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 - Baccouch, Mahboub

N1 - Funding Information:
This research was supported by the University Committee on Research and Creative Activity (UCRCA Proposal 2016-01-F) at the University of Nebraska at Omaha.
Publisher Copyright:
© 2017 Institute for Scientific Computing and Information.

PY - 2017

Y1 - 2017

N2 - 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(hp+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 Pp polynomials with p ≥ 1. Several numerical experiments are performed to validate the theoretical results.

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(hp+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 Pp polynomials with p ≥ 1. Several numerical experiments are performed to validate the theoretical results.

KW - A posteriori error estimation

KW - Local discontinuous Galerkin method

KW - Radau points

KW - Second-order wave equation

KW - Superconvergence

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M3 - Article

AN - SCOPUS:85019643566

VL - 14

SP - 355

EP - 380

JO - International Journal of Numerical Analysis and Modeling

JF - International Journal of Numerical Analysis and Modeling

SN - 1705-5105

IS - 3

ER -