A smoothing analysis for multigrid methods applied to tempered fractional problems

TL;DR

This paper develops and analyzes a multigrid solver for time-dependent space tempered fractional diffusion equations, demonstrating its effectiveness and expanding the understanding of suitable smoothing weights.

Contribution

It introduces a new smoothing analysis for multigrid methods applied to tempered fractional problems and shows the transferability of multigrid effectiveness from non-tempered to tempered cases.

Findings

The multigrid solver is computationally effective for tempered fractional diffusion.

The new smoothing analysis broadens the set of suitable Jacobi weights.

Effectiveness in the non-tempered case implies effectiveness in the tempered case.

Abstract

We consider the numerical solution of time-dependent space tempered fractional diffusion equations. The use of Crank-Nicolson in time and of second-order accurate tempered weighted and shifted Gr\"unwald difference in space leads to dense (multilevel) Toeplitz-like linear systems. By exploiting the related structure, we design an ad-hoc multigrid solver and multigrid-based preconditioners, all with weighted Jacobi as smoother. A new smoothing analysis is provided, which refines state-of-the-art results expanding the set of the suitable Jacobi weights. Furthermore, we prove that if a multigrid method is effective in the non-tempered case, then the same multigrid method is effective also in the tempered one. The numerical results confirm the theoretical analysis, showing that the resulting multigrid-based solvers are computationally effective for tempered fractional diffusion equations.