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

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→ 0, which correspond to the mass matrix dominant [R. E. Bank and T. Dupont, 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 𝒪(N 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→ 0.