A mixed virtual element method for the Brinkman problem

Abstract

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.

A new staggered DG method for the Brinkman problem robust in the Darcy and Stokes limits

“…Recently, some novel methods have been proposed to solve the Brinkman problem on general quadrilateral and polygonal meshes, which is tricky and usually requires special treatment in order to deliver robust results with respect to the rough grids. The development of numerical methods for Brinkman problem on general meshes is still in its infancy and the existing methods that have been successfully designed on general meshes for Brinkman problem include the weak Galerkin method, the virtual element method and the hybrid high-order methods [35,12,37,7].…”

A Hybrid High-Order discretisation of the Brinkman problem robust in the Darcy and Stokes limits

“…We proceed to bound the consistency error for a generic v h ∈ U k h,0 . For the sake of brevity, throughout the proof we let, for all T ∈ T h ,w T r k+1 S,T I k T w = π k+1 ε,T w (see (15)). We start by noting the following consistency property for the stabilisation term valid under Assumption 2, whose proof follows using the arguments of [28, Proposition 3.1]: For all T ∈ T h ,…”

“…Finally, new generation technologies have been recently proposed for the discretisation of problem (2). We cite, in particular, the isogeometric divergence-conforming B-splines of [33], the Weak Galerkin method of [40], the two-dimensional Virtual Element methods of [15,47] (see also the related work [7]), and the multiscale hybrid-mixed method of [5].…”

A virtual element method for a nonlocal FitzHugh–Nagumo model of cardiac electrophysiology

“…Although the VEM is very recent, it has been applied to a large number of problems; for instance, VEM for Stokes, Brinkman, Cahn-Hilliard, plates bending, advection-diffusion, Helmholtz, parabolic, and hyperbolic problems have been introduced in [4,5,15,17,24,19,21,26,27,30,51,54,55,56], VEM for spectral problems in [18,37,42,44], VEM for linear and non-linear elasticity in [6,9,13,36,57], whereas a posteriori error analysis have been developed in [16,20,28,43].…”