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Artificial conditions for the linear elasticity equations


Authors: Virginie Bonnaillie-Noël, Marc Dambrine, Frédéric Hérau and Grégory Vial
Journal: Math. Comp. 84 (2015), 1599-1632
MSC (2010): Primary 35J47, 35J57, 35P10, 35S15, 47A10, 47G30, 65N20
Published electronically: November 18, 2014
MathSciNet review: 3335885
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Abstract: In this paper, we consider the equations of linear elasticity in an exterior domain. We exhibit artificial boundary conditions on a circle, which lead to a non-coercive second order boundary value problem. In the particular case of an axisymmetric geometry, explicit computations can be performed in Fourier series proving the well-posedness except for a countable set of parameters. A perturbation argument allows us to consider near-circular domains. We complete the analysis by some numerical simulations.


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Virginie Bonnaillie-Noël
Affiliation: IRMAR - UMR6625, ENS Rennes, Univ. Rennes 1, CNRS, UEB, av. Robert Schuman, 35170 Bruz, France
Email: bonnaillie@math.cnrs.fr

Marc Dambrine
Affiliation: LMAP - UMR5142, Université de Pau et des Pays de l’Adour, av. de l’Université, BP 1155, 64013 Pau Cedex, France
Email: marc.dambrine@univ-pau.fr

Frédéric Hérau
Affiliation: LMJL - UMR6629, Université de Nantes, 2 rue de la Houssinière, BP 92208, 44322 Nantes Cedex 3, France
Email: frederic.herau@univ-nantes.fr

Grégory Vial
Affiliation: Université de Lyon, CNRS UMR 5208, École Centrale de Lyon, Institut Camille Jordan, 36 avenue Guy de Collongue, 69134 Écully cedex, France
Email: gregory.vial@ec-lyon.fr

DOI: https://doi.org/10.1090/S0025-5718-2014-02901-3
Keywords: Linear elasticity equations, singular perturbation, artificial boundary conditions, Ventcel condition, Dirichlet-to-Neumann map, spectral theory
Received by editor(s): December 10, 2012
Received by editor(s) in revised form: September 4, 2013, and October 7, 2013
Published electronically: November 18, 2014
Article copyright: © Copyright 2014 American Mathematical Society