Divergence-Free Virtual Element Method for the Stokes Equations with Damping on Polygonal Meshes

Abstract

In this paper, we construct, analyze, and numerically validate a family of divergence-free virtual elements for Stokes equations with nonlinear damping on polygonal meshes. The virtual element method is $\mathbf{H}^1$-conforming and exact divergence-free. By virtue of these properties and the topological degree argument, we rigorously prove the well-posedness of the proposed discrete scheme. The convergence analysis is carried out, which imply that the error estimate for the velocity in energy norm does not explicitly depend on the pressure. Numerical experiments on various polygonal meshes validate the accuracy of the theoretical analysis and the asymptotic pressure robustness of the proposed scheme.