Analysis of optimal error estimates and superconvergence of the discontinuous Galerkin method for convection-diffusion problems in one space dimension

Authors

Mahboub Baccouch, University of Nebraska at OmahaFollow
Helmi Temimi, Gulf University for Science & Technology

Document Type

Article

Publication Date

2016

Publication Title

International Journal of Numerical Analysis and Modeling

Volume

13

Issue

3

First Page

403

Last Page

434

Abstract

In this paper, we study the convergence and superconvergence properties of the discontinuous Galerkin (DG) method for a linear convection-diffusion problem in one-dimensional setting. We prove that the DG solution and its derivative exhibit optimal O(h^p+1) and O(h^p) convergence rates in the L 2 -norm, respectively, when p-degree piecewise polynomials with p ≥ 1 are used. We further prove that the p-degree DG solution and its derivative are O(h^2p) superconvergent at the downwind and upwind points, respectively. Numerical experiments demonstrate that the theoretical rates are optimal and that the DG method does not produce any oscillation. We observed optimal rates of convergence and superconvergence even in the presence of boundary layers when Shishkin meshes are used.

Comments

© 2016 Institute for Scientific Computing and Information

Recommended Citation

Baccouch, Mahboub and Temimi, Helmi, "Analysis of optimal error estimates and superconvergence of the discontinuous Galerkin method for convection-diffusion problems in one space dimension" (2016). Mathematics Faculty Publications. 10.
https://digitalcommons.unomaha.edu/mathfacpub/10