Post-processed Galerkin approximation of improved order for wave equations

We introduce and analyze a post-processing for continuous variational space-time discretizations of wave problems. The post-processing lifts the fully discrete approximation in time from a continuous to a continuously differentiable one. Further, it increases the order of convergence. The discretization in time is based on the Gauss-Lobatto quadrature formula which is essential for ensuring the improved convergence behavior. Error estimates of optimal order in various norms are proved. A bound of superconvergence at the discrete time nodes is included. To show the error estimates, a special approach is developed. First, error estimates for the time derivative of the post-processed solution are proved. In a second step these results are used to show the error estimates for the post-processed solution itself. The need for this approach comes through the structure of the wave equation. Stability properties of its solution preclude us from using absorption arguments for the control of certain error quantities. A further key ingredient of this work is the construction of a time-interpolate of the exact solution that is needed in an essential way. Finally, a conservation of energy property is shown for the post-processed solution which is an important feature for approximation schemes to wave equations. The error estimates are confirmed by numerical experiments.