Postprocessing Techniques of High-Order Galerkin Approximations to Nonlinear Second-Order Initial Value Problems with Applications to Wave Equations (Communications in Computational Physics, 2024)
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) timestepping methods for nonlinear second-order initial value problems, respectively. Wefirst 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 localpostprocessing techniques for the $C^1$- and $C^0$-$CG$ methods, respectively. The key ideaof the postprocessing techniques is to add a certain higher order generalized Jacobipolynomial of degree $k+1$ to the $C^1$- or $C^0$-$CG$ approximation of degree $k$ on each localtime step. We prove that, for problems with regular solutions, such postprocessingtechniques 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. Asapplications, we apply the superconvergent postprocessing techniques to the $C^1$- and $C^0$-$CG$ time discretization of nonlinear wave equations. Several numerical examplesare presented to verify the theoretical results.
Keywords: Galerkin time stepping; second-order initial value problem; superconvergent postprocessing.