Papers › Error Profile for Discontinuous Galerkin Time Stepping of Parabolic PDEs
Error Profile for Discontinuous Galerkin Time Stepping of Parabolic PDEs
William McLean, Kassem Mustapha
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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 L₂-norm of the DG error is of optimal order kʳ globally in time, and is, for r≥2, superconvergent of order k²ʳ⁻¹ at the nodes. We show that on the $n$th subinterval (tₙ₋₁,tₙ), 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 kʳ⁺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 tₙ₋₁ provides an accurate \emph{a posteriori} estimate for the maximum error over the subinterval (tₙ₋₁,tₙ). Furthermore, a simple post-processing step yields a \emph{continuous} piecewise polynomial of degree r with the optimal global convergence rate of order kʳ⁺¹. We illustrate these results with some numerical experiments.
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