A locally stabilized collocated finite volume method for the stationary Stokes problem

Abstract

In this paper, a novel discrete formulation for stabilizing the collocated finite volume method (CFVM) is introduced and analyzed for the stationary Stokes problem. This method is based on the lowest equal‐order approximation (piecewise constants) for both velocity and pressure unknowns. A macro‐volume (i.e., a small group of volumes) condition is introduced to build the local stabilized finite volume formulation for the problem. We prove that a stabilization term based on the jump of the discrete pressures over the interior edges of the macro‐volume gives a stable scheme. Moreover, the existence and uniqueness of the discrete solution are proved. The error estimate of the stabilized finite volume formulation is derived. Finally, some numerical tests are performed to exhibit the stability and effectiveness of the proposed method.