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 >= 1 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 >= 2, superconvergent of order k(2r-1) at the nodes. We show that on the nth 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 a posteriori estimate for the maximum error over the subinterval (t(n-1),t(n)). Furthermore, a simple post-processing step yields a 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.