A lowest-order divergence-free virtual element method for Navier-Stokes equations with damping on polygonal mesh

Abstract

This paper focuses on designing a lowest-order divergence-free virtual element method for solving Navier-Stokes equations with a nonlinear damping term on polygonal meshes. The exact divergence-free property of virtual space preserves the mass-conservation of the system. With the application of Helmholtz projection, we provide stability estimates regarding the velocity. An optimal convergence estimate is derived, showing that the error estimate for the velocity in energy norm is pressure-independent. Finally, we perform various numerical simulations to validate the accuracy of our theoretical findings.