Postprocessing Techniques of High-Order Galerkin Approximations to Nonlinear Second-Order Initial Value Problems with Applications to Wave Equations
Year: 2024
Author: Mingzhu Zhang, Lijun Yi
Communications in Computational Physics, Vol. 35 (2024), Iss. 3 : pp. 816–858
Abstract
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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cicp.OA-2023-0232
Communications in Computational Physics, Vol. 35 (2024), Iss. 3 : pp. 816–858
Published online: 2024-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 43
Keywords: Galerkin time stepping second-order initial value problem superconvergent postprocessing.