A divergence-free low-order stabilized finite element method for a generalized steady state Boussinesq problem

Alejandro Allendes, Gabriel R. Barrenechea, Cesar Narranjo

Research output: Contribution to journalArticle

Abstract

In this work we propose and analyze a new stabilized finite element method for the coupled Navier-Stokes/temperature (or Boussinesq) equations. The method is built using low order conforming elements for velocity and temperature, and piecewise constant elements for pressure. With the help of the lowest order Raviart-Thomas space, a lifting of the jumps of the discrete pressure is built in such a way that when this lifting is added to the conforming velocity field, the resulting velocity is solenoidal (at the price of being non-conforming). This field is then fed to the momentum and temperature equations, guaranteeing that the convective terms in these equations are antisymmetric, without the need of altering them, thus simplifying the analysis of the resulting method. Existence of solutions, discrete stability, and optimal convergence are proved for both the conforming velocity field, and its corresponding divergence-free non-conforming counterpart. Numerical results confirm the theoretical findings, as well as the gain provided by the solenoidal discrete velocity field over the conforming one.
LanguageEnglish
JournalComputer Methods in Applied Mechanics and Engineering
Early online date31 May 2018
DOIs
Publication statusE-pub ahead of print - 31 May 2018

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divergence
finite element method
velocity distribution
Finite element method
temperature
momentum
Temperature
Momentum

Keywords

  • Boussinesq problem
  • stabilized finite element method
  • divergence-free discrete velocity

Cite this

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title = "A divergence-free low-order stabilized finite element method for a generalized steady state Boussinesq problem",
abstract = "In this work we propose and analyze a new stabilized finite element method for the coupled Navier-Stokes/temperature (or Boussinesq) equations. The method is built using low order conforming elements for velocity and temperature, and piecewise constant elements for pressure. With the help of the lowest order Raviart-Thomas space, a lifting of the jumps of the discrete pressure is built in such a way that when this lifting is added to the conforming velocity field, the resulting velocity is solenoidal (at the price of being non-conforming). This field is then fed to the momentum and temperature equations, guaranteeing that the convective terms in these equations are antisymmetric, without the need of altering them, thus simplifying the analysis of the resulting method. Existence of solutions, discrete stability, and optimal convergence are proved for both the conforming velocity field, and its corresponding divergence-free non-conforming counterpart. Numerical results confirm the theoretical findings, as well as the gain provided by the solenoidal discrete velocity field over the conforming one.",
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author = "Alejandro Allendes and Barrenechea, {Gabriel R.} and Cesar Narranjo",
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AU - Allendes, Alejandro

AU - Barrenechea, Gabriel R.

AU - Narranjo, Cesar

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Y1 - 2018/5/31

N2 - In this work we propose and analyze a new stabilized finite element method for the coupled Navier-Stokes/temperature (or Boussinesq) equations. The method is built using low order conforming elements for velocity and temperature, and piecewise constant elements for pressure. With the help of the lowest order Raviart-Thomas space, a lifting of the jumps of the discrete pressure is built in such a way that when this lifting is added to the conforming velocity field, the resulting velocity is solenoidal (at the price of being non-conforming). This field is then fed to the momentum and temperature equations, guaranteeing that the convective terms in these equations are antisymmetric, without the need of altering them, thus simplifying the analysis of the resulting method. Existence of solutions, discrete stability, and optimal convergence are proved for both the conforming velocity field, and its corresponding divergence-free non-conforming counterpart. Numerical results confirm the theoretical findings, as well as the gain provided by the solenoidal discrete velocity field over the conforming one.

AB - In this work we propose and analyze a new stabilized finite element method for the coupled Navier-Stokes/temperature (or Boussinesq) equations. The method is built using low order conforming elements for velocity and temperature, and piecewise constant elements for pressure. With the help of the lowest order Raviart-Thomas space, a lifting of the jumps of the discrete pressure is built in such a way that when this lifting is added to the conforming velocity field, the resulting velocity is solenoidal (at the price of being non-conforming). This field is then fed to the momentum and temperature equations, guaranteeing that the convective terms in these equations are antisymmetric, without the need of altering them, thus simplifying the analysis of the resulting method. Existence of solutions, discrete stability, and optimal convergence are proved for both the conforming velocity field, and its corresponding divergence-free non-conforming counterpart. Numerical results confirm the theoretical findings, as well as the gain provided by the solenoidal discrete velocity field over the conforming one.

KW - Boussinesq problem

KW - stabilized finite element method

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