A fully-mixed finite element method for the steady state Oberbeck–Boussinesq system
The SMAI Journal of computational mathematics, Volume 6 (2020), pp. 125-157.

A new fully-mixed formulation is advanced for the stationary Oberbeck–Boussinesq problem when viscosity depends on both temperature and concentration of a solute. Following recent ideas in the context of mixed methods for Boussinesq and Navier–Stokes systems, the velocity gradient and the Bernoulli stress tensor are taken as additional field variables in the momentum and mass equilibrium equations. Similarly, the gradients of temperature and concentration together with a Bernoulli vector are considered as unknowns in the heat and mass transfer equations. Consequently, a dual-mixed approach with Dirichlet data is defined in each sub-system, and the well-known Banach and Brouwer theorems are combined with Babuška–Brezzi’s theory in each independent set of equations, yielding the solvability of the continuous and discrete schemes. We show that our development also applies to the case where the equations of thermal energy and solute transport are coupled through cross-diffusion. Appropriate finite element subspaces are specified, and optimal a priori error estimates are derived. Furthermore, a reliable and efficient residual-based a posteriori error estimator is proposed. Several numerical examples illustrate the performance of the fully-mixed scheme and of the adaptive refinement algorithm driven by the error estimator.

Published online:
DOI: 10.5802/smai-jcm.64
Classification: 65N30, 65N12, 65N15, 35Q79, 80A20, 76D05, 76R10
Keywords: Oberbeck–Boussinesq equations, fully–mixed formulation, fixed-point theory, finite element methods, a priori error analysis
Eligio Colmenares 1; Gabriel N. Gatica 2; Sebastián Moraga 3; Ricardo Ruiz-Baier 4

1 Departamento de Ciencias Básicas, Facultad de Ciencias, Universidad del Bío-Bío, Campus Fernando May, Chillán, Chile.
2 CI 2 MA and Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile.
3 Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, BC V5A 1S6, Canada.
4 School of Mathematics, Monash University, 9 Rainforest Walk, Clayton, VIC 3800, Australia.
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Eligio Colmenares; Gabriel N. Gatica; Sebastián Moraga; Ricardo Ruiz-Baier. A fully-mixed finite element method for the steady state Oberbeck–Boussinesq system. The SMAI Journal of computational mathematics, Volume 6 (2020), pp. 125-157. doi : 10.5802/smai-jcm.64. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.64/

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