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Numerical approximation schemes for multi-dimensional wave equations in asymmetric spaces


Authors: Vincent Lescarret and Enrique Zuazua
Journal: Math. Comp. 84 (2015), 119-152
MSC (2010): Primary 35A35, 35L05; Secondary 35S15
DOI: https://doi.org/10.1090/S0025-5718-2014-02887-1
Published electronically: September 4, 2014
MathSciNet review: 3266955
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Abstract: We develop finite difference numerical schemes for a model arising in multi-body structures, previously analyzed by H. Koch and E. Zuazua, constituted by two $ n$-dimensional wave equations coupled with a $ (n-1)$-dimensional one along a flexible interface.

That model, under suitable assumptions on the speed of propagation in each media, is well-posed in asymmetric spaces in which the regularity of solutions differs by one derivative from one medium to the other.

Here we consider a flat interface and analyze this property at a discrete level, for finite difference and mixed finite element methods on regular meshes parallel to the interface. We prove that those methods are well-posed in such asymmetric spaces uniformly with respect to the mesh-size parameters and we prove the convergence of the numerical solutions towards the continuous ones in these spaces.

In other words, these numerical methods that are well-behaved in standard energy spaces, preserve the convergence properties in these asymmetric spaces too.

These results are illustrated by several numerical experiments.


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Additional Information

Vincent Lescarret
Affiliation: Laboratoire des signaux et systèmes (L2S) Supélec - 3 rue Joliot-Curie, 91192 Gif-sur-Yvette cedex (France)
Email: Vincent.LESCARRET@lss.supelec.fr

Enrique Zuazua
Affiliation: BCAM - Basque Center for Applied Mathematics, Mazarredo, 14 E-48009 Bilbao-Basque Country-Spain, Ikerbasque, Basque Foundation for Science, Alameda Urquijo 36-5, Plaza Bizkaia, 48011, Bilbao-Basque Country-Spain
Email: zuazua@bcamath.org

DOI: https://doi.org/10.1090/S0025-5718-2014-02887-1
Received by editor(s): May 5, 2012
Received by editor(s) in revised form: May 5, 2013
Published electronically: September 4, 2014
Additional Notes: The first author thanks N. Burq for an enlightening discussion on the extension of the present analysis to the Schrödinger case
This work was partially supported by the Grant MTM2011-29306-C02-00 of the MICINN (Spain), project PI2010-04 of the Basque Government, the ERC Advanced Grant FP7-246775 NUMERIWAVES, the ESF Research Networking Program OPTPDE
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.