An Absolutely Stabilized Virtual Element Method for the Incompressible Stokes Equations

Abstract

In this paper, we propose an absolutely stabilized mixed virtual element method (mixed VEM) for the incompressible Stokes equations. We employ the $C^0$-conforming virtual element space for the velocity and discontinuous polynomial space for the pressure. The velocity-pressure pair contains "arbitrary-order" polynomials, and stability is guaranteed by the "absolutely stabilized finite element formulation" originally introduced in [21]. We establish error estimates in the energy norm for both velocity and pressure, as well as an $L^2$-error estimate for the velocity. Several numerical experiments validate the theoretical analysis.