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

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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.

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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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Authors and Affiliations

  1. GIMNAP, Departamento de Matemática, Facultad de Ciencias, Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile

    Verónica Anaya, David Mora & Ricardo Oyarzúa

  2. Centro de Investigación en Ingeniería Matemática (CI²MA), Universidad de Concepción, Casilla 160-C, Concepción, Chile

    David Mora & Ricardo Oyarzúa

  3. Institut des Sciences de la Terre, University of Lausanne, Géopolis Unil-Mouline, 1015, Lausanne, Switzerland

    Ricardo Ruiz-Baier

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

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Anaya, V., Mora, D., Oyarzúa, R. et al. A priori and a posteriori error analysis of a mixed scheme for the Brinkman problem. Numer. Math. 133, 781–817 (2016). https://doi.org/10.1007/s00211-015-0758-x

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