A multigrid method for linear systems arising from time-dependent two-dimensional space-fractional diffusion equations

In this paper, we study a V-cycle multigrid method for linear systems arising from time-dependent two-dimensional space-fractional diffusion equations. The coefficient matrices of the linear systems are structured such that their matrix-vector multiplications can be computed efficiently. The main advantage using the multigrid method is to handle the space-fractional diffusion equations on non-rectangular domains, and to solve the linear systems with non-constant coefficients more effectively. The main idea of the proposed multigrid method is to employ two banded splitting iteration schemes as pre-smoother and post-smoother. The pre-smoother and the post-smoother take banded splitting of the coefficient matrix under the x-dominant ordering and the y-dominant ordering, respectively. We prove the convergence of the proposed two banded splitting iteration schemes in the sense of infinity norm. Results of numerical experiments for time-dependent two-dimensional space-fractional diffusion equations on rectangular, L-shape and U-shape domains are reported to demonstrate that both computational time and iteration number required by the proposed method are significantly smaller than those of the other tested methods.