A fully-mixed finite element method for the steady state Oberbeck-Boussinesq system
type: Item
instance of: scholarly article
title: A fully-mixed finite element method for the steady state Oberbeck-Boussinesq system
zbMATH Open document ID: 1451.65187
author: Eligio Colmenares
author: Gabriel N. Gatica
author: Sebastián Moraga
author: Ricardo Ruiz-Baier
publication date: 2020-07-13T00:00:00Z
review text: The rough contents of this vast paper are as follows: 1. Introduction, 2. Governing equations, 3. Well-posedness of the continuous problem, 4. The Galerkin scheme, 5. A priori error analysis, 6. A posteriori error estimator and 7. Numerical results. The References contain 48 entries. The authors are concerned with the construction, the analysis and the Implementation of a high-order optimally convergent mixed finite element scheme for solving various nonlinearly coupled system from fluid dynamics. Thus, the stress and the velocity gradient in the fluid equations, the temperature gradient, the concentration gradient and a vector version of the Bernoulli tensor combining advective and diffusive heat and concentration fluxes are the field variables (unknowns). From mathematical point of view the well-known Banach and Brouwer theorems, combined with the application of the Babuška-Brezzi theory to each independent equation, lead to the solvability of the continuous and discrete schemes. A reliable and efficient residual-based a posteriori error estimator and its associated adaptive refinement algorithm are introduced and discussed. Five challenging numerical examples are carried out. These give technical details on the implementation of the proposed algorithm and at the same time they show the beauty of the numerical results obtained.
Mathematics Subject Classification ID: 65N30
Mathematics Subject Classification ID: 65N12
Mathematics Subject Classification ID: 65N15
Mathematics Subject Classification ID: 35Q79
Mathematics Subject Classification ID: 80A17
Mathematics Subject Classification ID: 76D05
Mathematics Subject Classification ID: 76R10
zbMATH DE Number: 7219867
zbMATH Keywords: steady Oberbeck-Boussinesq equations
zbMATH Keywords: fully-mixed formulation
zbMATH Keywords: fixed-point theory
zbMATH Keywords: finite element methods
zbMATH Keywords: a priori error analysis
zbMATH Keywords: numerical example
reviewed by: Calin Ioan Gheorghiu
describes a project that uses: FEniCS
describes a project that uses: Gmsh
MaRDI profile type: Publication
full work available at URL: smai-jcm.64
OpenAlex ID: W3033075557
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DOI: 10.5802/SMAI-JCM.64
published in: The SMAI journal of computational mathematics
description: scientific article; zbMATH DE number 7219867