In this article, we consider the convection-diffusion-reaction problem coupled the Darcy-Forchheimer problem by a non-linear external force depending on the concentration. We establish existence of a solution by using a Galerkin method and we prove uniqueness. We introduce and analyse a numerical scheme based on the finite element method. An optimal a priori error estimate is then derived for each numerical scheme. Numerical investigation are performed to confirm the theoretical accuracy of the discretization.
Keywords: Darcy-Forchheimer problem, convection-diffusion-reaction equation, finite element method, $$ error estimates
@article{M2AN_2021__55_6_2643_0,
author = {Sayah, Toni and Semaan, Georges and Triki, Faouzi},
title = {Finite element methods for the {Darcy-Forchheimer} problem coupled with the convection-diffusion-reaction problem},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {2643--2678},
year = {2021},
publisher = {EDP-Sciences},
volume = {55},
number = {6},
doi = {10.1051/m2an/2021066},
mrnumber = {4337455},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2021066/}
}
TY - JOUR AU - Sayah, Toni AU - Semaan, Georges AU - Triki, Faouzi TI - Finite element methods for the Darcy-Forchheimer problem coupled with the convection-diffusion-reaction problem JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2021 SP - 2643 EP - 2678 VL - 55 IS - 6 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2021066/ DO - 10.1051/m2an/2021066 LA - en ID - M2AN_2021__55_6_2643_0 ER -
%0 Journal Article %A Sayah, Toni %A Semaan, Georges %A Triki, Faouzi %T Finite element methods for the Darcy-Forchheimer problem coupled with the convection-diffusion-reaction problem %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2021 %P 2643-2678 %V 55 %N 6 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2021066/ %R 10.1051/m2an/2021066 %G en %F M2AN_2021__55_6_2643_0
Sayah, Toni; Semaan, Georges; Triki, Faouzi. Finite element methods for the Darcy-Forchheimer problem coupled with the convection-diffusion-reaction problem. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021) no. 6, pp. 2643-2678. doi: 10.1051/m2an/2021066
[1] , and , A second order accuracy in time for a full discretized time-dependent Navier-Stockes equations by a two-grid scheme. Numer. Math. 114 (2009) 189–231. | MR | Zbl | DOI
[2] , Sobolev Spaces. Academic Press, New York (1975). | MR | Zbl
[3] , and , Adaptive mesh refinement for a finite volume method for flow and transport of radionuclides in heterogeneous porous media. Oil Gas Sci. Technol. - Rev. IFP Energies nouvelles 69 (2014) 687–699. | DOI
[4] , , , , and , Finite element method for Darcy’s problem coupled with the heat equation. Numer. Math. 139 (2018) 315–348. | MR | DOI
[5] and , A local regularisation operation for triangular and quadrilateral finite elements. SIAM J. Numer. Anal. 35 (1998) 1893–1916. | MR | Zbl | DOI
[6] , and , Spectral discretization of Darcy’s equations coupled with the heat equation. IMA J. Numer. Anal. 36 (2015) 1193–1216. | MR | DOI
[7] , and , Couplage des équations dePlease check & approve edit of the journal title of ref [7] Navier-Stokes et de la chaleur: le modèle et son approximation par éléments finis. ESAIM:M2AN 29 (1995) 871–921. | MR | Zbl | Numdam | DOI
[8] , Analytical and numerical studies of the structure of steady separated flows. J. Fluid Mech. 24 (1996) 113–151. | DOI
[9] , , and , Full discretization of time dependent convection-diffusion-reaction equation coupled with the Darcy system. Calcolo 57 (2020) 4. | MR | DOI
[10] , , and , A posteriori error estimates for the time-dependent convection-diffusion-reaction equation coupled with the Darcy system. Calcolo (submitted). | MR
[11] , Basic error estimates for elliptic problems, in Handbook of Numerical Analysis, North-Holland, volume II of Finite Element Methods (Part 1). North-Holland, Amsterdam (1991) pp. 17–351. | MR | Zbl | DOI
[12] , Approximation by finite element functions using local regularization. RAIRO Anal. Numér. 9 (1975) 77–84. | MR | Zbl | Numdam
[13] , and , A coupled prediction scheme for solving the Navier-Stokes and convection-diffusion equations. SIAM J. Numer. Anal. 52 (2014) 2415–2439. | MR | Zbl | DOI
[14] , and , New numerical studies for Darcy’s problem coupled with the heat equation. Comput. Appl. Math. 39 (2020) 1. | MR | DOI
[15] , , and , A posteriori error estimates for Darcy’s problem coupled with the heat equation. ESAIM: M2AN 53 (2019) 2121–2159. | MR | Numdam | DOI
[16] , and , Numerical solutions of 2-D steady incompressible driven cavity flow at high Reynolds numbers. Int. J. Numer. Methods Fluids 48 (2005) 747–774. | Zbl | DOI
[17] , Regularity of the solution of Darcy-Forchheimer’s equation. Nonlinear Anal. Theory Methods 13 (1989) 1025–1045. | MR | Zbl | DOI
[18] and , Singular limits in thermodynamics of viscous fluids. Adv. Math. Fluid Mech. Birkhäuser, Basel (2009). | MR | Zbl
[19] , Wasserbewegung durch Boden. Z. Ver. Deutsh. Ing. 45 (1901) 1782–1788.
[20] and , Équations de Navier-Stokes couplées à des équations de la chaleur: Résolution par une méthode de point fixe en dimension infinie. Ann. Sci. Math. Québec 13 (1989) 1–17. | MR | Zbl
[21] and , Elliptic partial differential equations of second order, Vol. 224 of Classics in Mathematics. Springer, Berlin (2001). | MR | Zbl
[22] and , Two-grid finite-element schemes for the transient Navier-Stokes problem. ESAIM: M2AN 35 (2001) 945–980. | MR | Zbl | Numdam | DOI
[23] and , Finite element methods for Navier-Stokes equations, in Theory and Algorithms, SCM 5. Springer, Berlin (1986). | MR | Zbl
[24] and , Numerical discretization of a Darcy-Forchheimer model. Numer. Math. 110 (2008) 161–198. | MR | Zbl | DOI
[25] , development in FreeFem++. J. Numer. Math. 20 (2012) 251–266. | MR | Zbl | DOI
[26] , and , Comparison between different numerical discretizations for a Darcy-Forchheimer model. Electron. Trans. Numer. Anal. 34 (2009) 187–203. | MR | Zbl
[27] , Numerical solution of the Navier-Stokes equations for the flow in a two-dimensional cavity. J. Phys. Soc. Jpn 16 (1961) 2307–2315. | MR | Zbl | DOI
[28] , and , An analysis of a mixed finite element method for a Darcy-Forchsheimer model. Math. Comput. Model. 57 (2013) 2325–2338. | MR | Zbl | DOI
[29] and , On the derivation of the Forchheimer equation by means of the averaging theorem. Transp. Porous Media 7 (1992) 255–264. | DOI
[30] , Les Méthodes directes en théorie des équations elliptiques. Masson, Paris (1967). | MR | Zbl
[31] , Theoretical derivation of Darcy’s law. Acta Mech. 25 (1977) 153–170. | Zbl | DOI
[32] and , Mixed element method for two-dimensional Darcy-Forchheimer model. J. Sci. Comput. 52 (2012) 563–587. | MR | Zbl | DOI
[33] , Convergence analysis of numerical schemes for the Darcy-Forchheimer problem. Mediterr. J. Math. (2021) (submitted).
[34] and , Driven cavity flows by efficient numerical techniques. J. Comput. Phys. 49 (1983) 310–333. | Zbl | DOI
[35] and , Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54 (1990) 483–493. | MR | Zbl | DOI
[36] , Darcy-Forchheimer natural, forced and mixed convection heat transfer in non-Newtonian power-law fluid-saturated porous media. Transp. Porous Med. 11 (1993) 219–241. | DOI
[37] , Flow in porous media I: A theoretical derivation of Darcy’s law. Transp. Porous Med. 1 (1986) 3–25. | DOI
Cité par Sources :
