(untitled) (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) 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.