Spectral Analysis of Matrices in B-Spline Galerkin Methods for Riesz Fractional Equations

Polynomial B-spline collocation discretizations for Riesz fractional diffusion equations over uniform meshes have recently appeared in the literature and a spectral study of the related coefficient matrices has been performed. Here we focus on the coefficient matrices obtained by the Galerkin approach. For an arbitrary polynomial degree p we show that, as for collocation, the resulting coefficient matrices possess a Toeplitz-like structure. The derivation of their spectral distribution is simpler compared to collocation, due to the symmetry of the coefficient matrices in this case and by leveraging the generalized locally Toeplitz theory. We see that, like for second-order differential problems, also in the fractional context the spectral distribution in the Galerkin formulation with B-splines of degree p is the same as in the collocation formulation with B-splines of degree 2p + 1. As a consequence, the Galerkin matrices are poorly conditioned in both low and high frequencies similar to the collocation ones. Finally, we numerically observe that the approximation order of the Galerkin approach for smooth solutions does not depend on the fractional derivative order as for collocation and that it coincides with p + 1 as for non-fractional diffusion problems.