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 infinity -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.