Superconvergence of a Discontinuous Galerkin Method for Fractional Diffusion and Wave Equations

We consider an initial-boundary value problem for ∂ₜu-∂ₜ-α∇²u=f(t), that is, for a fractional diffusion (-1<α<0) or wave (0<α<1) equation. A numerical solution is found by applying a piecewise-linear, discontinuous Galerkin (DG) method in time combined with a piecewise-linear, conforming finite element method in space. The time mesh is graded appropriately near $t=0$, but the spatial mesh is quasi-uniform. Previously, we proved that the error, measured in the spatial L₂-norm, is of order k2+α₋+h²(k), uniformly in t, where k is the maximum time step, h is the maximum diameter of the spatial finite elements, α₋=min(α,0)≤0, and (k)=max(1,|log k|). Here, we prove convergence of order k3+2α₋(k)+h² at each time level tₙ for -1<α<1. Thus, if -1/2<α<1, then the DG solution is superconvergent, which generalizes a known result for the classical heat equation (i.e., the case α=0). A simple postprocessing step employing Lagrange interpolation leads to superconvergence for any t. Numerical experiments indicate that our theoretical error bound is pessimistic if α<0. Ignoring logarithmic factors, we observe that the error in the DG solution at t=tₙ, and after postprocessing at all t, is of order k3+α₋+h² for -1<α<1.