2022/08/07 by William McLean, Kassem Mustapha, McLean, William +1
Engineering · Mathematics · #65J08 #65M15 #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2208.03846
openalex publication_date 2022/08/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 nth 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 a posteriori estimate for the maximum error over the subinterval (tn-1,tn). Furthermore, a simple post-processing step yields a continuous piecewise polynomial of degree r with the optimal global convergence rate of order kr+1. We illustrate these results with some numerical experiments.