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New Mixed Finite Element Methods for the Coupled Convective Brinkman-Forchheimer and Double-Diffusion Equations

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Abstract

In this paper we introduce and analyze new Banach spaces-based mixed finite element methods for the stationary nonlinear problem arising from the coupling of the convective Brinkman-Forchheimer equations with a double diffusion phenomenon. Besides the velocity and pressure variables, the symmetric stress and the skew-symmetric vorticity tensors are introduced as auxiliary unknowns of the fluid. Thus, the incompressibility condition allows to eliminate the pressure, which, along with the velocity gradient and the shear stress, can be computed afterwards via postprocessing formulae depending on the velocity and the aforementioned new tensors. Regarding the diffusive part of the coupled model, and additionally to the temperature and concentration of the solute, their gradients and pseudoheat/pseudodiffusion vectors are incorporated as further unknowns as well. The resulting mixed variational formulation, settled within a Banach spaces framework, consists of a nonlinear perturbation of, in turn, a nonlinearly perturbed saddle-point scheme, coupled with a usual saddle-point system. A fixed-point strategy, combined with classical and recent solvability results for suitable linearizations of the decoupled problems, including in particular, the Banach-Nečas-Babuška theorem and the Babuška-Brezzi theory, are employed to prove, jointly with the Banach fixed-point theorem, the well-posedness of the continuous and discrete formulations. Both PEERS and AFW elements of order \(\ell \geqslant 0\) for the fluid variables, and piecewise polynomials of degree \(\leqslant \ell \) together with Raviart-Thomas elements of order \(\ell \) for the unknowns of the diffusion equations, constitute feasible choices for the Galerkin scheme. In turn, optimal a priori error estimates, including those for the postprocessed unknowns, are derived, and corresponding rates of convergence are established. Finally, several numerical experiments confirming the latter and illustrating the good performance of the proposed methods, are reported.

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References

  1. Adams, S., Cockburn, B.: A mixed finite element for elasticity in three dimensions. J. Sci. Comput. 25, 515–521 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alnaes, M.S., Blechta, J., Hake, J., Johansson, A., Kehlet, B., Logg, A., Richardson, C., Ring, J., Rognes, M.E., Wells, G.N.: The FEniCS project version 1.5. Arch. Numer. Softw. 3, 9–23 (2015)

    Google Scholar 

  3. Alzahrani, A.K.: Importance of Darcy-Forchheimer porous medium in \(3\)D convective flow of carbon nanotubes. Phys. Lett. A 382(40), 2938–2943 (2018)

    Article  MathSciNet  Google Scholar 

  4. Arnold, D.N., Brezzi, F., Douglas, J.: PEERS: a new mixed finite element method for plane elasticity. Jpn. J. Appl. Math. 1, 347–367 (1984)

    Article  MATH  Google Scholar 

  5. Arnold, D.N., Falk, R.S., Winther, R.: Mixed finite element methods for linear elasticity with weakly imposed symmetry. Math. Comput. 76(260), 1699–1723 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Arnold, D.N., Winther, R.: Mixed finite elements for elasticity. Numer. Math. 92(3), 401–419 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Benavides, G.A., Caucao, S., Gatica, G.N., Hopper, A.A.: A new non-augmented and momentum-conserving fully-mixed finite element method for a coupled flow-transport problem. Calcolo 59(1), 6 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bhatti, M.M., Zeeshan, A., Ellahi, R., Shit, G.C.: Mathematical modeling of heat and mass transfer effects on MHD peristaltic propulsion of two-phase flow through a Darcy-Brinkman-Forchheimer porous medium. Adv. Powder Technol. 29(5), 1189–1197 (2018)

    Article  Google Scholar 

  9. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics, vol. 15. Springer-Verlag, New York (1991)

    MATH  Google Scholar 

  10. Camaño, J., García, C., Oyarzúa, R.: Analysis of a momentum conservative mixed-FEM for the stationary Navier-Stokes problem. Numer. Methods Partial Differ. Equ. 37(5), 2895–2923 (2021)

    Article  MathSciNet  Google Scholar 

  11. Camaño, J., Muñoz, C., Oyarzúa, R.: Numerical analysis of a dual-mixed problem in non-standard Banach spaces. Electron. Trans. Numer. Anal. 48, 114–130 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  12. Caucao, S., Colmenares, E., Gatica, G.N., Inzunza, C.: A Banach spaces-based fully-mixed finite element method for the stationary chemotaxis-Navier-Stokes problem. Comput. Math. Appl. 145, 65–89 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  13. Caucao, S., Esparza, J.: An augmented mixed FEM for the convective Brinkman-Forchheimer problem: a priori and a posteriori error analysis. J. Comput. Appl. Math. 438, 115517 (2024)

    Article  MathSciNet  MATH  Google Scholar 

  14. Caucao, S., Gatica, G.N., Gatica, L.F.: A Banach spaces-based mixed finite element method for the stationary convective Brinkman-Forchheimer problem. Calcolo. https://doi.org/10.1007/s10092-023-00544-2

  15. Caucao, S., Gatica, G.N., Ortega, J.P.: A fully-mixed formulation in Banach spaces for the coupling of the steady Brinkman-Forchheimer and double-diffusion equations. ESAIM Math. Model. Numer. Anal. 55(6), 2725–2758 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  16. Caucao, S., Gatica, G.N., Ortega, J.P.: A posteriori error analysis of a Banach spaces-based fully mixed FEM for double-diffusive convection in a fluid-saturated porous medium. Comput. Geosci. 27(2), 289–316 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  17. Caucao, S., Gatica, G.N., Oyarzúa, R., Sánchez, N.: A fully-mixed formulation for the steady double-diffusive convection system based upon Brinkman-Forchheimer equations. J. Sci. Comput. 85(2), 44 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  18. Caucao, S., Oyarzúa, R., Villa-Fuentes, S., Yotov, I.: A three-field Banach spaces-based mixed formulation for the unsteady Brinkman-Forchheimer equations. Comput. Methods Appl. Mech. Eng. 394, 114895 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  19. Celebi, A.O., Kalantarov, V.K., Ugurlu, D.: Continuous dependence for the convective Brinkman-Forchheimer equations. Appl. Anal. 84(9), 877–888 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Colmenares, E., Gatica, G.N., Moraga, S.: A Banach spaces-based analysis of a new fully-mixed finite element method for the Boussinesq problem. ESAIM Math. Model. Numer. Anal. 54(5), 1525–1568 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  21. Correa, C.I., Gatica, G.N.: On the continuous and discrete well-posedness of perturbed saddle-point formulations in Banach spaces. Comput. Math. Appl. 117, 14–23 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  22. Correa, C.I., Gatica, G.N., Ruiz-Baier, R.: New mixed finite element methods for the coupled Stokes and Poisson-Nernst-Planck equations in Banach spaces. ESAIM Math. Model. Numer. Anal. 57(3), 1511–1551 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ern, A., Guermond, J.-L.: Theory and Practice of Finite Elements. Applied Mathematical Sciences, vol. 159. Springer-Verlag, New York (2004)

    Book  MATH  Google Scholar 

  24. Faulkner, J., Hu, B.X., Kish, S., Hua, F.: Laboratory analog and numerical study of ground water flow and solute transport in a karst aquifer with conduit and matrix domains. J. Contam. Hydrol. 110(1–2), 34–44 (2009)

    Article  Google Scholar 

  25. Gatica, G.N.: A simple introduction to the mixed finite element method. In: Theory and Applications. Springer Briefs in Mathematics. Springer, Cham (2014)

    Google Scholar 

  26. Gatica, G.N., Inzunza, C.: On the well-posedness of Banach spaces-based mixed formulations for the nearly incompressible Navier-Lamé and Stokes equations. Comput. Math. Appl. 102, 87–94 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  27. Gatica, G.N., Núñez, N., Ruiz-Baier, R.: New non-augmented mixed finite element methods for the Navier-Stokes-Brinkman equations using Banach spaces. J. Numer. Math. (2023). https://doi.org/10.1515/jnma-2022-0073

    Article  Google Scholar 

  28. Gatica, G.N., Núñez, N., Ruiz-Baier, R.: Mixed-primal methods for natural convection driven phase change with Navier-Stokes-Brinkman equations. J. Sci. Comput. 95(3), 79 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  29. Gatica, G.N., Oyarzúa, R., Ruiz-Baier, R., Sobral, Y.: Banach spaces-based analysis of a fully-mixed finite element method for the steady-state model of fluidized beds. Comput. Math. Appl. 84, 244–276 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  30. GDR-MiDi. On dense granular flows. Eur. J. Phys. E 14, 341–365 (2004)

  31. Glowinski, R., Marrocco, A.: Sur l’approximation, par éléments finis d’ordre un, et la résolution, par pénalisations-dualité d’une classe de problémes de Dirichlet non lineaires. Rev. Fr. Autom. Inform. Rech. Opér. Anal Numér. 9(2), 41–76 (1975)

    MATH  Google Scholar 

  32. Hecht, F.: New development in FreeFem++. J. Numer. Math. 20, 251–265 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  33. Kaloni, P.N., Guo, J.: Steady nonlinear double-diffusive convection in a porous medium based upon the Brinkman-Forchheimer model. J. Math. Anal. Appl. 204(1), 138–155 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  34. Liu, D., Li, K.: Mixed finite element for two-dimensional incompressible convective Brinkman-Forchheimer equations. Appl. Math. Mech. (Engl. Ed.) 40(6), 889–910 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  35. Lonsing, M., Verfuürth, R.: On the stability of BDMS and PEERS elements. Numer. Math. 99(1), 131–140 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  36. Ôtani, M., Uchida, S.: Global solvability of some double-diffusive convection system coupled with Brinkman-Forchheimer equations. Lib. Math. 33(1), 79–107 (2013)

    MathSciNet  MATH  Google Scholar 

  37. Safi, S., Benissaad, S.: Double-diffusive convection in an anisotropic porous layer using the Darcy-Brinkman-Forchheimer formulation. Arch. Mech. 70(1), 89–102 (2018)

    MathSciNet  MATH  Google Scholar 

  38. Zhao, C., You, Y.: Approximation of the incompressible convective Brinkman-Forchheimer equations. J. Evol. Equ. 12(4), 767–788 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  39. Zhuang, Y.J., Yu, H.Z., Zhu, Q.Y.: A thermal non-equilibrium model for \(3\)D double diffusive convection of power-law fluids with chemical reaction in the porous medium. Int. J. Heat Mass Transf. 115–B, 670–694 (2017)

    Article  Google Scholar 

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Authors and Affiliations

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

    Sergio Carrasco & Gabriel N. Gatica

  2. GIANuC² and Departamento de Matemática y Física Aplicadas, Universidad Católica de la Santísima Concepción, Casilla 297, Concepción, Chile

    Sergio Caucao

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  1. Sergio Carrasco
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This research was partially supported by ANID-Chile through Centro de Modelamiento Matemático (FB210005), Anillo of Computational Mathematics for Desalination Processes (ACT210087), and Fondecyt project 11220393; by Centro de Investigación en Ingeniería Matemática (CI\(^2\)MA), Universidad de Concepción; and by Grupo de Investigación en Análisis Numérico y Cálculo Científico (GIANuC\(^2\)), Universidad Católica de la Santísima Concepción.

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Carrasco, S., Caucao, S. & Gatica, G.N. New Mixed Finite Element Methods for the Coupled Convective Brinkman-Forchheimer and Double-Diffusion Equations. J Sci Comput 97, 61 (2023). https://doi.org/10.1007/s10915-023-02371-7

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  • DOI: https://doi.org/10.1007/s10915-023-02371-7

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