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

Caucao, Sergio; Discacciati, Marco; Gatica, Gabriel N.; Oyarzúa, Ricardo

doi:10.1051/m2an/2020009

Caucao, Sergio ; Discacciati, Marco ; Gatica, Gabriel N. ; Oyarzúa, Ricardo

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 54 (2020) no. 5, pp. 1689-1723.

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Résumé

In this work we present and analyse a mixed finite element method for the coupling of fluid flow with porous media flow. The flows are governed by the Navier–Stokes and the Darcy–Forchheimer 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–Forchheimer region, which yields the introduction of the trace of the porous medium pressure as a suitable Lagrange multiplier. The well-posedness of the problem is achieved by combining a fixed-point strategy, classical results on nonlinear monotone operators and the well-known Schauder and Banach fixed-point theorems. As for the associated Galerkin scheme we employ Bernardi–Raugel and Raviart–Thomas elements for the velocities, and piecewise constant elements for the pressures and the Lagrange multiplier, whereas its existence and uniqueness of solution is established similarly to its continuous counterpart, using in this case the Brouwer and Banach fixed-point theorems, respectively. We show stability, convergence, and a priori error estimates for the associated Galerkin scheme. Finally, we report some numerical examples confirming the predicted rates of convergence, and illustrating the performance of the method.

DOI : 10.1051/m2an/2020009

Classification : 65N30, 65N12, 65N15, 74F10, 76D05, 76S05
Mots-clés : Navier–Stokes problem, Darcy–Forchheimer problem, pressure-velocity formulation, fixed-point theory, mixed finite element methods, $$ error analysis

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     title = {A conforming mixed finite element method for the {Navier{\textendash}Stokes/Darcy{\textendash}Forchheimer} coupled problem},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1689--1723},
     publisher = {EDP-Sciences},
     volume = {54},
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     year = {2020},
     doi = {10.1051/m2an/2020009},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2020009/}
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Caucao, Sergio; Discacciati, Marco; Gatica, Gabriel N.; Oyarzúa, Ricardo. A conforming mixed finite element method for the Navier–Stokes/Darcy–Forchheimer coupled problem. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 54 (2020) no. 5, pp. 1689-1723. doi : 10.1051/m2an/2020009. http://www.numdam.org/articles/10.1051/m2an/2020009/

Bibliographie
Cité par

[1] R.A. Adams and J.J.F. Fournier, Sobolev Spaces, 2nd edition. In: Vol. 140 of Pure and Applied Mathematics (Amsterdan). Elsevier/Academic Press, Amsterdam (2003). | Zbl

[2] M. Amara, D. Capatina and L. Lizaik, Coupling of Darcy-Forchheimer and compressible Navier-Stokes equations with heat transfer. SIAM J. Sci. Comput. 31 (2008/09) 1470–1499. | DOI | Zbl

[3] T. Arbogast and D.S. Brunson, A computational method for approximating a Darcy-Stokes system governing a vuggy porous medium. Comput. Geosci. 11 (2007) 207–218. | DOI | Zbl

[4] T. Arbogast and H.L. Lehr, Homogenization of a Darcy-Stokes system modeling vuggy porous media. Comput. Geosci. 10 (2006) 291–302. | DOI | Zbl

[5] M.M. Arzanfudi, S. Saeid, R. Al-Khoury and L.J. Sluys, Multidomain-staggered coupling technique for Darcy-Navier Stokes multiphase flow: an application to CO₂ geosequestration. Finite Elem. Anal. Des. 121 (2016) 52–63. | DOI

[6] K. Aziz and A. Settari, Petroleum Reservoir Simulation. Applied Science Publishers LTD, London (1979).

[7] L. Badea, M. Discacciati and A. Quarteroni, Numerical analysis of the Navier–Stokes/Darcy coupling. Numer. Math. 115 (2010) 195–227. | DOI | Zbl

[8] A. Bagchi and F.A. Kulacki, Natural Convection in Superposed Fluid-Porous Layers. Springer Briefs in Applied Sciences and Technology. Springer, New York (2014). | DOI | Zbl

[9] G.S. Beavers and D.D. Joseph, Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30 (1967) 197–207. | DOI

[10] C. Bernardi and G. Raugel, Analysis of some finite elements for the Stokes problem. Math. Comput. 44 (1985) 71–79. | DOI | Zbl

[11] H. Brezis and P. Mironescu, Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces. J. Evol. Equ. 1 (2001) 387–404. | DOI | Zbl

[12] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. In: Vol 15 of Springer Series in Computational Mathematics. Springer-Verlag, New York (1991). | Zbl

[13] J. Camaño, G.N. Gatica, R. Oyarzúa and G. Tierra, An augmented mixed finite element method for the Navier-Stokes equations with variable viscosity. SIAM J. Numer. Anal. 54 (2016) 1069–1092. | DOI

[14] C. Canuto and F. Cimolin, A sweating model for the internal ventilation of a motorcycle helmet. Comput. Fluids 43 (2011) 29–37. | DOI | Zbl

[15] S. Caucao, G.N. Gatica, R. Oyarzúa and I. Šebestová, A fully-mixed finite element method for the Navier–Stokes/Darcy coupled problem with nonlinear viscosity. J. Numer. Math. 25 (2017) 55–88. | DOI

[16] A. Çeşmelioğlu and B. Rivière, Analysis of time-dependent Navier-Stokes flow coupled with Darcy flow. J. Numer. Math. 16 (2008) 249–280. | DOI | Zbl

[17] P.G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications. Society for Industrial and Applied Mathematics, Philadelphia, PA (2013). | DOI | Zbl

[18] F. Cimolin and M. Discacciati, Navier–Stokes/Forchheimer models for filtration through porous media. Appl. Numer. Math. 72 (2013) 205–224. | DOI | Zbl

[19] T. Davis, Algorithm 832: UMFPACK V4.3 – an unsymmetric-pattern multifrontal method. ACM Trans. Math. Softw. 30 (2004) 196–199. | DOI | Zbl

[20] M. Discacciati, Domain Decomposition Methods for the Coupling of Surface and Groundwater Flows. École Polytechnique Fédérale de Lausanne, Switzerland (2004).

[21] M. Discacciati and R. Oyarzúa, A conforming mixed finite element method for the Navier–Stokes/Darcy coupled problem. Numer. Math. 135 (2017) 571–606. | DOI

[22] M. Discacciati, E. Miglio and A. Quarteroni, Mathematical and numerical models for coupling surface and groundwater flows. Appl. Numer. Math. 43 (2002) 57–74. | DOI | Zbl

[23] A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements. In: Vol. 159 of Applied Mathematical Sciences. Springer-Verlag, New York (2004). | DOI | Zbl

[24] V.J. Ervin, E.W. Jenkins and S. Sun, Coupled generalized nonlinear Stokes flow with flow through a porous medium. SIAM J. Numer. Anal. 47 (2009) 929–952. | DOI | Zbl

[25] M. Farhloul, A mixed finite element method for a nonlinear Dirichlet problem. IMA J. Numer. Anal. 18 (1998) 121–132. | DOI | Zbl

[26] J. Faulkner, B.X. Hu, S. Kish and F. Hua, Laboratory analog and numerical study of groundwater flow and solute transport in a karst aquifer with conduit and matrix domains. J. Contam. Hydrol. 110 (2009) 34–44. | DOI

[27] G.N. Gatica, A Simple Introduction to the Mixed Finite Element Method. Theory and Applications. Springer Briefs in Mathematics. Springer, Cham (2014). | DOI | Zbl

[28] G.N. Gatica, S. Meddahi and R. Oyarzúa, A conforming mixed finite-element method for the coupling of fluid flow with porous media flow. IMA J. Numer. Anal. 29 (2009) 86–108. | DOI | Zbl

[29] G.N. Gatica, R. Oyarzúa and F.-J. Sayas, Convergence of a family of Galerkin discretizations for the Stokes-Darcy coupled problem. Numer. Methods Partial Differ. Equ. 27 (2011) 721–748. | DOI | Zbl

[30] G.N. Gatica, R. Oyarzúa and F.-J. Sayas, Analysis of fully-mixed finite element methods for the Stokes-Darcy coupled problem. Math. Comput. 80 (2011) 1911–1948. | DOI | Zbl

[31] G.N. Gatica, R. Oyarzúa and F.-J. Sayas, A twofold saddle point approach for the coupling of fluid flow with nonlinear porous media flow. IMA J. Numer. Anal. 32 (2012) 845–887. | DOI | Zbl

[32] J. Geng, $W^{(1, p)}$ estimates for elliptic problems with Neumann boundary conditions in Lipschitz domains. Adv. Math. 229 (2012) 2427–2448. | DOI | Zbl

[33] V. Girault and P.A. Raviart, Finite Element Methods for Navier–Stokes Equations. Theory and algorithms. In: Vol. 5 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin (1986). | DOI | Zbl

[34] V. Girault and M.F. Wheeler, Numerical discretization of a Darcy-Forchheimer model. Numer. Math. 110 (2008) 161–198. | DOI | Zbl

[35] R. Glowinski and A. Marroco, Sur l’approximation, par éléments finis d’ordre un, et la résolution, par pénalisation-dualité, d’une classe de problèmes de Dirichlet non linéaires, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge Anal. Numér. 9 (1975) 41–76. | Numdam | Zbl

[36] P. Grisvard, Théorèmes de traces relatifs à un polyèdre. C. R. Acad. Sci. Paris Sér. A 278 (1974) 1581–1583. | Zbl

[37] P. Grisvard, Elliptic problems in nonsmooth domains. In: Vol. 24 of Monographs and Studies in Mathematics. Pitman (Advanced Publishing Program), Boston, MA (1985). | Zbl

[38] P. Grisvard, Problèmes aux limites dans les polygones. Mode d’emploi. EDF Bull. Direction Études Rech. Sér. C Math. Inform. 1 (1986) 21–59. | Zbl

[39] F. Hecht, New development in FreeFem++. J. Numer. Math. 20 (2012) 251–265. | DOI | Zbl

[40] W. Jäger and A. Mikelić, On the interface boundary condition of Beavers, Joseph, and Saffman. SIAM J. Appl. Math. 60 (2000) 1111–1127. | DOI | Zbl

[41] L.I.G. Kovasznay, Laminar flow behind a two-dimensional grid. Proc. Cambridge Philos. Soc. 44 (1948) 58–62. | DOI | Zbl

[42] H. Manouzi and M. Farhloul, Mixed finite element analysis of a non-linear three-fields Stokes model. IMA J. Numer. Anal. 21 (2001) 143–164. | DOI | Zbl

[43] M. Moraiti, On the quasistatic approximation in the Stokes-Darcy model of groundwater-surface water flows. J. Math. Anal. Appl. 394 (2012) 796–808. | DOI | Zbl

[44] H. Pan and H. Rui, Mixed element method for two-dimensional Darcy-Forchheimer model. J. Sci. Comput. 52 (2012) 563–587. | DOI | Zbl

[45] C. Pozrikidis and D.A. Farrow, A model of fluid flow in solid tumors. Ann. Biomed. Eng. 31 (2003) 181–194. | DOI

[46] A. Quarteroni and A. Valli, Numerical approximation of partial differential equations. In: Vol. 23 of Springer Series in Computational Mathematics. Springer–Verlag, Berlin (1994). | DOI | Zbl

[47] P.-A. Raviart and J.M. Thomas, A mixed finite element method for 2nd order elliptic problems. In: Vol. 606 of Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975). Lecture Notes in Math. Springer, Berlin (1977) 292–315. | Zbl

[48] D. Ruth and H. Ma, On the derivation of the Forchheimer equation by means of the averaging theorem. Transp. Porous Media 7 (1992) 255–264. | DOI

[49] P. Saffman, On the boundary condition at the interface of a porous medium. Stud. Appl. Math. 50 (1971) 77–84. | DOI | Zbl

[50] B. Scheurer, Existence et approximation de points selles pour certains problèmes non linéaires. RAIRO Anal. Numér. 11 (1977) 369–400. | DOI | Numdam | Zbl

[51] R.E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. In: Vol. 49 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (1997). | Zbl

[52] M. Sugihara-Seki and B.M. Fu, Blood flow and permeability in microvessels. Fluid Dynam. Res. 37 (2005) 82–132. | DOI | Zbl

[53] T. Tang, Z. Li, J.M. Mcdonough and P.D. Hislop, Numerical investigation of the “poor man’s Navier-Stokes equations” with Darcy and Forchheimer terms. Int. J. Bifur. Chaos Appl. Sci. Eng. 26 (2016) 1650086. | DOI | Zbl

[54] R. Temam, Navier–Stokes Equations. Theory and Numerical Analysis. In: Vol. 2 of Studies in Mathematics and its Applications. North-Holland Publishing Co., Amsterdam-New York-Oxford (1977). | Zbl

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