Optimal error estimates of spectral Galerkin method for mixed diffusion equations

We present a highly accurate and efficient spectral Galerkin method for the advection-diffusion-reaction equations with fractional lower-order terms in one dimension. We first prove sharp a priori regularity estimates of solutions in weighted Sobolev spaces. Then we obtain optimal convergence orders of the spectral Galerkin method. We also describe its implementation and present an efficient solver with a quasi-optimal complexity. Finally, we perform the numerical experiments and report numerical observations to support the theoretical predictions.