Volume 34, Issue 5
Optimal a Posteriori Error Estimates of the Local Discontinuous Galerkin Method for Convection-Diffusion Problems in One Space Dimension

J. Comp. Math., 34 (2016), pp. 511-531.

Published online: 2016-10

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• Abstract

In this paper, we derive optimal order a posteriori error estimates for the local discontinuous Galerkin (LDG) method for linear convection-diffusion problems in one space dimension. One of the key ingredients in our analysis is the recent optimal superconvergence result in [Y. Yang and C.-W. Shu, J. Comp. Math., 33 (2015), pp. 323-340]. 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. These results are used 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. We further prove that, for smooth solutions, the a posteriori LDG error estimates, which were constructed by the author in an earlier paper, converge, at a fixed time, to the true spatial errors in the L²-norm at O($h^{p+2}$) rate. Finally, we prove that the global effectivity indices in the L²-norm converge to unity at O(h) rate. These 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.

• Keywords

Local discontinuous Galerkin method Convection-diffusion problems Superconvergence Radau polynomials A posteriori error estimation

65M15 65M60 65M50 65N30 65N50.

@Article{JCM-34-511, author = {Baccouch , Mahboub }, title = {Optimal a Posteriori Error Estimates of the Local Discontinuous Galerkin Method for Convection-Diffusion Problems in One Space Dimension}, journal = {Journal of Computational Mathematics}, year = {2016}, volume = {34}, number = {5}, pages = {511--531}, abstract = { In this paper, we derive optimal order a posteriori error estimates for the local discontinuous Galerkin (LDG) method for linear convection-diffusion problems in one space dimension. One of the key ingredients in our analysis is the recent optimal superconvergence result in [Y. Yang and C.-W. Shu, J. Comp. Math., 33 (2015), pp. 323-340]. 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. These results are used 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. We further prove that, for smooth solutions, the a posteriori LDG error estimates, which were constructed by the author in an earlier paper, converge, at a fixed time, to the true spatial errors in the L²-norm at O($h^{p+2}$) rate. Finally, we prove that the global effectivity indices in the L²-norm converge to unity at O(h) rate. These 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.}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1603-m2015-0317}, url = {http://global-sci.org/intro/article_detail/jcm/9810.html} }
TY - JOUR T1 - Optimal a Posteriori Error Estimates of the Local Discontinuous Galerkin Method for Convection-Diffusion Problems in One Space Dimension AU - Baccouch , Mahboub JO - Journal of Computational Mathematics VL - 5 SP - 511 EP - 531 PY - 2016 DA - 2016/10 SN - 34 DO - http://doi.org/10.4208/jcm.1603-m2015-0317 UR - https://global-sci.org/intro/article_detail/jcm/9810.html KW - Local discontinuous Galerkin method KW - Convection-diffusion problems KW - Superconvergence KW - Radau polynomials KW - A posteriori error estimation AB - In this paper, we derive optimal order a posteriori error estimates for the local discontinuous Galerkin (LDG) method for linear convection-diffusion problems in one space dimension. One of the key ingredients in our analysis is the recent optimal superconvergence result in [Y. Yang and C.-W. Shu, J. Comp. Math., 33 (2015), pp. 323-340]. 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. These results are used 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. We further prove that, for smooth solutions, the a posteriori LDG error estimates, which were constructed by the author in an earlier paper, converge, at a fixed time, to the true spatial errors in the L²-norm at O($h^{p+2}$) rate. Finally, we prove that the global effectivity indices in the L²-norm converge to unity at O(h) rate. These 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.