Post-processed Galerkin approximation of improved order for wave equations

TL;DR

This paper presents a novel post-processing technique for finite element approximations of wave equations that enhances convergence order, achieves superconvergence at discrete times, and preserves energy, supported by rigorous error analysis and numerical validation.

Contribution

It introduces a new post-processing method that increases the order of convergence and superconvergence for wave equation approximations, with a novel error analysis approach.

Findings

Enhanced convergence order in time discretization.

Superconvergence at discrete time nodes.

Energy conservation property of the post-processed solution.

Abstract

We introduce and analyze a post-processing for a family of variational space-time approximations to wave problems. The discretization in space and time is based on continuous finite element methods. The post-processing lifts the fully discrete approximations in time from continuous to continuously differentiable ones. Further, it increases the order of convergence of the discretization in time which can be be exploited nicely, for instance, for a-posteriori error control. The convergence behavior is shown by proving error estimates of optimal order in various norms. A bound of superconvergence at the discrete times nodes is included. To show the error estimates, a special approach is developed. Firstly, error estimates for the time derivative of the post-processed solution are proved. Then, in a second step these results are used to establish the desired error estimates for the post-processed solution itself. The need for this approach comes through the structure of the wave equation providing only stability estimates that 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 new time-interpolate of the exact solution that is needed in an essential way for deriving the error estimates. Finally, a conservation of energy property is shown for the post-processed solution which is a key feature for approximation schemes to wave equations. The error estimates given in this work are confirmed by numerical experiments.