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://www.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://www.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://www.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://www.numdam.org/articles/10.1051/m2an/2015015/
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