Uniform convergence of multigrid finite element method for time-dependent Riesz tempered fractional problem

TL;DR

This paper establishes the uniform convergence of a multigrid finite element method for time-dependent Riesz tempered fractional problems, proving the convergence rate is independent of mesh size and time step, supported by numerical experiments.

Contribution

It provides the first proof of the V-cycle multigrid finite element method's convergence rate as the time step approaches zero for fractional problems.

Findings

Convergence rate of V-cycle multigrid method is independent of mesh size and time step.

Numerical experiments verify convergence with O(N log N) complexity.

First proof of multigrid convergence rate as time step tends to zero for fractional problems.

Abstract

In this article a theoretical framework for the Galerkin finite element approximation to the time-dependent Riesz tempered fractional problem is provided without the fractional regularity assumption. Because the time-dependent problems should become easier to solve as the time step $\tau \rightarrow 0$, which correspond to the mass matrix dominant [R. E. Bank and T. Dupont, {\em Math. Comp.}, 153 (1981), pp. 35--51]. Based on the introduced and analysis of the fractional $\tau$-norm, the uniform convergence estimates of the V-cycle multigrid method with the time-dependent fractional problem is strictly proved, which means that the convergence rate of the V-cycle MGM is independent of the mesh size $h$ and the time step $\tau$. The numerical experiments are performed to verify the convergence with only $\mathcal{O}(N \mbox{log} N)$ complexity by the fast Fourier transform method, where $N$ is the number of the grid points. To the best of our knowledge, this is the first proof for the convergence rate of the V-cycle multigrid finite element method with $\tau\rightarrow 0$.