Multigrid Methods for Space Fractional Partial Differential Equations

TL;DR

This paper introduces optimal multigrid methods for efficiently solving algebraic systems from finite element discretizations of space fractional PDEs, with convergence independent of mesh size and level.

Contribution

The paper presents the first multigrid algorithms specifically designed for SFPDEs that are proven to be optimal without regularity assumptions.

Findings

Multigrid methods achieve mesh-independent convergence rates.

Theoretical analysis confirms optimality of the proposed methods.

Numerical experiments support the theoretical convergence results.

Abstract

We propose some multigrid methods for solving the algebraic systems resulting from finite element approximations of space fractional partial differential equations (SFPDEs). It is shown that our multigrid methods are optimal, which means the convergence rates of the methods are independent of the mesh size and mesh level. Moreover, our theoretical analysis and convergence results do not require regularity assumptions of the model problems. Numerical results are given to support our theoretical findings.