Analysis of a posteriori error estimates of the discontinuous Galerkin method for nonlinear ordinary differential equations

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• 作者：Mahboub Baccouch ^{mbaccouch@unomaha.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Discontinuous Galerkin method ; Nonlinear ordinary differential equations ; Superconvergence ; A posteriori error estimation ; Adaptive mesh refinement
• 刊名：Applied Numerical Mathematics
• 出版年：2016
• 出版时间：August 2016
• 年：2016
• 卷：106
• 期：Complete
• 页码：129-153
• 全文大小：1907 K

文摘

We develop and analyze a new residual-based a posteriori error estimator for the discontinuous Galerkin (DG) method for nonlinear ordinary differential equations (ODEs). The a posteriori   DG error estimator under investigation is computationally simple, efficient, and asymptotically exact. It is obtained by solving a local residual problem with no boundary condition on each element. We first prove that the DG solution exhibits an optimal the MathML source">O(h^p+1)$\mathcal{O}(h^{p+1})$ convergence rate in the the MathML source">L²$L^2$-norm when p  -degree piecewise polynomials with the MathML source">p≥1$p\ge 1$ are used. We further prove that the DG solution is ce4ec688465d5039d1" title="Click to view the MathML source">O(h^2p+1)$\mathcal{O}(h^{2p+1})$ superconvergent at the downwind points. We use these results to prove that the p  -degree DG solution is the MathML source">O(h^p+2)$\mathcal{O}(h^{p+2})$ super close to a particular projection of the exact solution. This superconvergence result allows us to show that the true error can be divided into a significant part and a less significant part. The significant part of the discretization error for the DG solution is proportional to the the MathML source">(p+1)$(p+1)$-degree right Radau polynomial and the less significant part converges at the MathML source">O(h^p+2)$\mathcal{O}(h^{p+2})$ rate in the the MathML source">L²$L^2$-norm. Numerical experiments demonstrate that the theoretical rates are optimal. Based on the global superconvergent approximations, we construct asymptotically exact a posteriori   error estimates and prove that they converge to the true errors in the the MathML source">L²$L^2$-norm under mesh refinement. The order of convergence is proved to be the MathML source">p+2$p+2$. Finally, we prove that the global effectivity index in the the MathML source">L²$L^2$-norm converges to unity at the MathML source">O(h)$\mathcal{O}(h)$ rate. Several numerical examples are provided to illustrate the global superconvergence results and the convergence of the proposed estimator under mesh refinement. A local adaptive procedure that makes use of our local a posteriori error estimate is also presented.