Volume 17, Issue 6
Asymptotically Exact a Posteriori Error Estimates for the Local Discontinuous Galerkin Method for Nonlinear KdV Equations in One Space Dimension

​Mahboub Baccouch

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Int. J. Numer. Anal. Mod., 17 (2020), pp. 767-793.

Published online: 2020-10

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  • Abstract

In this paper, we develop and analyze an implicit $a$ $posteriori$ error estimates for the local discontinuous Galerkin (LDG) method for nonlinear third-order Korteweg-de Vries (KdV) equations in one space dimension. First, we show that the LDG error on each element can be split into two parts. The first part is proportional to the $(p+1)$-degree right Radau polynomial and the second part converges with order $p$ $+$ $\frac{3}{2}$ in the $L^2$-norm, when piecewise polynomials of degree at most $p$ are used. These results allow us to construct $a$ $posteriori$ LDG error estimates. The proposed error estimates are computationally simple and are obtained by solving a local steady problem with no boundary conditions on each element. Furthermore, we prove that, for smooth solutions, these $a$ $posteriori$ error estimates converge at a fixed time to the exact spatial errors in the $L^2$-norm under mesh refinement. The order of convergence is proved to be $p$ $+$ $\frac{3}{2}$. Finally, we prove that the global effectivity index converges to unity at $\mathcal{O}(h^{\frac{1}{2}})$ rate. Several numerical examples are provided to illustrate the global superconvergence results and the convergence of the proposed error estimator.

  • Keywords

Local discontinuous Galerkin method, nonlinear KdV equations, superconvergence, $a$ $posteriori$ error estimation.

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@Article{IJNAM-17-767, author = {​Mahboub Baccouch , }, title = {Asymptotically Exact a Posteriori Error Estimates for the Local Discontinuous Galerkin Method for Nonlinear KdV Equations in One Space Dimension}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2020}, volume = {17}, number = {6}, pages = {767--793}, abstract = {

In this paper, we develop and analyze an implicit $a$ $posteriori$ error estimates for the local discontinuous Galerkin (LDG) method for nonlinear third-order Korteweg-de Vries (KdV) equations in one space dimension. First, we show that the LDG error on each element can be split into two parts. The first part is proportional to the $(p+1)$-degree right Radau polynomial and the second part converges with order $p$ $+$ $\frac{3}{2}$ in the $L^2$-norm, when piecewise polynomials of degree at most $p$ are used. These results allow us to construct $a$ $posteriori$ LDG error estimates. The proposed error estimates are computationally simple and are obtained by solving a local steady problem with no boundary conditions on each element. Furthermore, we prove that, for smooth solutions, these $a$ $posteriori$ error estimates converge at a fixed time to the exact spatial errors in the $L^2$-norm under mesh refinement. The order of convergence is proved to be $p$ $+$ $\frac{3}{2}$. Finally, we prove that the global effectivity index converges to unity at $\mathcal{O}(h^{\frac{1}{2}})$ rate. Several numerical examples are provided to illustrate the global superconvergence results and the convergence of the proposed error estimator.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/18350.html} }
TY - JOUR T1 - Asymptotically Exact a Posteriori Error Estimates for the Local Discontinuous Galerkin Method for Nonlinear KdV Equations in One Space Dimension AU - ​Mahboub Baccouch , JO - International Journal of Numerical Analysis and Modeling VL - 6 SP - 767 EP - 793 PY - 2020 DA - 2020/10 SN - 17 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/18350.html KW - Local discontinuous Galerkin method, nonlinear KdV equations, superconvergence, $a$ $posteriori$ error estimation. AB -

In this paper, we develop and analyze an implicit $a$ $posteriori$ error estimates for the local discontinuous Galerkin (LDG) method for nonlinear third-order Korteweg-de Vries (KdV) equations in one space dimension. First, we show that the LDG error on each element can be split into two parts. The first part is proportional to the $(p+1)$-degree right Radau polynomial and the second part converges with order $p$ $+$ $\frac{3}{2}$ in the $L^2$-norm, when piecewise polynomials of degree at most $p$ are used. These results allow us to construct $a$ $posteriori$ LDG error estimates. The proposed error estimates are computationally simple and are obtained by solving a local steady problem with no boundary conditions on each element. Furthermore, we prove that, for smooth solutions, these $a$ $posteriori$ error estimates converge at a fixed time to the exact spatial errors in the $L^2$-norm under mesh refinement. The order of convergence is proved to be $p$ $+$ $\frac{3}{2}$. Finally, we prove that the global effectivity index converges to unity at $\mathcal{O}(h^{\frac{1}{2}})$ rate. Several numerical examples are provided to illustrate the global superconvergence results and the convergence of the proposed error estimator.

​Mahboub Baccouch. (2020). Asymptotically Exact a Posteriori Error Estimates for the Local Discontinuous Galerkin Method for Nonlinear KdV Equations in One Space Dimension. International Journal of Numerical Analysis and Modeling. 17 (6). 767-793. doi:
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