Error Profile for Discontinuous Galerkin Time Stepping of Parabolic PDEs
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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.
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