TY - JOUR
T1 - Postprocessing Techniques of High-Order Galerkin Approximations to Nonlinear Second-Order Initial Value Problems with Applications to Wave Equations
AU - Zhang , Mingzhu
AU - Yi , Lijun
JO - Communications in Computational Physics
VL - 3
SP - 816
EP - 858
PY - 2024
DA - 2024/04
SN - 35
DO - http://doi.org/10.4208/cicp.OA-2023-0232
UR - https://global-sci.org/intro/article_detail/cicp/23060.html
KW - Galerkin time stepping, second-order initial value problem, superconvergent postprocessing.
AB - <p style="text-align: justify;">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.</p>