Abstract
In arXiv:1112.4297, we studied the propagation, observation, and control properties of the 1D wave equation on a bounded interval semi-discretized in space using the quadratic classical finite element approximation. It was shown that the discrete wave dynamics consisting of the interaction of nodal and midpoint components leads to the existence of two different eigenvalue branches in the spectrum: an acoustic one, of physical nature, and an optic one, of spurious nature. The fact that both dispersion relations have critical points where the corresponding group velocities vanish produces numerical wave packets whose energy is concentrated in the interior of the domain, without propagating, and for which the observability constant blows up as the mesh size goes to zero. This extends to the quadratic finite element setting the fact that the classical property of continuous waves being observable from the boundary fails for the most classical approximations on uniform meshes (finite differences, linear finite elements, etc.). As a consequence, the numerical controls of minimal norm may blow up as the mesh size parameter tends to zero. To cure these high-frequency pathologies, in arXiv:1112.4297 we designed a filtering mechanism consisting in taking piecewise linear and continuous initial data (so that the curvature component vanishes at the initial time) with nodal components given by a bi-grid algorithm. The aim of this article is to implement this filtering technique and to show numerically its efficiency.
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Acknowledgements
Both authors were partially supported by ERC Advanced Grant FP7-246775 NUMERIWAVES, Grants MTM2008-03541 and MTM2011-29306 of MICINN Spain, Project PI2010-04 of the Basque Government, and ESF Research Networking Programme OPTPDE. Additionally, the first author was supported by Grant PN-II-ID-PCE-2011-3-0075 of CNCS-UEFISCDI Romania.
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Marica, A., Zuazua, E. (2013). On the Quadratic Finite Element Approximation of 1D Waves: Propagation, Observation, Control, and Numerical Implementation. In: de Moura, C., Kubrusly, C. (eds) The Courant–Friedrichs–Lewy (CFL) Condition. Birkhäuser, Boston. https://doi.org/10.1007/978-0-8176-8394-8_6
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DOI: https://doi.org/10.1007/978-0-8176-8394-8_6
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