SUPERCONVERGENCE OF LOCAL DISCONTINUOUS GALERKIN METHODS FOR ONE-DIMENSIONAL LINEAR PARABOLIC EQUATIONS

摘要

In this paper, we study superconvergence properties of the local discontinuous Galerkin method for one-dimensional linear parabolic equations when alternating fluxes are used. We prove, for any polynomial degree k, that the numerical fluxes converge at a rate of 2k + 1 (or 2k + 1/2) for all mesh nodes and the domain average under some suitable initial discretization. We further prove a k+1th superconvergence rate for the derivative approximation and a k + 2th superconvergence rate for the function value approximation at the Radau points. Numerical experiments demonstrate that in most cases, our error estimates are optimal, i.e., the error bounds are sharp.