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

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 i...

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Bibliographic Details
Published in:Numerische Mathematik 2015-10, Vol.131 (2), p.369-404
Main Authors: Gigante, Giacomo, Vergara, Christian
Format: Article
Language:English
Subjects:
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Numerical Analysis
Numerical and Computational Physics
Simulation
Theoretical
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Summary: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.
ISSN:0029-599X
0945-3245
DOI:10.1007/s00211-014-0693-2