Superconvergence of a discontinuous Galerkin method for fractional diffusion and wave equations

We consider an initial-boundary value problem for ${\mathrm{\partial }}_{t}u-{\mathrm{\partial }}_t^{-\alpha }{\mathrm{\nabla }}^{2}u=f(t)$, 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 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 quasiuniform. Previously, we proved that the error, measured in the spatial $L_2$-norm, is of order $k^{2+{\alpha }_{-}}+h^2\mathrm{\ell }(k)$, uniformly in $t$, where $k$ is the maximum time step, $h$ is the maximum diameter of the spatial finite elements, ${\alpha }_{-}=\min (\alpha ,0)\le 0$ and $\mathrm{\ell }(k)=\max (1,\mathrm{\mid }\log k\mathrm{\mid })$. Here, we generalize a known result for the classical heat equation (i.e., the case $\alpha =0$) by showing that at each time level $t_n$ the solution is superconvergent with respect to $k$: the error is of order $(k^{3+2{\alpha }_{-}}+h^2)\mathrm{\ell }(k)$. Moreover, a simple postprocessing step employing Lagrange interpolation yields a superconvergent approximation 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 $k^{3+{\alpha }_{-}}+h^2$.