TY - JOUR
T1 - Analysis of Sharp Superconvergence of Local Discontinuous Galerkin Method for One-Dimensional Linear Parabolic Equations
AU - Yang , Yang
AU - Shu , Chi-Wang
JO - Journal of Computational Mathematics
VL - 3
SP - 323
EP - 340
PY - 2015
DA - 2015/06
SN - 33
DO - http://doi.org/10.4208/jcm.1502-m2014-0001
UR - https://global-sci.org/intro/article_detail/jcm/9845.html
KW - Superconvergence, Local discontinuous Galerkin method, Parabolic equation,
Initial discretization, Error estimates, Radau points.
AB - <p style="text-align: justify;">In this paper, we study the superconvergence of the error for the local discontinuous Galerkin (LDG) finite element method for one-dimensional linear parabolic equations when the alternating flux is used. We prove that if we apply piecewise $k$-th degree polynomials, the error between the LDG solution and the exact solution is ($k$+2)-th order superconvergent at the Radau points with suitable initial discretization. Moreover, we also prove the LDG solution is ($k$+2)-th order superconvergent for the error to a particular projection of the exact solution. Even though we only consider periodic boundary condition, this boundary condition is not essential, since we do not use Fourier analysis. Our analysis is valid for arbitrary regular meshes and for $P^k$ polynomials with arbitrary $k$ ≥ 1. We perform numerical experiments to demonstrate that the superconvergence rates proved in this paper are sharp.</p>