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Numerical Analysis of Penalty Stabilized Finite Element Discretizations of Evolution Navier–Stokes Equations

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Abstract

We perform in this paper the numerical analysis of some penalty stabilized solvers for the unsteady Navier–Stokes equations. We consider low-order and high-order methods. The low-order method is a pure penalty method, while the high-order one is a projection-stabilized method. We perform their numerical analysis (stability and convergence) for solutions that only need to bear the natural regularity. In this analysis, the stability is based upon specific inf-sup conditions. No local orthogonality properties are needed for the projection-interpolation operator. The convergence is based upon the representation of the stabilizing terms by means of bubble finite element spaces. We include some numerical tests for realistic flows that confirm the theoretical expectations.

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Acknowledgments

The work of all three authors has been partially supported by Proyectos de Excelencia de la Junta de Andalucia—FEDER funds P07-FQM-02538 and P12-FQM-454. The work of M. Restelli has been partially supported by Regione Lombardia and CILEA Consortium through a “LISA” 2012 project as well as by the CINECA Consortium award N. HP10C9F0EL, 2012.

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Authors and Affiliations

  1. IMUS & Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de correos 1160, 41080, Sevilla, Spain

    T. Chacón Rebollo

  2. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de correos 1160, 41080, Sevilla, Spain

    M. Gómez Mármol & M. Restelli

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  1. T. Chacón Rebollo
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  2. M. Gómez Mármol
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  3. M. Restelli
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Correspondence to T. Chacón Rebollo.

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Chacón Rebollo, T., Gómez Mármol, M. & Restelli, M. Numerical Analysis of Penalty Stabilized Finite Element Discretizations of Evolution Navier–Stokes Equations. J Sci Comput 63, 885–912 (2015). https://doi.org/10.1007/s10915-014-9918-x

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