Volume 54, Number 6, November-December 2020
|Page(s)||2099 - 2123|
|Published online||12 October 2020|
Galerkin–collocation approximation in time for the wave equation and its post-processing
Helmut Schmidt University, Faculty of Mechanical Engineering, Holstenhofweg 85, 22043 Hamburg, Germany
2 Technical University of Dresden, Institute of Numerical Mathematics, 01062 Dresden, Germany
* Corresponding author: firstname.lastname@example.org
Accepted: 1 May 2020
We introduce and analyze families of Galerkin–collocation discretization schemes in time for the wave equation. Their conceptual basis is the establishment of a 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 in terms of less complex algebraic systems. Firstly, continuously differentiable in time discrete solutions are studied. Optimal order error estimates are proved. Then, the concept of Galerkin–collocation approximation is extended to twice continuously differentiable in time discrete solutions. A direct link between the two families by a computationally cheap post-processing is presented. A key ingredient of the proposed methods is the application of quadrature rules involving derivatives. The performance properties of the schemes are illustrated by numerical experiments.
Mathematics Subject Classification: 65M60 / 65M12
Key words: Wave equation / variational time discretization / collocation methods / space-time finite element methods / error estimates / post-processing
© EDP Sciences, SMAI 2020
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