TY - JOUR
T1 - Analysis of Optimal Error Estimates and Superconvergence of the Discontinuous Galerkin Method for Convection-Diffusion Problems in One Space Dimension
AU - M. Baccouch & H. Temimi
JO - International Journal of Numerical Analysis and Modeling
VL - 3
SP - 403
EP - 434
PY - 2016
DA - 2016/05
SN - 13
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/446.html
KW - Discontinuous Galerkin method, convection-diffusion problems, singularly pertur-
bed problems, superconvergence, upwind and downwind points, Shishkin meshes.
AB - <p style="text-align: justify;">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 $\mathcal{O}(h^{p+1})$ and $\mathcal{O}(h^p)$ convergence rates in the $L^2$-norm, respectively, when $p$-degree piecewise polynomials with $p\geqslant 1$ are used. We further prove that the $p$-degree DG solution and its derivative are $\mathcal{O}(h^{2p})$ 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.</p>