An augmented mixed-primal finite element method for a coupled flow-transport problem

Alvarez, Mario; Gatica, Gabriel N.; Ruiz–Baier, Ricardo

doi:10.1051/m2an/2015015

Alvarez, Mario ^1,² ; Gatica, Gabriel N.² ; Ruiz–Baier, Ricardo ³

¹ Sección de Matemática, Sede Occidente, Universidad de Costa Rica, San Ramón de Alajuela, Costa Rica
² CI2 MA and Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile
³ Institute of Earth Sciences, Géopolis UNIL-Mouline, University of Lausanne, 1015 Lausanne, Switzerland

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 5, pp. 1399-1427.

Résumé

In this paper we analyze the coupling of a scalar nonlinear convection-diffusion problem with the Stokes equations where the viscosity depends on the distribution of the solution to the transport problem. An augmented variational approach for the fluid flow coupled with a primal formulation for the transport model is proposed. The resulting Galerkin scheme yields an augmented mixed-primal finite element method employing Raviart−Thomas spaces of order $k$ for the Cauchy stress, and continuous piecewise polynomials of degree $\leq k + 1$ for the velocity and also for the scalar field. The classical Schauder and Brouwer fixed point theorems are utilized to establish existence of solution of the continuous and discrete formulations, respectively. In turn, suitable estimates arising from the connection between a regularity assumption and the Sobolev embedding and Rellich−Kondrachov compactness theorems, are also employed in the continuous analysis. Then, sufficiently small data allow us to prove uniqueness and to derive optimal a priori error estimates. Finally, we report a few numerical tests confirming the predicted rates of convergence, and illustrating the performance of a linearized method based on Newton−Raphson iterations; and we apply the proposed framework in the simulation of thermal convection and sedimentation-consolidation processes.

MR Zbl | 3 citations dans Numdam

DOI : 10.1051/m2an/2015015

Classification : 65N30, 65N12, 76R05, 76D07, 65N15
Mots-clés : Stokes equations, nonlinear transport problem, augmented mixed-primal formulation, fixed point theory, thermal convection, sedimentation-consolidation process, finite element methods, a priori error analysis

Affiliations des auteurs :

Alvarez, Mario ^{1, 2} ; Gatica, Gabriel N. ² ; Ruiz–Baier, Ricardo ³

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     title = {An augmented mixed-primal finite element method for a coupled flow-transport problem},
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     pages = {1399--1427},
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Alvarez, Mario; Gatica, Gabriel N.; Ruiz–Baier, Ricardo. An augmented mixed-primal finite element method for a coupled flow-transport problem. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 5, pp. 1399-1427. doi : 10.1051/m2an/2015015. https://www.numdam.org/articles/10.1051/m2an/2015015/

Bibliographie
Cité par

R.A. Adams and J.J.F. Fournier, Sobolev Spaces. Academic Press. Elsevier Ltd (2003). | MR | Zbl

M. Alvarez, G.N. Gatica and R. Ruiz-Baier, Mixed-primal finite element approximation of a steady sedimentation-consolidation system. In preparation (2015). | MR

P.R. Amestoy, I.S. Duff and J.-Y. L’Excellent, Multifrontal parallel distributed symmetric and unsymmetric solvers. Comput. Methods Appl. Mech. Engrg. 184 (2000) 501–520. | DOI | Zbl

C. Amrouche, C. Bernardi, M. Dauge and V. Girault, Vector potentials in three-dimensional non-smooth domains. Math. Methods Appl. Sci. 21 (1998) 823–864. | DOI | MR | Zbl

F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag (1991). | MR | Zbl

R. Bürger, R. Ruiz-Baier and H. Torres, A stabilized finite volume element formulation for sedimentation-consolidation processes. SIAM J. Sci. Comput. 34 (2012) B265–B289. | DOI | MR | Zbl

R. Bürger, W.L. Wendland and F. Concha, Model equations for gravitational sedimentation-consolidation processes. ZAMM Z. Angew. Math. Mech. 80 (2000) 79–92. | DOI | MR | Zbl

M.C. Bustos, F. Concha, R. Bürger and E.M. Tory, Sedimentation and Thickening. Kluwer Academic Publishers, Dordrecht (1999). | MR | Zbl

Z. Cai, B. Lee and P. Wang, Least-squares methods for incompressible Newtonian fluid flow: linear stationary problems. SIAM J. Numer. Anal. 42 (2004) 843–859. | DOI | MR | Zbl

P. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland (1978). | MR | Zbl

P. Ciarlet, Linear and Nonlinear Functional Analysis with Applications. Society for Industrial and Applied Mathematics, Philadelphia, PA (2013). | MR | Zbl

E. Colmenares, G.N. Gatica and R. Oyarzúa, Analysis of an augmented mixed-primal formulation for the stationary Boussinesq problem. Centro de Investigación en Ingeniería Matemática, Universidad de Concepción, (2014). Preprint 2015-07. Available at http://www.ci2ma.udec.cl/publicaciones/prepublicaciones/. | MR

C. Cox, H. Lee and D. Szurley, Finite element approximation of the non-isothermal Stokes-Oldroyd equations. Int. J. Numer. Anal. Model. 4 (2007) 425–440. | MR | Zbl

G. De Vahl Davis, Natural convection of air in a square cavity: A benchmark numerical solution. Int. J. Numer. Meth. Fluids 3 (1983) 249–264. | DOI | Zbl

M. Farhloul and A. Zine, A dual mixed formulation for non-isothermal Oldroyd-Stokes problem. Math. Model. Nat. Phenom. 6 (2011) 130–156. | DOI | MR | Zbl

M. Farhloul, S. Nicaise and L. Paquet, A mixed formulation of Boussinesq equations: Analysis of nonsingular solutions. Math. Comput. 69 (2000) 965–986. | DOI | MR | Zbl

L.E. Figueroa, G.N. Gatica and N. Heuer, A priori and a posteriori error analysis of an augmented mixed finite element method for incompressible fluid flows. Comput. Methods Appl. Mech. Engrg. 198 (2008) 280–291. | DOI | MR | Zbl

L.E. Figueroa, G.N. Gatica and A. Márquez, Augmented mixed finite element methods for the stationary Stokes equations. SIAM J. Sci. Comput. 31 (2008/09) 1082–1119. | DOI | MR | Zbl

T. Fusegi and J.M. Hyun, A numerical study of 3D natural convection in a cube: effects of the horizontal thermal boundary conditions. Fluid Dyn. Res. 8 (1991) 221–230. | DOI

G.N. Gatica, Analysis of a new augmented mixed finite element method for linear elasticity allowing $R T_{0} - P_{1} - P_{0}$ approximations. ESAIM: M2AN 40 (2006) 1–28. | DOI | Numdam | MR | Zbl

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

G.N. Gatica and G.C. Hsiao, On the coupled BEM and FEM for a nonlinear exterior Dirichlet problem in $R^{2}$ . Numer. Math. 61 (1992) 171–214. | DOI | MR | Zbl

G.N. Gatica and W. Wendland, Coupling of mixed finite elements and boundary elements for linear and nonlinear elliptic problems. Appl. Anal. 63 (1996) 39–75. | DOI | MR | Zbl

G.N. Gatica, A. Márquez and M.A. Sánchez, Analysis of a velocity-pressure-pseudostress formulation for the stationary Stokes equations. Comput. Methods Appl. Mech. Engrg. 199 (2010) 1064–1079. | DOI | MR | Zbl

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 | MR | Zbl

Y. Le Pentrec, and G. Lauriat, Effects of the heat transfer at the side walls on natural convection in cavities. J. Heat Trans. 112 (1990) 370–378. | DOI

J. Nečas, Introduction to the Theory of Nonlinear Elliptic Equations. Reprint of the 1983 edition. A Wiley-Interscience Publication. John Wiley & Sons, Ltd., Chichester (1986). | MR | Zbl

R. Oyarzúa, T. Qin and D. Schötzau, An exactly divergence-free finite element method for a generalized Boussinesq problem. IMA J. Numer. Anal. 34 (2014) 1104–1135. | DOI | MR | Zbl

A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations. Vol. 23 of Springer Ser. Comput. Math. Springer-Verlag Berlin Heidelberg (1994). | MR | Zbl

R. Ruiz-Baier and H. Torres, Numerical solution of a multidimensional sedimentation problem using finite volume-element methods. Appl. Numer. Math. 95 (2015) 280–291. | DOI | MR | Zbl

S.B. Savage, Gravity flow of cohesionless granular materials in chutes and channels. J. Fluid Mech. 92 (1979) 53–96. | DOI | Zbl

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