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

We consider an initial-boundary value problem for ∂tu−∂ t ∇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 piecewiselinear, conforming finite element method in space. The time mesh is graded appropriately near t = 0, but the spatial mesh is quasiuniform. Previously, we proved that the error, measured in the spatial L2-norm, is of order k 2+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− + 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.