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A conforming mixed finite element method for the Navier–Stokes/Darcy coupled problem

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Abstract

In this paper we develop the a priori analysis of a mixed finite element method for the coupling of fluid flow with porous media flow. Flows are governed by the Navier–Stokes and Darcy equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers-Joseph-Saffman law. We consider the standard mixed formulation in the Navier–Stokes domain and the dual-mixed one in the Darcy region, which yields the introduction of the trace of the porous medium pressure as a suitable Lagrange multiplier. The finite element subspaces defining the discrete formulation employ Bernardi-Raugel and Raviart-Thomas elements for the velocities, piecewise constants for the pressures, and continuous piecewise linear elements for the Lagrange multiplier. We show stability, convergence, and a priori error estimates for the associated Galerkin scheme. Finally, several numerical results illustrating the good performance of the method and confirming the theoretical rates of convergence are reported.

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Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Loughborough University, Epinal Way, Loughborough, LE11 3TU, UK

    Marco Discacciati

  2. GIMNAP-Departamento de Matemática, Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile

    Ricardo Oyarzúa

  3. CI²MA, Universidad de Concepción, Casilla 160-C, Concepción, Chile

    Ricardo Oyarzúa

Authors
  1. Marco Discacciati
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  2. Ricardo Oyarzúa
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Corresponding author

Correspondence to Ricardo Oyarzúa.

Additional information

Marco Discacciati acknowledges funding from the European Union Seventh Framework Programme (FP7/2007-2013) under Grant Agreement No. 294229. Ricardo Oyarzúa acknowledges funding from CONICYT-Chile through project Fondecyt 11121347, project Anillo ACT1118 (ANANUM) and by Universidad del Bío-Bío through DIUBB project GI 151408/VC.

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Discacciati, M., Oyarzúa, R. A conforming mixed finite element method for the Navier–Stokes/Darcy coupled problem. Numer. Math. 135, 571–606 (2017). https://doi.org/10.1007/s00211-016-0811-4

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