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Spectral conditions for admissibility and observability of wave systems: applications to finite element schemes

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Abstract

In this article, we derive uniform admissibility and observability properties for the finite element space semi-discretizations of \({\ddot u+A_0 u=0 }\), where A 0 is an unbounded self-adjoint positive definite operator with compact resolvent. To address this problem, we present a new spectral approach based on several spectral criteria for admissibility and observability of such systems. Our approach provides very general admissibility and observability results for finite element approximation schemes of \({\ddot u+A_{0}u =0}\), which stand in any dimension and for any regular mesh (in the sense of finite elements). Our results can be combined with previous works to derive admissibility and observability properties for full discretizations of \({\ddot u+A_0 u=0}\). We also present applications of our results to controllability and stabilization problems.

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  1. Laboratoire de Mathématiques de Versailles, Université de Versailles Saint-Quentin-en-Yvelines, 45 avenue des États Unis, 78035, Versailles Cedex, France

    Sylvain Ervedoza

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  1. Sylvain Ervedoza
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Correspondence to Sylvain Ervedoza.

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The author was partially supported by the “Agence Nationale de la Recherche” (ANR), Project C-QUID, number BLAN-3-139579.

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Ervedoza, S. Spectral conditions for admissibility and observability of wave systems: applications to finite element schemes. Numer. Math. 113, 377–415 (2009). https://doi.org/10.1007/s00211-009-0235-5

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