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A primal-mixed formulation for the strong coupling of quasi-Newtonian fluids with porous media

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Abstract

In this work we analyze a primal-mixed finite element method for the coupling of quasi-Newtonian fluids with porous media in 2D and 3D. The flows are governed by a class of nonlinear Stokes and linear Darcy equations, respectively, and the transmission conditions on the interface between the fluid and the porous medium are given by mass conservation, balance of normal forces and the Beavers-Joseph-Saffman law. We apply a primal formulation in the Stokes domain and a mixed formulation in the Darcy formulation. The “strong coupling” concept means that the conservation of mass across the interface is introduced as an essential condition in the space where the velocity unknowns live. In this way, under some assumptions on the nonlinear kinematic viscosity, a generalization of the Babuška-Brezzi theory is utilized to show the well posedness of the primal-mixed formulation. Then, we introduce a Galerkin scheme in which the discrete conservation of mass is imposed approximately through an orthogonal projector. The unique solvability of this discrete system and its Strang-type error estimate follow from the generalized Babuška-Brezzi theory as well. In particular, the feasible finite element subspaces include Bernadi-Raugel elements for the Stokes flow, and either the Raviart-Thomas elements of lowest order or the Brezzi-Douglas-Marini elements of first order for the Darcy flow, which yield nonconforming and conforming Galerkin schemes, respectively. In turn, piecewise constant functions are employed to approximate in both cases the global pressure field in the Stokes and Darcy domain. Finally, several numerical results illustrating the good performance of both discrete methods and confirming the theoretical rates of convergence, are provided.

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Author notes

  1. Sebastián Domínguez

    Present address: Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, BC, V5A 1S6, Canada

Authors and Affiliations

  1. CI2MA and Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile

    Sebastián Domínguez & Gabriel N. Gatica

  2. Departamento de Construcción e Ingeniería de Fabricación, Universidad de Oviedo, Oviedo, España

    Antonio Márquez

  3. Departamento de Matemática, Universidad de Oviedo, Oviedo, España

    Salim Meddahi

Authors
  1. Sebastián Domínguez
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  2. Gabriel N. Gatica
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  3. Antonio Márquez
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  4. Salim Meddahi
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Corresponding author

Correspondence to Gabriel N. Gatica.

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Communicated by: Axel Voigt

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Domínguez, S., Gatica, G.N., Márquez, A. et al. A primal-mixed formulation for the strong coupling of quasi-Newtonian fluids with porous media. Adv Comput Math 42, 675–720 (2016). https://doi.org/10.1007/s10444-015-9439-7

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  • DOI: https://doi.org/10.1007/s10444-015-9439-7

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