- Journal Home
- Volume 37 - 2025
- Volume 36 - 2024
- Volume 35 - 2024
- Volume 34 - 2023
- Volume 33 - 2023
- Volume 32 - 2022
- Volume 31 - 2022
- Volume 30 - 2021
- Volume 29 - 2021
- Volume 28 - 2020
- Volume 27 - 2020
- Volume 26 - 2019
- Volume 25 - 2019
- Volume 24 - 2018
- Volume 23 - 2018
- Volume 22 - 2017
- Volume 21 - 2017
- Volume 20 - 2016
- Volume 19 - 2016
- Volume 18 - 2015
- Volume 17 - 2015
- Volume 16 - 2014
- Volume 15 - 2014
- Volume 14 - 2013
- Volume 13 - 2013
- Volume 12 - 2012
- Volume 11 - 2012
- Volume 10 - 2011
- Volume 9 - 2011
- Volume 8 - 2010
- Volume 7 - 2010
- Volume 6 - 2009
- Volume 5 - 2009
- Volume 4 - 2008
- Volume 3 - 2008
- Volume 2 - 2007
- Volume 1 - 2006
Commun. Comput. Phys., 35 (2024), pp. 816-858.
Published online: 2024-04
Cited by
- BibTex
- RIS
- TXT
The aim of this paper is to propose and analyze two postprocessing techniques for improving the accuracy of the $C^1$- and $C^0$-continuous Galerkin (CG) time stepping methods for nonlinear second-order initial value problems, respectively. We first derive several optimal a priori error estimates and nodal superconvergent estimates for the $C^1$- and $C^0$-$CG$ methods. Then we propose two simple but efficient local postprocessing techniques for the $C^1$- and $C^0$-$CG$ methods, respectively. The key idea of the postprocessing techniques is to add a certain higher order generalized Jacobi polynomial of degree $k+1$ to the $C^1$- or $C^0$-$CG$ approximation of degree $k$ on each local time step. We prove that, for problems with regular solutions, such postprocessing techniques improve the global convergence rates for the $L^2$-, $H^1$- and $L^∞$-error estimates of the $C^1$- and $C^0$-$CG$ methods with quasi-uniform meshes by one order. As applications, we apply the superconvergent postprocessing techniques to the $C^1$- and $C^0$-$CG$ time discretization of nonlinear wave equations. Several numerical examples are presented to verify the theoretical results.
}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2023-0232}, url = {http://global-sci.org/intro/article_detail/cicp/23060.html} }The aim of this paper is to propose and analyze two postprocessing techniques for improving the accuracy of the $C^1$- and $C^0$-continuous Galerkin (CG) time stepping methods for nonlinear second-order initial value problems, respectively. We first derive several optimal a priori error estimates and nodal superconvergent estimates for the $C^1$- and $C^0$-$CG$ methods. Then we propose two simple but efficient local postprocessing techniques for the $C^1$- and $C^0$-$CG$ methods, respectively. The key idea of the postprocessing techniques is to add a certain higher order generalized Jacobi polynomial of degree $k+1$ to the $C^1$- or $C^0$-$CG$ approximation of degree $k$ on each local time step. We prove that, for problems with regular solutions, such postprocessing techniques improve the global convergence rates for the $L^2$-, $H^1$- and $L^∞$-error estimates of the $C^1$- and $C^0$-$CG$ methods with quasi-uniform meshes by one order. As applications, we apply the superconvergent postprocessing techniques to the $C^1$- and $C^0$-$CG$ time discretization of nonlinear wave equations. Several numerical examples are presented to verify the theoretical results.