Numerische Mathematik

, Volume 133, Issue 4, pp 781–817 | Cite as

A priori and a posteriori error analysis of a mixed scheme for the Brinkman problem

  • Verónica Anaya
  • David Mora
  • Ricardo Oyarzúa
  • Ricardo Ruiz-Baier
Article
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Abstract

This paper deals with the analysis of new mixed finite element methods for the Brinkman equations formulated in terms of velocity, vorticity and pressure. Employing the Babuška–Brezzi theory, it is proved that the resulting continuous and discrete variational formulations are well-posed. In particular, we show that Raviart–Thomas elements of order \(k \ge 0\) for the approximation of the velocity field, piecewise continuous polynomials of degree \(k+1\) for the vorticity, and piecewise polynomials of degree k for the pressure, yield unique solvability of the discrete problem. On the other hand, we also show that families of finite elements based on Brezzi–Douglas–Marini elements of order \(k+1\) for the approximation of velocity, piecewise continuous polynomials of degree \(k+2\) for the vorticity, and piecewise polynomials of degree k for the pressure ensure the well-posedness of the associated Galerkin scheme. We note that these families provide exactly divergence-free approximations of the velocity field. We establish a priori error estimates in the natural norms with constants independent of the viscosity and we carry out the reliability and efficiency analysis of a residual-based a posteriori error estimator. Finally, we report several numerical experiments illustrating the behaviour of the proposed schemes and confirming our theoretical results on unstructured meshes. Additional examples of cases not covered by our theory are also presented.

Mathematics Subject Classification

65N30 65N12 76D07 65N15 
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Notes

Acknowledgments

Stimulating discussions with Prof. Gabriel N. Gatica are gratefully acknowledged. In addition, V. Anaya was partially supported by CONICYT-Chile through FONDECYT postdoctorado No. 3120197, by project Inserción de Capital Humano Avanzado en la Academia No. 79112012, and DIUBB through Project 120808 GI/EF. D. Mora was partially supported by CONICYT-Chile through FONDECYT Project No. 1140791, by DIUBB through Project 120808 GI/EF, and Anillo ANANUM, ACT1118, CONICYT (Chile). R. Oyarzúa was partially supported by CONICYT-Chile through FONDECYT Project No. 1121347, by DIUBB through Project 120808 GI/EF, and Anillo ANANUM, ACT1118, CONICYT (Chile). R. Ruiz-Baier acknowledges support by the Swiss National Science Foundation through the Grant SNSF PP00P2-144922. This work was advanced during a visit of V. Anaya and D. Mora to the Institut des Sciences de la Terre, University of Lausanne.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Verónica Anaya
    • 1
  • David Mora
    • 1
    • 2
  • Ricardo Oyarzúa
    • 1
    • 2
  • Ricardo Ruiz-Baier
    • 3
  1. 1.GIMNAP, Departamento de Matemática, Facultad de CienciasUniversidad del Bío-BíoConcepciónChile
  2. 2.Centro de Investigación en Ingeniería Matemática (CI²MA)Universidad de ConcepciónConcepciónChile
  3. 3.Institut des Sciences de la TerreUniversity of Lausanne, Géopolis Unil-MoulineLausanneSwitzerland

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