a very crucial role, since it can be used for designing fast and robust iterative solvers. In particular, we employ the obtained spectral information to define a Galerkin multigrid method based on the classical linear interpolation as grid transfer operator and damped-Jacobi as smoother, and to prove the linear convergence rate of the corresponding two-grid method. The theoretical analysis suggests that the proposed grid transfer operator is strong enough for working also with the V-cycle method and the geometric multigrid. On this basis, we introduce two computationally favourable variants of the proposed multigrid method and we use them as preconditioners for Krylov methods. Several numerical results confirm that the resulting preconditioning strategies still keep a linear convergence rate.