The Lowest-Order Stabilized Virtual Element Method for the Stokes Problem

Abstract

In this paper, we develop and analyze two stabilized mixed virtual element schemes for the Stokes problem based on the lowest-order velocity-pressure pairs (i.e., a piecewise constant approximation for pressure and an approximation with an accuracy order $k = 1$ for velocity). By applying local pressure jump and projection stabilization, we ensure the well-posedness of our discrete schemes and obtain the corresponding optimal $H^1$- and $L^2$-error estimates. The proposed schemes offer a number of attractive computational properties, such as, the use of polygonal/polyhedral meshes (including non-convex and degenerate elements), yielding a symmetric linear system that involves neither the calculations of higher-order derivatives nor additional coupling terms, and being parameter-free in the local pressure projection stabilization. Finally, we present the matrix implementations of the essential ingredients of our stabilized virtual element methods and investigate two- and three-dimensional numerical experiments for incompressible flow to show the performance of these numerical schemes.