Stefano Serra-Capizzano
University of Insubria, Italy
Fractional partial differential equations (FDEs) are a generalization of classical partial differential equations, used to model anomalous diffusion phenomena. Several discretization schemes (finite differences, finite volumes, etc.) combined with (semi)-implicit methods lead to a Toeplitz-like matrix-sequence. In the constant diffusion coeffcients case such a matrix-sequence reduces to a Toeplitz one, then exploiting well-known results on Toeplitz sequences, we are able to describe its asymptotic eigenvalue behavior. In the case of nonconstant diffusion coeffcients, we show that the resulting matrix-sequence is a Generalized Locally Toeplitz (GLT) sequence and then we use the GLT machinery to study its singular value/eigenvalue distribution as the matrix size diverges.
The new spectral information is employed for analyzing preconditioned Krylov and multigrid methods recently appeared in the literature, with both positive and negative results. Moreover, such spectral analysis guides the design of new preconditioning and multigrid strategies. We propose new structure preserving preconditioners with minimal bandwidth (and thus with effcient computational cost) and multigrid methods for related 1D and 2D problems. Numerical results confirm the theoretical analysis and the effectiveness of the new proposals