A C1-virtual element method of high order for the Brinkman equations in stream function formulation with pressure recovery

Abstract

In this paper, we propose and analyze a $C^1$-virtual element method of high order to solve the Brinkman problem formulated in terms of the stream function. The velocity is obtained as a simple post-process from stream function and a novel strategy is written to recover the fluid pressure. We establish optimal a priori error estimates for the stream function, velocity and pressure with constants independent of the viscosity. Finally, we report some numerical test illustrating the behavior of the virtual scheme and supporting our theoretical results on different families of polygonal meshes.

Divergence-Free Virtual Element Method for the Stokes Equations with Damping on Polygonal Meshes

“…For a thorough description of the VEM, see [3,19,41]. The (divergence-free) VEM are used to discretize the Stokes problem [12,24,29,40,42], the Brinkman problem [33,44,51], the Navier-Stokes problem [25,56] and so on [21,28,30,43,53]. So far, we have not found any literature on divergence-free VEM for the Stokes equations with a nonlinear damping term, which constitutes another motivating and innovative aspect of this paper.…”