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We consider the numerical solution of time-dependent space temperedfractional diffusion equations. The use of Crank-Nicolson in time and ofsecond-order accurate tempered weighted and shifted Gr\"unwald difference inspace leads to dense (multilevel) Toeplitz-like linear systems. By exploitingthe related structure, we design an ad-hoc multigrid solver and multigrid-basedpreconditioners, all with weighted Jacobi as smoother. A new smoothing analysisis provided, which refines state-of-the-art results expanding the set of thesuitable Jacobi weights. Furthermore, we prove that if a multigrid method iseffective in the non-tempered case, then the same multigrid method is effectivealso in the tempered one. The numerical results confirm the theoreticalanalysis, showing that the resulting multigrid-based solvers arecomputationally effective for tempered fractional diffusion equations.

A smoothing analysis for multigrid methods applied to tempered fractional problems