An Optimal a Posteriori Error Estimates of the Local Discontinuous Galerkin Method for the Second-Order Wave Equation in One Space Dimension
Year: 2017
International Journal of Numerical Analysis and Modeling, Vol. 14 (2017), Iss. 3 : pp. 355–380
Abstract
In this paper, we provide the optimal convergence rate of a posteriori error estimates for the local discontinuous Galerkin (LDG) method for the second-order wave equation in one space dimension. One of the key ingredients in our analysis is the recent optimal superconvergence result in [W. Cao, D. Li and Z. Zhang, Commun. Comput. Phys. 21 (1) (2017) 211-236]. We first prove that the LDG solution and its spatial derivative, respectively, converge in the $L^2$-norm to $(p+1)$-degree right and left Radau interpolating polynomials under mesh refinement. The order of convergence is proved to be $p+2$, when piecewise polynomials of degree at most $p$ are used. We use these results to show that the leading error terms on each element for the solution and its derivative are proportional to $(p+1)$-degree right and left Radau polynomials. These new results enable us to construct residual-based a posteriori error estimates of the spatial errors. We further prove that, for smooth solutions, these a posteriori LDG error estimates converge, at a fixed time, to the true spatial errors in the $L^2$-norm at $\mathcal{O}(h^{p+2})$ rate. Finally, we show that the global effectivity indices in the $L^2$-norm converge to unity at $\mathcal{O}(h)$ rate. The current results improve upon our previously published work in which the order of convergence for the a posteriori error estimates and the global effectivity index are proved to be $p+3/2$ and $1/2$, respectively. Our proofs are valid for arbitrary regular meshes using $P^p$ polynomials with $p\geq1$. Several numerical experiments are performed to validate the theoretical results.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2017-IJNAM-10012
International Journal of Numerical Analysis and Modeling, Vol. 14 (2017), Iss. 3 : pp. 355–380
Published online: 2017-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 26
Keywords: Local discontinuous Galerkin method second-order wave equation superconvergence Radau points a posteriori error estimation.