A Posteriori Error Analysis for an Ultra-Weak Discontinuous Galerkin Approximations of Nonlinear Second-Order Two-Point Boundary-Value Problems
Year: 2023
Author: Mahboub Baccouch
International Journal of Numerical Analysis and Modeling, Vol. 20 (2023), Iss. 5 : pp. 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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/ijnam2023-1027
International Journal of Numerical Analysis and Modeling, Vol. 20 (2023), Iss. 5 : pp. 618–646
Published online: 2023-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 29
Keywords: Second-order boundary-value problems ultra-weak discontinuous Galerkin method superconvergence a posteriori error estimation Baccouch polynomials.