On the convergence of a low order Lagrange finite element approach for natural convection problems

The purpose of this article is to study the convergence of a low order finite element approximation for a natural convection problem. We prove that the discretization based on P1 polynomials for every variable (velocity, pressure and temperature) is well-posed if used with a penalty term in the dive...

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Bibliographic Details
Published inarXiv.org
Main Authors Danaila, Ionut, Luddens, Francky, Legrand, Cécile
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 26.07.2022
Subjects
Boussinesq equations
Convergence
Divergence
Free convection
Polynomials
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Summary:The purpose of this article is to study the convergence of a low order finite element approximation for a natural convection problem. We prove that the discretization based on P1 polynomials for every variable (velocity, pressure and temperature) is well-posed if used with a penalty term in the divergence equation, to compensate the loss of an inf-sup condition. With mild assumptions on the pressure regularity, we recover convergence for the Navier-Stokes-Boussinesq system, provided the penalty term is chosen in accordance with the mesh size. We express conditions to obtain optimal order of convergence. We illustrate theoretical convergence results with extensive examples. The computational cost that can be saved by this approach is also assessed.
ISSN:2331-8422