SUPERCONVERGENCE OF A DISCONTINUOUS GALERKIN METHOD FOR FRACTIONAL DIFFUSION AND WAVE EQUATIONS

We consider an initial-boundary value problem for partial derivative(t)u - partial derivative(-alpha)(t)del(2)u - f(t), that is, for a fractional diffusion (-1 < alpha < 0) or wave (0 < alpha < 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 L-2-norm, is of order k(2+alpha-) + h(2)l(k), uniformly in t, where k is the maximum time step, h is the maximum diameter of the spatial finite elements, alpha(-) = min(alpha,0) <= 0, and l(k) = max(1, vertical bar log k vertical bar). Here, we prove convergence of order k(3+2 alpha-) l(k) + h(2) at each time level t(n) for -1 < alpha < 1. Thus, if -1/2 < alpha < 1, then the DG solution is superconvergent, which generalizes a known result for the classical heat equation (i.e., the case alpha = 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 alpha < 0. Ignoring logarithmic factors, we observe that the error in the DG solution at t = t(n), and after postprocessing at all t, is of order k(3+alpha-) + h(2) for -1 < alpha < 1.