A projection-based stabilized virtual element method for the unsteady incompressible Brinkman equations

Abstract

This paper is devoted to a stabilized mixed virtual element method (mixed VEM) for the unsteady incompressible Brinkman equations. We employ the pairs of C0-conforming virtual element spaces containing the “equal-order” polynomials to approximate the velocity and pressure variables, and replace the time derivative by a backward Euler difference quotient. The numerical stability is guaranteed with a new projection-based stabilization term, which is simple without the projection of second derivatives or the coupling terms. We also establish the error estimates for both the semi-discrete and fully-discrete schemes with respect to the viscosity coefficient. Finally, we carry out several numerical experiments to validate the theoretical analysis.