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Volume 20, Issue 5
A Posteriori Error Analysis for an Ultra-Weak Discontinuous Galerkin Approximations of Nonlinear Second-Order Two-Point Boundary-Value Problems

Mahboub Baccouch

Int. J. Numer. Anal. Mod., 20 (2023), pp. 618-646.

Published online: 2023-09

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  • Abstract

In this paper, we present and analyze a posteriori error estimates in the $L^2$-norm of an ultra-weak discontinuous Galerkin (UWDG) method for nonlinear second-order boundary-value problems for ordinary differential equations of the form $u′′=f(x, u).$ We first use the superconvergence results proved in the first part of this paper (J. Appl. Math. Comput. 69, 1507-1539, 2023) to prove that the UWDG solution converges, in the $L^2$-norm, towards a special $p$-degree interpolating polynomial, when piecewise polynomials of degree at most $p ≥ 2$ are used. The order of convergence is proved to be $p + 2.$ We then show that the UWDG error on each element can be divided into two parts. The dominant part is proportional to a special $(p+1)$-degree Baccouch polynomial, which can be written as a linear combination of Legendre polynomials of degrees $p − 1,$ $p,$ and $p + 1.$ The second part converges to zero with order $p + 2$ in the $L^2$-norm. These results allow us to construct a posteriori UWDG error estimates. The proposed error estimates are computationally simple and are obtained by solving a local problem with no boundary conditions on each element. Furthermore, we prove that, for smooth solutions, these a posteriori error estimates converge to the exact errors in the $L^2$-norm under mesh refinement. The order of convergence is proved to be $p + 2.$ Finally, we prove that the global effectivity index converges to unity at $\mathcal{O}(h)$ rate. Numerical results are presented exhibiting the reliability and the efficiency of the proposed error estimator.

  • AMS Subject Headings

65L10, 65L60, 65L70

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-20-618, author = {Baccouch , Mahboub}, title = {A Posteriori Error Analysis for an Ultra-Weak Discontinuous Galerkin Approximations of Nonlinear Second-Order Two-Point Boundary-Value Problems}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2023}, volume = {20}, number = {5}, pages = {618--646}, abstract = {

In this paper, we present and analyze a posteriori error estimates in the $L^2$-norm of an ultra-weak discontinuous Galerkin (UWDG) method for nonlinear second-order boundary-value problems for ordinary differential equations of the form $u′′=f(x, u).$ We first use the superconvergence results proved in the first part of this paper (J. Appl. Math. Comput. 69, 1507-1539, 2023) to prove that the UWDG solution converges, in the $L^2$-norm, towards a special $p$-degree interpolating polynomial, when piecewise polynomials of degree at most $p ≥ 2$ are used. The order of convergence is proved to be $p + 2.$ We then show that the UWDG error on each element can be divided into two parts. The dominant part is proportional to a special $(p+1)$-degree Baccouch polynomial, which can be written as a linear combination of Legendre polynomials of degrees $p − 1,$ $p,$ and $p + 1.$ The second part converges to zero with order $p + 2$ in the $L^2$-norm. These results allow us to construct a posteriori UWDG error estimates. The proposed error estimates are computationally simple and are obtained by solving a local problem with no boundary conditions on each element. Furthermore, we prove that, for smooth solutions, these a posteriori error estimates converge to the exact errors in the $L^2$-norm under mesh refinement. The order of convergence is proved to be $p + 2.$ Finally, we prove that the global effectivity index converges to unity at $\mathcal{O}(h)$ rate. Numerical results are presented exhibiting the reliability and the efficiency of the proposed error estimator.

}, issn = {2617-8710}, doi = {https://doi.org/10.4208/ijnam2023-1027}, url = {http://global-sci.org/intro/article_detail/ijnam/22005.html} }
TY - JOUR T1 - A Posteriori Error Analysis for an Ultra-Weak Discontinuous Galerkin Approximations of Nonlinear Second-Order Two-Point Boundary-Value Problems AU - Baccouch , Mahboub JO - International Journal of Numerical Analysis and Modeling VL - 5 SP - 618 EP - 646 PY - 2023 DA - 2023/09 SN - 20 DO - http://doi.org/10.4208/ijnam2023-1027 UR - https://global-sci.org/intro/article_detail/ijnam/22005.html KW - Second-order boundary-value problems, ultra-weak discontinuous Galerkin method, superconvergence, a posteriori error estimation, Baccouch polynomials. AB -

In this paper, we present and analyze a posteriori error estimates in the $L^2$-norm of an ultra-weak discontinuous Galerkin (UWDG) method for nonlinear second-order boundary-value problems for ordinary differential equations of the form $u′′=f(x, u).$ We first use the superconvergence results proved in the first part of this paper (J. Appl. Math. Comput. 69, 1507-1539, 2023) to prove that the UWDG solution converges, in the $L^2$-norm, towards a special $p$-degree interpolating polynomial, when piecewise polynomials of degree at most $p ≥ 2$ are used. The order of convergence is proved to be $p + 2.$ We then show that the UWDG error on each element can be divided into two parts. The dominant part is proportional to a special $(p+1)$-degree Baccouch polynomial, which can be written as a linear combination of Legendre polynomials of degrees $p − 1,$ $p,$ and $p + 1.$ The second part converges to zero with order $p + 2$ in the $L^2$-norm. These results allow us to construct a posteriori UWDG error estimates. The proposed error estimates are computationally simple and are obtained by solving a local problem with no boundary conditions on each element. Furthermore, we prove that, for smooth solutions, these a posteriori error estimates converge to the exact errors in the $L^2$-norm under mesh refinement. The order of convergence is proved to be $p + 2.$ Finally, we prove that the global effectivity index converges to unity at $\mathcal{O}(h)$ rate. Numerical results are presented exhibiting the reliability and the efficiency of the proposed error estimator.

Mahboub Baccouch. (2023). A Posteriori Error Analysis for an Ultra-Weak Discontinuous Galerkin Approximations of Nonlinear Second-Order Two-Point Boundary-Value Problems. International Journal of Numerical Analysis and Modeling. 20 (5). 618-646. doi:10.4208/ijnam2023-1027
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