Volume 14, Issue 3
An Optimal a Posteriori Error Estimates of the Local Discontinuous Galerkin Method for the Second-Order Wave Equation in One Space Dimension.

Mahboub Baccouch

Int. J. Numer. Anal. Mod., 14 (2017), pp. 355-380

Published online: 2017-06

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  • 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 L²-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 L²-norm at O(h^{p+2}) rate. Finally, we show that the global effectivity indices in the L²-norm converge to unity at 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 ≥ 1. Several numerical experiments are performed to validate the theoretical results.

  • Keywords

Local discontinuous Galerkin method second-order wave equation superconvergence Radau points a posteriori error estimation

  • AMS Subject Headings

65M15 65M60 65M50 65N30 65N50

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COPYRIGHT: © Global Science Press

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@Article{IJNAM-14-355, author = {Mahboub Baccouch}, title = {An Optimal a Posteriori Error Estimates of the Local Discontinuous Galerkin Method for the Second-Order Wave Equation in One Space Dimension.}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2017}, volume = {14}, number = {3}, pages = {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 L²-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 L²-norm at O(h^{p+2}) rate. Finally, we show that the global effectivity indices in the L²-norm converge to unity at 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 ≥ 1. Several numerical experiments are performed to validate the theoretical results.}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/10012.html} }
TY - JOUR T1 - An Optimal a Posteriori Error Estimates of the Local Discontinuous Galerkin Method for the Second-Order Wave Equation in One Space Dimension. AU - Mahboub Baccouch JO - International Journal of Numerical Analysis and Modeling VL - 3 SP - 355 EP - 380 PY - 2017 DA - 2017/06 SN - 14 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/10012.html KW - Local discontinuous Galerkin method KW - second-order wave equation KW - superconvergence KW - Radau points KW - a posteriori error estimation AB - 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 L²-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 L²-norm at O(h^{p+2}) rate. Finally, we show that the global effectivity indices in the L²-norm converge to unity at 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 ≥ 1. Several numerical experiments are performed to validate the theoretical results.
Mahboub Baccouch. (1970). An Optimal a Posteriori Error Estimates of the Local Discontinuous Galerkin Method for the Second-Order Wave Equation in One Space Dimension.. International Journal of Numerical Analysis and Modeling. 14 (3). 355-380. doi:
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