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A Schwarz Method for a Rayleigh–Bénard Problem

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Abstract

The aim of this work is to study a Schwarz domain decomposition numerical method for a stationary Rayleigh–Bénard convection problem. The model equations are the stationary version of the incompressible Navier–Stokes equations coupled with a heat equation under Boussinesq approximation. The problem is defined in a rectangular domain. The nonlinear stationary problem is dealt with a Newton method. Each step in the Newton method is solved with a Schwarz domain decomposition method with the domain partitioned into several subdomains with appropriate interface conditions. Their convergence properties are studied theoretically in a simplified domain divided in two subdomains including two artificial parameters in the equations. The numerical resolution of the problem confirms the theoretical results. The convergence rate is less than one when overlap is considered. Convergence is achieved for large values of the aspect ratio, which are inabordable for the standard Legendre collocation method. Convergence is optimal for some values of the parameters. Other advantages of this methodology compared with standard methods are parallelization and high order.

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Acknowledgements

This work was partially supported by Research Grants MTM2015-68818-R (MINECO, Spanish Government) and PEII-2014-006-A (Junta de Comunidades de Castilla-La Mancha), which include RDEF funds.

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Authors and Affiliations

  1. Departamento de Matemáticas, Facultad de Ciencias y Tecnologías Químicas, Universidad de Castilla-La Mancha, 13071, Ciudad Real, Spain

    H. Herrero & F. Pla

  2. Departamento de Informática, Escuela Superior de Ingeniería, Universidad de Almería, 04120, Almería, Spain

    M. Ruiz-Ferrández

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  1. H. Herrero
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  2. F. Pla
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  3. M. Ruiz-Ferrández
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Correspondence to H. Herrero.

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Herrero, H., Pla, F. & Ruiz-Ferrández, M. A Schwarz Method for a Rayleigh–Bénard Problem. J Sci Comput 78, 376–392 (2019). https://doi.org/10.1007/s10915-018-0771-1

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  • DOI: https://doi.org/10.1007/s10915-018-0771-1

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