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Mixed Methods for a Stream-Function – Vorticity Formulation of the Axisymmetric Brinkman Equations

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Abstract

This paper is devoted to the numerical analysis of a family of finite element approximations for the axisymmetric, meridian Brinkman equations written in terms of the stream-function and vorticity. A mixed formulation is introduced involving appropriate weighted Sobolev spaces, where well-posedness is derived by means of the Babuška–Brezzi theory. We introduce a suitable Galerkin discretization based on continuous piecewise polynomials of degree \(k\ge 1\) for all the unknowns, where its solvability is established using the same framework as the continuous problem. Optimal a priori error estimates are derived, which are robust with respect to the fluid viscosity, and valid also in the pure Darcy limit. A few numerical examples are presented to illustrate the convergence and performance of the proposed schemes.

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Acknowledgments

The first author was supported by CONICYT-Chile through FONDECYT Project 11160706 and by DIUBB through Projects 165608-3/R and 151408 GI/VC. The second author was partially supported by CONICYT-Chile through FONDECYT project 1140791, by DIUBB through project 151408 GI/VC and by Anillo ANANUM, ACT1118, CONICYT (Chile). The fourth author acknowledges the support by the Elsevier Mathematical Sciences Sponsorship Fund MSSF-2016.

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Authors and Affiliations

  1. GIMNAP, Departamento de Matemática, Universidad del Bío-Bío, Avenida Collao 1202, Concepción, Chile

    Verónica Anaya & David Mora

  2. Centro de Investigación en Ingeniería Matemática (CI2MA), Universidad de Concepción, Víctor Lamas 1290, Concepción, Chile

    David Mora

  3. Departamento de Matemáticas y Estadística, Universidad de Córdoba, Carrera 6 No. 76-103, Montería, Colombia

    Carlos Reales

  4. Mathematical Institute, Oxford University, A. Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, OX2 6GG, Oxford, UK

    Ricardo Ruiz-Baier

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  1. Verónica Anaya
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  2. David Mora
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  3. Carlos Reales
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  4. Ricardo Ruiz-Baier
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Correspondence to Carlos Reales.

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Anaya, V., Mora, D., Reales, C. et al. Mixed Methods for a Stream-Function – Vorticity Formulation of the Axisymmetric Brinkman Equations. J Sci Comput 71, 348–364 (2017). https://doi.org/10.1007/s10915-016-0302-x

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  • DOI: https://doi.org/10.1007/s10915-016-0302-x

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