Multigrid for two-sided fractional differential equations discretized by finite volume elements on graded meshes

TL;DR

This paper develops a multigrid preconditioner for finite volume discretizations of two-sided fractional diffusion equations on graded meshes, improving computational efficiency and accuracy near boundary singularities.

Contribution

It introduces a parameter-free multigrid preconditioner tailored for Toeplitz-like matrices from graded meshes in fractional PDEs, enhancing solver performance.

Findings

Multigrid preconditioner achieves good convergence with graded meshes.

Power graded meshes outperform composite meshes in approximation accuracy.

Solver demonstrates wide applicability across different mesh types.

Abstract

It is known that the solution of a conservative steady-state two-sided fractional diffusion problem can exhibit singularities near the boundaries. As consequence of this, and due to the conservative nature of the problem, we adopt a finite volume elements discretization approach over a generic non-uniform mesh. We focus on grids mapped by a smooth function which consist in a combination of a graded mesh near the singularity and a uniform mesh where the solution is smooth. Such a choice gives rise to Toeplitz-like discretization matrices and thus allows a low computational cost of the matrix-vector product and a detailed spectral analysis. The obtained spectral information is used to develop an ad-hoc parameter free multigrid preconditioner for GMRES, which is numerically shown to yield good convergence results in presence of graded meshes mapped by power functions that accumulate points near the singularity. The approximation order of the considered graded meshes is numerically compared with the one of a certain composite mesh given in literature that still leads to Toeplitz-like linear systems and is then still well-suited for our multigrid method. Several numerical tests confirm that power graded meshes result in lower approximation errors than composite ones and that our solver has a wide range of applicability.