- Journal Home
- Volume 39 - 2021
- Volume 38 - 2020
- Volume 37 - 2019
- Volume 36 - 2018
- Volume 35 - 2017
- Volume 34 - 2016
- Volume 33 - 2015
- Volume 32 - 2014
- Volume 31 - 2013
- Volume 30 - 2012
- Volume 29 - 2011
- Volume 28 - 2010
- Volume 27 - 2009
- Volume 26 - 2008
- Volume 25 - 2007
- Volume 24 - 2006
- Volume 23 - 2005
- Volume 22 - 2004
- Volume 21 - 2003
- Volume 20 - 2002
- Volume 19 - 2001
- Volume 18 - 2000
- Volume 17 - 1999
- Volume 16 - 1998
- Volume 15 - 1997
- Volume 14 - 1996
- Volume 13 - 1995
- Volume 12 - 1994
- Volume 11 - 1993
- Volume 10 - 1992
- Volume 9 - 1991
- Volume 8 - 1990
- Volume 7 - 1989
- Volume 6 - 1988
- Volume 5 - 1987
- Volume 4 - 1986
- Volume 3 - 1985
- Volume 2 - 1984
- Volume 1 - 1983
Optimal a Posteriori Error Estimates of the Local Discontinuous Galerkin Method for Convection-Diffusion Problems in One Space Dimension
- BibTex
- RIS
- TXT
@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.
Mahboub Baccouch. (2019). Optimal a Posteriori Error Estimates of the Local Discontinuous Galerkin Method for Convection-Diffusion Problems in One Space Dimension.
Journal of Computational Mathematics. 34 (5).
511-531.
doi:10.4208/jcm.1603-m2015-0317
Copy to clipboard