OPTIMAL REGULARITY AND ERROR ESTIMATES OF A SPECTRAL GALERKIN METHOD FOR FRACTIONAL ADVECTION-DIFFUSION-REACTION EQUATIONS

We investigate a spectral Galerkin method for the fractional advection-diffusion-reaction equations in one dimension. We first prove sharp regularity estimates of solutions in non-weighted and weighted Sobolev spaces. Then we obtain optimal convergence orders of the spectral Galerkin methods for both fractional advection-diffusion and diffusion-reaction equations. We also present an iterative solver with a quasi-optimal complexity. Numerical results are presented to verify the theoretical analysis.