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 for the Cauchy stress, and continuous piecewise polynomials of degree 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.
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
@article{M2AN_2015__49_5_1399_0, author = {Alvarez, Mario and Gatica, Gabriel N. and Ruiz{\textendash}Baier, Ricardo}, title = {An augmented mixed-primal finite element method for a coupled flow-transport problem}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {1399--1427}, publisher = {EDP-Sciences}, volume = {49}, number = {5}, year = {2015}, doi = {10.1051/m2an/2015015}, mrnumber = {3423229}, zbl = {1329.76157}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/m2an/2015015/} }
TY - JOUR AU - Alvarez, Mario AU - Gatica, Gabriel N. AU - Ruiz–Baier, Ricardo TI - An augmented mixed-primal finite element method for a coupled flow-transport problem JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2015 SP - 1399 EP - 1427 VL - 49 IS - 5 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2015015/ DO - 10.1051/m2an/2015015 LA - en ID - M2AN_2015__49_5_1399_0 ER -
%0 Journal Article %A Alvarez, Mario %A Gatica, Gabriel N. %A Ruiz–Baier, Ricardo %T An augmented mixed-primal finite element method for a coupled flow-transport problem %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2015 %P 1399-1427 %V 49 %N 5 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2015015/ %R 10.1051/m2an/2015015 %G en %F M2AN_2015__49_5_1399_0
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. http://archive.numdam.org/articles/10.1051/m2an/2015015/
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
Multifrontal parallel distributed symmetric and unsymmetric solvers. Comput. Methods Appl. Mech. Engrg. 184 (2000) 501–520. | DOI | Zbl
, and ,Vector potentials in three-dimensional non-smooth domains. Math. Methods Appl. Sci. 21 (1998) 823–864. | DOI | MR | Zbl
, , and ,F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag (1991). | MR | Zbl
A stabilized finite volume element formulation for sedimentation-consolidation processes. SIAM J. Sci. Comput. 34 (2012) B265–B289. | DOI | MR | Zbl
, and ,Model equations for gravitational sedimentation-consolidation processes. ZAMM Z. Angew. Math. Mech. 80 (2000) 79–92. | DOI | MR | Zbl
, and ,M.C. Bustos, F. Concha, R. Bürger and E.M. Tory, Sedimentation and Thickening. Kluwer Academic Publishers, Dordrecht (1999). | MR | Zbl
Least-squares methods for incompressible Newtonian fluid flow: linear stationary problems. SIAM J. Numer. Anal. 42 (2004) 843–859. | DOI | MR | Zbl
, and ,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
Finite element approximation of the non-isothermal Stokes-Oldroyd equations. Int. J. Numer. Anal. Model. 4 (2007) 425–440. | MR | Zbl
, and ,Natural convection of air in a square cavity: A benchmark numerical solution. Int. J. Numer. Meth. Fluids 3 (1983) 249–264. | DOI | Zbl
,A dual mixed formulation for non-isothermal Oldroyd-Stokes problem. Math. Model. Nat. Phenom. 6 (2011) 130–156. | DOI | MR | Zbl
and ,A mixed formulation of Boussinesq equations: Analysis of nonsingular solutions. Math. Comput. 69 (2000) 965–986. | DOI | MR | Zbl
, and ,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
, and ,Augmented mixed finite element methods for the stationary Stokes equations. SIAM J. Sci. Comput. 31 (2008/09) 1082–1119. | DOI | MR | Zbl
, and ,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
and ,Analysis of a new augmented mixed finite element method for linear elasticity allowing 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
On the coupled BEM and FEM for a nonlinear exterior Dirichlet problem in . Numer. Math. 61 (1992) 171–214. | DOI | MR | Zbl
and ,Coupling of mixed finite elements and boundary elements for linear and nonlinear elliptic problems. Appl. Anal. 63 (1996) 39–75. | DOI | MR | Zbl
and ,Analysis of a velocity-pressure-pseudostress formulation for the stationary Stokes equations. Comput. Methods Appl. Mech. Engrg. 199 (2010) 1064–1079. | DOI | MR | Zbl
, and ,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
, and ,Effects of the heat transfer at the side walls on natural convection in cavities. J. Heat Trans. 112 (1990) 370–378. | DOI
, and ,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
An exactly divergence-free finite element method for a generalized Boussinesq problem. IMA J. Numer. Anal. 34 (2014) 1104–1135. | DOI | MR | Zbl
, and ,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
Numerical solution of a multidimensional sedimentation problem using finite volume-element methods. Appl. Numer. Math. 95 (2015) 280–291. | DOI | MR | Zbl
and ,Gravity flow of cohesionless granular materials in chutes and channels. J. Fluid Mech. 92 (1979) 53–96. | DOI | Zbl
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