Two-Dimensional Semi-linear Riesz Space Fractional Diffusion Equations in Convex Domains: GLT Spectral Analysis and Multigrid Solvers

The current work is devoted to the design of fast numerical methods for solving large linear systems, stemming from time-dependent Riesz space fractional diffusion equations, with a nonlinear source term in the convex (non Cartesian) domain. The problem is simpler than the distributed version in [9] and hence it is easier and more elegant to show that the sequence of coefficient matrices (as the finesse parameters decrease to zero) is a Generalized Locally Toeplitz (GLT) sequence and to compute its GLT symbol. From this study we recover important spectral information that we use for designing fast multigrid methods and for discussing the convergence speed of our multigrid solver. Numerical experiments are presented and critically discussed.