A mixed virtual element method for the Brinkman problem

In this paper, we introduce and analyze a mixed virtual element method (mixed-VEM) for the two-dimensional Brinkman model of porous media flow with non-homogeneous Dirichlet boundary conditions. More precisely, we employ a dual-mixed formulation in which the only unknown is given by the pseudostress, whereas the velocity and pressure are computed via postprocessing formulae. We first recall the corresponding variational formulation, and then summarize the main mixed-VEM ingredients that are required for our discrete analysis. In particular, in order to define a calculable discrete bilinear form, whose continuous version involves deviatoric tensors, we propose two well-known alternatives for the local projector onto a suitable polynomial subspace, which allows the explicit integration of these terms. Next, we show that the global discrete bilinear form satisfies the hypotheses required by the Lax–Milgram lemma. In this way, we conclude the well-posedness of our mixed-VEM scheme and derive the associated a priori error estimates for the virtual solution as well as for the fully computable projection of it. Furthermore, we also introduce a second element-by-element postprocessing formula for the pseudostress, which yields an optimally convergent approximation of this unknown with respect to the broken ℍ(div)-norm. Finally, several numerical results illustrating the good performance of the method and confirming the theoretical rates of convergence are presented.