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

TL;DR

This paper demonstrates that a discontinuous Galerkin method for fractional diffusion and wave equations achieves superconvergence in time, with improved error rates at each time level, supported by theoretical analysis and numerical experiments.

Contribution

It extends superconvergence results to fractional PDEs, showing enhanced accuracy of DG methods with a simple postprocessing step.

Findings

Superconvergence in time at each time level with order $(k^{3+2eta}+h^2)\

Postprocessing yields globally superconvergent approximations for all times

Numerical results suggest the theoretical bounds are conservative for $eta<0$

Abstract

We consider an initial-boundary value problem for $\partial_tu-\partial_t^{-\alpha}\nabla^2u=f(t)$, 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 $k^{2+\alpha_-}+h^2\ell(k)$, uniformly in $t$, where $k$ is the maximum time step, $h$ is the maximum diameter of the spatial finite elements, $\alpha_-=\min(\alpha,0)\le0$ and $\ell(k)=\max(1,|\log k|)$. 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 $(k^{3+2\alpha_-}+h^2)\ell(k)$. 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 $k^{3+\alpha_-}+h^2$.