A Fully-Mixed Formulation for the Steady Double-Diffusive Convection System Based upon Brinkman–Forchheimer Equations


We propose and analyze a new mixed finite element method for the problem of steady double-diffusive convection in a fluid-saturated porous medium. More precisely, the model is described by the coupling of the Brinkman–Forchheimer and double-diffusion equations, in which the originally sought variables are the velocity and pressure of the fluid, and the temperature and concentration of a solute. Our approach is based on the introduction of the further unknowns given by the fluid pseudostress tensor, and the pseudoheat and pseudodiffusive vectors, thus yielding a fully-mixed formulation. Furthermore, since the nonlinear term in the Brinkman–Forchheimer equation requires the velocity to live in a smaller space than usual, we partially augment the variational formulation with suitable Galerkin type terms, which forces both the temperature and concentration scalar fields to live in \(\mathrm {L}^4\). As a consequence, the aforementioned pseudoheat and pseudodiffusive vectors live in a suitable \(\mathrm {H}(\mathrm {div})\)-type Banach space. The resulting augmented scheme is written equivalently as a fixed point equation, so that the well-known Schauder and Banach theorems, combined with the Lax–Milgram and Banach–Nečas–Babuška theorems, allow to prove the unique solvability of the continuous problem. As for the associated Galerkin scheme we utilize Raviart–Thomas spaces of order \(k\ge 0\) for approximating the pseudostress tensor, as well as the pseudoheat and pseudodiffusive vectors, whereas continuous piecewise polynomials of degree \(\le k + 1\) are employed for the velocity, and piecewise polynomials of degree \(\le k\) for the temperature and concentration fields. In turn, the existence and uniqueness of the discrete solution is established similarly to its continuous counterpart, applying in this case the Brouwer and Banach fixed-point theorems, respectively. Finally, we derive optimal a priori error estimates and provide several numerical results confirming the theoretical rates of convergence and illustrating the performance and flexibility of the method.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3


  1. 1.

    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)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Ambartsumyan, I., Khattatov, E., Nguyen, T., Yotov, I.: Flow and transport in fractured poroelastic media. GEM Int. J. Geomath. 10(1), 34 (2019)

    MathSciNet  Article  Google Scholar 

  3. 3.

    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 

  4. 4.

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

    Google Scholar 

  5. 5.

    Brinkman, H.C.: A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. 1, 27–34 (1949)

    Article  Google Scholar 

  6. 6.

    Bürger, R., Méndez, P.E., Ruiz-Baier, R.: On \({\mathbf{H}}(\div )\)-conforming methods for double-diffusion equations in porous media. SIAM J. Numer. Anal. 57(3), 1318–1343 (2019)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Camaño, J., García, C., Oyarzúa, R.: Analysis of a conservative mixed-FEM for the stationary Navier–Stokes problem. Preprint 2018-25, Centro de Investigación en Ingeniería Matemática (\(\text{CI}^2\)MA), Universidad de Concepción, Chile (2018)

  8. 8.

    Camaño, J., Gatica, G.N., Oyarzúa, R., Tierra, G.: An augmented mixed finite element method for the Navier–Stokes equations with variable viscosity. SIAM J. Numer. Anal. 54(2), 1069–1092 (2016)

    MathSciNet  Article  Google Scholar 

  9. 9.

    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)

    MathSciNet  Article  Google Scholar 

  10. 10.

    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. (in preparation)

  11. 11.

    Caucao, S., Gatica, G.N., Oyarzúa, R.: Analysis of an augmented fully-mixed formulation for the coupling of the Stokes and heat equations. ESAIM Math. Model. Numer. Anal. 52(5), 1947–1980 (2018)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Caucao, S., Oyarzúa, R., Villa-Fuentes, S.: A new mixed-FEM for steady-state natural convection models allowing conservation of momentum and thermal energy. Calcolo (to appear)

  13. 13.

    Celebi, A.O., Kalantarov, V.K., Ugurlu, D.: On continuous dependence on coefficients of the Brinkman–Forchheimer equations. Appl. Math. Lett. 19(8), 801–807 (2006)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Ciarlet, P.G.: Linear and Nonlinear Functional Analysis with Applications. Society for Industrial and Applied Mathematics, Philadelphia (2013)

    Google Scholar 

  15. 15.

    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)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Colmenares, E., Gatica, G.N., Moraga, S., Ruiz-Baier, R.: A fully-mixed finite element method for the steady state Oberbeck–Boussinesq system. SMAI J. Comput. Math. 6, 125–157 (2020)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Colmenares, E., Gatica, G.N., Oyarzúa, R.: An augmented fully-mixed finite element method for the stationary Boussinesq problem. Calcolo 54(1), 167–205 (2017)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Colmenares, E., Gatica, G.N., Oyarzúa, R.: Fixed point strategies for mixed variational formulations of the stationary Boussinesq problem. C. R. Math. Acad. Sci. Paris 354(1), 57–62 (2016)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Davis, T.A.: Algorithm 832: UMFPACK V43—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Softw. 30, 196–199 (2004)

    Article  Google Scholar 

  20. 20.

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

    Google Scholar 

  21. 21.

    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 

  22. 22.

    Forchheimer, P.: Wasserbewegung durch boden. Z. Ver. Deutsch Ing. 45, 1782–1788 (1901)

    Google Scholar 

  23. 23.

    Gatica, G.N.: A Simple Introduction to the Mixed Finite Element Method. Theory and Applications. Springer Briefs in Mathematics. Springer, Cham (2014)

    Google Scholar 

  24. 24.

    Gatica, G.N., Gatica, L.F., Márquez, A.: Analysis of a pseudostress-based mixed finite element method for the Brinkman model of porous media flow. Numer. Math. 126(4), 635–677 (2014)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Gatica, L.F., Oyarzúa, R., Sánchez, N.: A priori and a posteriori error analysis of an augmented mixed-FEM for the Navier–Stokes–Brinkman problem. Comput. Math. Appl. 75(7), 2420–2444 (2018)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Girault, V., Raviart, P.A.: Finite Element Methods for Navier–Stokes Equations. Theory and Algorithms. Springer Series in Computational Mathematics, vol. 5. Springer, Berlin (1986)

  27. 27.

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

    MathSciNet  Article  Google Scholar 

  28. 28.

    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)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Kufner, A., Jhon, O., Fučík, S.: Function Spaces. Monographs and Textbooks on Mechanics of Solids and Fluids; Mechanics: Analysis. Noordhoff International Publishing, Leyden (1977)

    Google Scholar 

  30. 30.

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

    MathSciNet  MATH  Google Scholar 

  31. 31.

    Payne, L.E., Straughan, B.: Convergence and continuous dependence for the Brinkman–Forchheimer equations. Stud. Appl. Math. 102(4), 419–439 (1999)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer Series in Computational Mathematics, vol. 23. Springer, Berlin (1994)

    Google Scholar 

  33. 33.

    Roberts, J.E., Thomas, J.M.: Mixed and Hybrid Methods. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, vol. II, Finite Element Methods (Part 1), North-Holland, Amsterdam (1991)

  34. 34.

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

    MathSciNet  MATH  Google Scholar 

  35. 35.

    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 

Download references

Author information



Corresponding author

Correspondence to Sergio Caucao.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was partially supported by CONICYT-Chile through the Project AFB170001 of the PIA Program: Concurso Apoyo a Centros Científicos y Tecnológicos de Excelencia con Financiamiento Basal, the Project PAI77190084 of the PAI Program: Convocatoria Nacional Subvención a la Instalación en la Academia, and Fondecyt Project 11121347; by Centro de Investigación en Ingeniería Matemática (\(\hbox {CI}^2\)MA), Universidad de Concepción; and by Universidad del Bío-Bío through DIUBB Project 151408 GI/VC.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Caucao, S., Gatica, G.N., Oyarzúa, R. et al. A Fully-Mixed Formulation for the Steady Double-Diffusive Convection System Based upon Brinkman–Forchheimer Equations. J Sci Comput 85, 44 (2020). https://doi.org/10.1007/s10915-020-01305-x

Download citation


  • Brinkman–Forchheimer equations
  • Double-diffusive convection system
  • Stress–velocity formulation
  • Fixed point theory
  • Mixed finite element methods
  • A priori error analysis

Mathematics Subject Classification

  • 65N30
  • 65N12
  • 65N15
  • 35Q79
  • 80A20
  • 76R05
  • 76D07