Main innovation: it results in a speed up for solving space fractional diffusion equations. Also, it enables a seamless and efficient extension to handle time-dependent cases.
This work deals with space/time multigrid preconditioners for fractional diffusion equations (FDEs). We started by considering an FDE with non-smooth solutions and built tailored multigrid preconditioners. Then, we considered the case of tempered FDEs. We provided a spectral study and exploited it to build tailored multigrid solvers. Finally, we dealt with a new method for solving time-dependent n-dimensional PDEs through a particular approach, which consists in computing the solution in parallel over many coarse anisotropic meshes and merging everything together to form a solution over a finer mesh.
Publications
- D. Ahmad, M. Donatelli, M. Mazza, Stefano Serra-Capizzano, K. Trotti, A smoothing analysis for multigrid methods applied to tempered fractional problems. arXiv preprint arXiv:2210.05031 [math.NA].
- M. Donatelli, R. Krause, M. Mazza, K. Trotti, Multigrid for two-sided fractional differential equations discretized by finite volume elements on graded meshes. arXiv preprint arXiv:2209.08841 [math.NA].
- K. Trotti, A domain splitting strategy for the solution of PDEs. arXiv preprint arXiv:2303.01163 [math.NA].