A Mixed Virtual Element Method for the Boussinesq Problem on Polygonal meshes (Journal of Computational Mathematics, 2021)
In this work we introduce and analyze a mixed virtual element method (mixed-VEM)for the two-dimensional stationary Boussinesq problem. The continuous formulation isbased 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 bythe aforementioned pseudostress tensor, the velocity and the temperature, whereas thepressure is computed via a postprocessing formula. In addition, an augmented approachtogether with a fixed point strategy is used to analyze the well-posedness of the resultingcontinuous formulation. Regarding the discrete problem, we follow the approach employedin a previous work dealing with the Navier-Stokes equations, and couple it with a VEMfor the convection-diffusion equation modelling the temperature. More precisely, we usea mixed-VEM for the scheme associated with the fluid equations in such a way that thepseudostress 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 virtualelement subspace of $H^1$. In this way, we make use of the $L^2$-orthogonal projectors ontosuitable polynomial spaces, which allows the explicit integration of the terms that appear inthe bilinear and trilinear forms involved in the scheme for the fluid equations. On the otherhand, in order to manipulate the bilinear form associated to the heat equations, we definea suitable projector onto a space of polynomials to deal with the fact that the diffusiontensor, which represents the thermal conductivity, is variable. Next, the correspondingsolvability analysis is performed using again appropriate fixed-point arguments. Further,Strang-type estimates are applied to derive the a priori error estimates for the componentsof the virtual element solution as well as for the fully computable projections of them andthe postprocessed pressure. The corresponding rates of convergence are also established.Finally, several numerical examples illustrating the performance of the mixed-VEM schemeand confirming these theoretical rates are presented.
Keywords: Boussinesq problem; Pseudostress-based formulation; Augmented formulation; Mixed virtual element method; High-order approximations.