Superconvergence error analysis of discontinuous Galerkin method with interior penalties for 2D elliptic convection – diffusion – reaction problems

Abstract

This article focuses on constructing and analyzing a non-symmetric interior penalty Galerkin method (NIPG) on Shishkin mesh for solving singularly perturbed 2D elliptic boundary-value problems (BVPs). We use piecewise Lagrange interpolation at Gaussian points to improve the order of convergence of the interpolation error. We then study the superconvergence properties of the NIPG method and prove $\mathcal{O}(N^{-1}\ln N{)}^{k+1}$ order of convergence in the discrete energy–norm. Various numerical experiments are provided to validate the theoretical results.

Keywords:

• Singularly perturbed 2D elliptic BVPs
• convection-diffusion problem
• Shishkin mesh
• discontinuous Galerkin method with interior penalty
• superconvergence

2020 Mathematics Subject Classifications:

• 65N12
• 65N15
• 65N30

Acknowledgments

The authors wish to acknowledge the referee for his/her valuable comments, which really helped to improve the presentation.

Disclosure statement

No potential conflict of interest was reported by the author(s).