Error Profile for Discontinuous Galerkin Time Stepping of Parabolic PDEs

We consider the time discretization of a linear parabolic problem by the discontinuous Galerkin (DG) method using piecewise polynomials of degree at most $r-1$ in $t$, for $r\ge1$ and with maximum step size~$k$. It is well known that the spatial $L_2$-norm of the DG error is of optimal order $k^r$ globally in time, and is, for $r\ge2$, superconvergent of order $k^{2r-1}$ at the nodes. We show that on the $n$th subinterval $(t_{n-1},t_n)$, the dominant term in the DG error is proportional to the local right Radau polynomial of degree $r$. This error profile implies that the DG error is of order $k^{r+1}$ at the right-hand Gauss--Radau quadrature points in each interval. We show that the norm of the jump in the DG solution at the left end point $t_{n-1}$ provides an accurate \emph{a posteriori} estimate for the maximum error over the subinterval $(t_{n-1},t_n)$. Furthermore, a simple post-processing step yields a \emph{continuous} piecewise polynomial of degree $r$ with the optimal global convergence rate of order $k^{r+1}$. We illustrate these results with some numerical experiments.