Postprocessing Techniques of High-Order Galerkin Approximations to Nonlinear Second-Order Initial Value Problems with Applications to Wave Equations

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.