Superconvergence of Local Discontinuous Galerkin method for one-dimensional linear parabolic equations

TL;DR

This paper investigates the superconvergence properties of the local discontinuous Galerkin method for 1D linear parabolic equations, demonstrating high convergence rates for fluxes, derivatives, and function values with numerical validation.

Contribution

It establishes new superconvergence results for the LDG method with alternating fluxes, including convergence rates at mesh nodes, Radau points, and for derivatives.

Findings

Numerical fluxes converge at rate 2k+1 or 2k+1/2.

Derivative approximation superconverges at rate k+1.

Function value approximation superconverges at rate k+2.

Abstract

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 $k+1$th superconvergence rate for the derivative approximation and a $k+2$th 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.