Edge Patch-Wise Local Projection Stabilized Nonconforming FEM for the Oseen Problem

Abstract

In finite element approximation of the Oseen problem, one needs to handle two major difficulties, namely, the lack of stability due to convection dominance and the incompatibility between the approximating finite element spaces for the velocity and the pressure. These difficulties are addressed in this article by using an edge patch-wise local projection (EPLP) stabilization technique. The article analyses the EPLP stabilized nonconforming finite element methods for the Oseen problem. For approximating the velocity, the lowest-order Crouzeix–Raviart (CR) nonconforming finite element space is considered; whereas for approximating the pressure, two discrete spaces are considered, namely, the piecewise constant polynomial space and the lowest-order CR finite element space. The proposed discrete weak formulation is a combination of the standard Galerkin method, EPLP stabilization and weakly imposed boundary condition by using Nitsche’s technique. The resulting bilinear form satisfies an inf-sup condition with respect to EPLP norm, which leads to the well-posedness of the discrete problem. A priori error analysis assures the optimal order of convergence in both the cases, that is, order one in the case of piecewise constant approximation and ¾ in the case of CR-finite element approximation for pressure. The numerical experiments illustrate the theoretical findings.