Superconvergence of local discontinuous Galerkin methods with generalized alternating fluxes for 1D linear convection-diffusion equations

TL;DR

This paper demonstrates superconvergence properties of local discontinuous Galerkin methods with generalized alternating fluxes for 1D linear convection-diffusion equations, achieving higher accuracy at specific points and averages.

Contribution

It proves new superconvergence results for LDG methods with generalized fluxes, including higher order convergence for cell averages and numerical traces.

Findings

Proves $(2k+1)$th order superconvergence for cell averages and traces.

Establishes $k+2$ and $k+1$ order superconvergence at Radau points.

Numerical experiments confirm theoretical superconvergence results.

Abstract

This paper investigates superconvergence properties of the local discontinuous Galerkin methods with generalized alternating fluxes for one-dimensional linear convection-diffusion equations. By the technique of constructing some special correction functions, we prove the $(2k+1)$th order superconvergence for the cell averages, and the numerical traces in the discrete $L^2$ norm. In addition, superconvergence of order $k+2$ and $k+1$ are obtained for the error and its derivative at generalized Radau points. All theoretical findings are confirmed by numerical experiments.