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Analysis and optimization of the generalized Schwarz method for elliptic problems with application to fluid–structure interaction

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Abstract

We propose a unified convergence analysis of the generalized Schwarz method applied to a linear elliptic problem for a general interface (flat, cylindrical or spherical) in any dimension. In particular, we provide the exact convergence set of the interface symbols related to the operators involved in the transmission conditions. We also provide a general procedure to obtain estimates of the optimized interface symbols within the constants. We apply such general results to a simple fluid–structure interaction model problem given by the interaction between an incompressible, inviscid fluid and the wave equation. Finally, we assess the effectiveness of the theoretical findings through three-dimensional numerical experiments in the haemodynamic context, obtained by solving the coupling between the Navier–Stokes equations and the linear infinitesimal elasticity.

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Acknowledgments

Giacomo Gigante has been partially supported by the Italian PRIN 2010-2011 “Real and complex manifolds: geometry, topology and harmonic analysis”. Christian Vergara has been partially supported by the Italian MIUR PRIN09 project no. 2009Y4RC3B_001.

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  1. Dipartimento di Ingegneria Gestionale, dell’Informazione e della Produzione, Università degli Studi di Bergamo, Viale Marconi 5, 24044, Dalmine, BG, Italy

    Giacomo Gigante & Christian Vergara

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  1. Giacomo Gigante
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  2. Christian Vergara
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Correspondence to Christian Vergara.

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Gigante, G., Vergara, C. Analysis and optimization of the generalized Schwarz method for elliptic problems with application to fluid–structure interaction. Numer. Math. 131, 369–404 (2015). https://doi.org/10.1007/s00211-014-0693-2

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  • DOI: https://doi.org/10.1007/s00211-014-0693-2

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