@Article{JCM-39-392,
author = {Gatica , Gabriel N.Munar , Mauricio and Sequeira , Filánder A.},
title = {A Mixed Virtual Element Method for the Boussinesq Problem on Polygonal meshes},
journal = {Journal of Computational Mathematics},
year = {2021},
volume = {39},
number = {3},
pages = {392--427},
abstract = {<p style="text-align: justify;">In this work we introduce and analyze a mixed virtual element method (mixed-VEM)
for the two-dimensional stationary Boussinesq problem. The continuous formulation is
based on the introduction of a pseudostress tensor depending nonlinearly on the velocity,
which allows to obtain an equivalent model in which the main unknowns are given by
the aforementioned pseudostress tensor, the velocity and the temperature, whereas the
pressure is computed via a postprocessing formula. In addition, an augmented approach
together with a fixed point strategy is used to analyze the well-posedness of the resulting
continuous formulation. Regarding the discrete problem, we follow the approach employed
in a previous work dealing with the Navier-Stokes equations, and couple it with a VEM
for the convection-diffusion equation modelling the temperature. More precisely, we use
a mixed-VEM for the scheme associated with the fluid equations in such a way that the
pseudostress and the velocity are approximated on virtual element subspaces of $\mathbb{H}$(div) and
$\boldsymbol{H}^1$, respectively, whereas a VEM is proposed to approximate the temperature on a virtual
element subspace of $H^1$. In this way, we make use of the $L^2$-orthogonal projectors onto
suitable polynomial spaces, which allows the explicit integration of the terms that appear in
the bilinear and trilinear forms involved in the scheme for the fluid equations. On the other
hand, in order to manipulate the bilinear form associated to the heat equations, we define
a suitable projector onto a space of polynomials to deal with the fact that the diffusion
tensor, which represents the thermal conductivity, is variable. Next, the corresponding
solvability analysis is performed using again appropriate fixed-point arguments. Further,
Strang-type estimates are applied to derive the a priori error estimates for the components
of the virtual element solution as well as for the fully computable projections of them and
the postprocessed pressure. The corresponding rates of convergence are also established.
Finally, several numerical examples illustrating the performance of the mixed-VEM scheme
and confirming these theoretical rates are presented.</p>},
issn = {1991-7139},
doi = {https://doi.org/10.4208/jcm.2001-m2019-0187},
url = {http://global-sci.org/intro/article_detail/jcm/18742.html}
}