### Abstract

Language | English |
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Journal | Computer Methods in Applied Mechanics and Engineering |

Early online date | 31 May 2018 |

DOIs | |

Publication status | E-pub ahead of print - 31 May 2018 |

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### Keywords

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

### Cite this

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**A divergence-free low-order stabilized finite element method for a generalized steady state Boussinesq problem.** / Allendes, Alejandro; Barrenechea, Gabriel R.; Narranjo, Cesar.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A divergence-free low-order stabilized finite element method for a generalized steady state Boussinesq problem

AU - Allendes, Alejandro

AU - Barrenechea, Gabriel R.

AU - Narranjo, Cesar

PY - 2018/5/31

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

KW - divergence-free discrete velocity

UR - https://www.journals.elsevier.com/computer-methods-in-applied-mechanics-and-engineering

U2 - 10.1016/j.cma.2018.05.020

DO - 10.1016/j.cma.2018.05.020

M3 - Article

JO - Computer Methods in Applied Mechanics end Engineering

T2 - Computer Methods in Applied Mechanics end Engineering

JF - Computer Methods in Applied Mechanics end Engineering

SN - 0045-7825

ER -