Uniform convergence of multigrid finite element method for time-dependent Riesz tempered fractional problem
This work provides the first proof of convergence for V-cycle multigrid finite element methods for time-dependent fractional problems as the time step approaches zero, benefiting researchers in numerical analysis of fractional PDEs.
The authors provide a theoretical framework for Galerkin finite element approximation to time-dependent Riesz tempered fractional problems without fractional regularity assumptions, and prove uniform convergence of the V-cycle multigrid method independent of mesh size and time step. Numerical experiments verify convergence with O(N log N) complexity.
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 $τ\rightarrow 0$, 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 $τ$-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 $τ$. 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 $τ\rightarrow 0$.