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Post-processed Galerkin approximation of improved order for wave equations

Authors: M. Bause, U. Köcher, F. A. Radu and F. Schieweck
Journal: Math. Comp. 89 (2020), 595-627
MSC (2010): Primary 65M60, 65M12; Secondary 35L05
Published electronically: August 30, 2019
MathSciNet review: 4044443
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Abstract: 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.

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Additional Information

M. Bause
Affiliation: Faculty of Mechanical Engineering, Helmut Schmidt University, Holstenhofweg 85, 22043 Hamburg, Germany

U. Köcher
Affiliation: Faculty of Mechanical Engineering, Helmut Schmidt University, Holstenhofweg 85, 22043 Hamburg, Germany

F. A. Radu
Affiliation: Department of Mathematics, University of Bergen, Allégaten 41, 50520 Bergen, Norway

F. Schieweck
Affiliation: Faculty of Mathematics, University of Magdeburg, Universitätsplatz 2, 39106 Magdeburg, Germany

Keywords: Wave equation, space-time finite element methods, variational time discretization, post-processing, error estimates, superconvergence
Received by editor(s): March 7, 2018
Received by editor(s) in revised form: December 10, 2018
Published electronically: August 30, 2019
Additional Notes: The first author is the corresponding author.
This work was supported by the German Academic Exchange Service (DAAD) under the grant ID 57238185, by the Research Council of Norway under the grant ID 255510, and the Toppforsk projekt under the grant ID 250223.
Article copyright: © Copyright 2019 American Mathematical Society