Multigrid preconditioners for anisotropic space-fractional diffusion equations

We focus on a two-dimensional time-space diffusion equation with fractional derivatives in space. The use of Crank-Nicolson in time and finite differences in space leads to dense Toeplitz-like linear systems. Multigrid strategies that exploit such structure are particularly effective when the fractional orders are both close to 2. We seek to investigate how structure-based multigrid approaches can be efficiently extended to the case where only one of the two fractional orders is close to 2, i.e., when the fractional equation shows an intrinsic anisotropy. Precisely, we design a multigrid (block-banded–banded-block) preconditioner whose grid transfer operator is obtained with a semi-coarsening technique and that has relaxed Jacobi as smoother. The Jacobi relaxation parameter is estimated by using an automatic symbol-based procedure. A further improvement in the robustness of the proposed multigrid method is attained using the V-cycle with semi-coarsening as smoother inside an outer full-coarsening. Several numerical results confirm that the resulting multigrid preconditioner is computationally effective and outperforms current state of the art techniques.