Optimal convergence of local discontinuous Galerkin methods for convection-diffusion equations

TL;DR

This paper improves the theoretical understanding of the local discontinuous Galerkin method for convection-diffusion equations, aligning convergence estimates with numerical results, especially for solutions with limited regularity.

Contribution

It establishes new approximation results for Gauss-Radau projections, enabling optimal convergence analysis for singular solutions in LDG methods.

Findings

Theoretical convergence rates match numerical experiments for singular solutions.

New approximation results improve understanding of DG scheme performance.

Framework applies to various types of singular solutions.

Abstract

The $hp$ local discontinuous Galerkin (LDG) method proposed by Castillo et al. [Math. Comp.,~71 (238): 455-478, 2002] has been shown to be an efficient approach for solving convection-diffusion equations. However, theoretical analysis indicates that, for solutions with limited spatial regularity, the method exhibits suboptimal convergence in $p$, suffering a loss of one order, comparing to numerical experiments. The purpose of this paper is to close the gap between theoretical estimates and numerical evidence. This is accomplished by establishing new approximation results for the associated Gauss-Radau projections of functions in suitable function spaces that can optimally characterize the regularity of singular solutions. We show that such a framework arises aturally and enables the study of various types of singular solutions, with full consistency between theoretical analysis and numerical results. This investigation sheds light on the resolution of the suboptimality in $p$ observed in the literature for several other types of DG schemes in different settings.