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 piecewiselinear, 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 κ^2+α- + h²ℓ(κ), uniformly in t, where κ is the maximum time step, h is the maximum diameter of the spatial finite elements, α_- = min(α, 0) ≤ 0, and ℓ(κ) = max(1, | log κ|). Here, we prove convergence of order κ^3+2α-ℓ( κ) + h₂ at each time level t_n 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 κ^3+α- + h² for -1 < α < 1.