A mixed virtual element discretization for the generalized Oseen problem

Abstract: In this paper we introduce a mixed virtual element method to approximate the solution for the two dimensional generalized Oseen problem. We introduce the pseudostress as an additional unknown, which allows to eliminate the pressure from the system; the pressure can be recovered via a post-process of the pseudostress tensor. We prove existence and uniqueness of the continuous solution via a fixed point argument. Under standard mesh assumptions, we develop a virtual element method to approximate both the tensor and the velocity field, and we show that it is stable. Furthermore, we provide a priori error estimates for the method and validate them through a series of numerical tests using different polygonal meshes.