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Volume 13, Issue 3
Analysis of Optimal Error Estimates and Superconvergence of the Discontinuous Galerkin Method for Convection-Diffusion Problems in One Space Dimension

M. Baccouch & H. Temimi

Int. J. Numer. Anal. Mod., 13 (2016), pp. 403-434.

Published online: 2016-05

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  • 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 $\mathcal{O}(h^{p+1})$ 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.

  • Keywords

Discontinuous Galerkin method, convection-diffusion problems, singularly pertur- bed problems, superconvergence, upwind and downwind points, Shishkin meshes.

  • AMS Subject Headings

65M15, 65M60, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-13-403, author = {M. Baccouch and H. Temimi}, title = {Analysis of Optimal Error Estimates and Superconvergence of the Discontinuous Galerkin Method for Convection-Diffusion Problems in One Space Dimension}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2016}, volume = {13}, number = {3}, pages = {403--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 $\mathcal{O}(h^{p+1})$ 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.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/446.html} }
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 -

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.

M. Baccouch and H. Temimi. (2016). Analysis of Optimal Error Estimates and Superconvergence of the Discontinuous Galerkin Method for Convection-Diffusion Problems in One Space Dimension. International Journal of Numerical Analysis and Modeling. 13 (3). 403-434. doi:
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