Post-processed Galerkin approximation of improved order for wave equations

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.

Comparative study of continuously differentiable Galerkin time discretizations for the wave equation

“…Error estimates of optimal order in space and time and superconvergence of the non post-processed approximation in the discrete time nodes t n are proved in [3]. In a post-processing step the thus obtained solution is then lifted to a continuously differentiable in time approximation.…”

“…for n > 1 and c n−1 (w τ,h ) := ∂ t w τ,h|I n (t n−1 ) − ∂ t L τ w τ,h (0) for n = 1, where ∂ t L τ w τ,h (0) is supposed to be given; cf. [3].…”

“…In a post-processing step the thus obtained solution is then lifted to a continuously differentiable in time approximation. Error estimates of optimal order in space and time and superconvergence of the non post-processed approximation in the discrete time nodes t n are proved in [3]. The continuous Galerkin-Petrov approximation cGP(k) of (1.1) is defined as follows.…”

“…) is supposed to be given; cf. [3]. The conceptual basis of our second family of scheme is the establishment of a direct connection between the Galerkin method for the time discretization and the classical collocation methods, with the perspective of achieving the accuracy of the former with reduced computational costs provided by the latter; cf.…”

“…The discretization of (1.1) by space-time finite element methods has attracted researchers' interest; cf., e.g., [3]. We present and compare numerically two different Galerkin time discretizations with continuously differentiable in time discrete solutions.…”