Abstract
In this paper, we consider the boundary stabilization problem associated to the 1 − d wave equation with both variable density and diffusion coefficients and to its finite difference semi-discretizations. It is well-known that, for the finite difference semi-discretization of the constant coefficients wave equation on uniform meshes (Tébou and Zuazua, Adv. Comput. Math. 26:337–365, 2007) or on some non-uniform meshes (Marica and Zuazua, BCAM, 2013, preprint), the discrete decay rate fails to be uniform with respect to the mesh-size parameter. We prove that, under suitable regularity assumptions on the coefficients and after adding an appropriate artificial viscosity to the numerical scheme, the decay rate is uniform as the mesh-size tends to zero. This extends previous results in Tébou and Zuazua (Adv. Comput. Math. 26:337–365, 2007) on the constant coefficient wave equation. The methodology of proof consists in applying the classical multiplier technique at the discrete level, with a multiplier adapted to the variable coefficients.
Mathematics Subject Classification (2010). Primary 49K40; Secondary 35L05
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Acknowledgements
This work is supported by the Advanced Grant NUMERIWAVES/FP7-246775 of the European Research Council Executive Agency, FA9550-14-1-0214 of the EOARD-AFOSR, MTM2011-29306-C02-00 and SEV-2013-0323 Grants of the MINECO Spain, the PI2010-04 Grant and the BERC 2014-2017 Program of the Basque Government. The details of this work were elaborated during the postdoc stay of the first author at Institute of Mathematics and Scientific Computing and University of Graz, funded by the MOBIS - Mathematical Optimization and Applications in Biomedical Sciences grant of the FWF - Austrian Science Fund. Additionally, the first author’s work was supported by two grants of the Romanian Ministry of National Education (CNCS-UEFISCDI), i.e., projects PN-II-ID-PCE-2012-4-0021 Variable Exponent Analysis: Partial Differential Equations and Calculus of Variations and PN-II-ID-PCE-2011-3-0075 Analysis, control and numerical approximations of PDEs. Both authors thank the CIMI - Toulouse for the hospitality and support during the preparation of this work in the context of the Excellence Chair in PDE, Control and Numerics.
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Marica, A., Zuazua, E. (2014). Boundary Stabilization of Numerical Approximations of the 1-D Variable Coefficients Wave Equation: A Numerical Viscosity Approach. In: Hoppe, R. (eds) Optimization with PDE Constraints. Lecture Notes in Computational Science and Engineering, vol 101. Springer, Cham. https://doi.org/10.1007/978-3-319-08025-3_9
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DOI: https://doi.org/10.1007/978-3-319-08025-3_9
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