A new class of stabilized virtual element methods for the time-dependent Oseen equations.

In this paper, we extend the divergence-free virtual element method for the time-dependent Oseen equations on polygonal meshes. We employ the standard virtual element method with suitable local projection stabilizations (tensor ??????2 projection, ??????1 projection and vector ??????2 projection) to cope with the convection-dominated case without disrupting the divergence-free property for space discretization and the Crank-Nicolson method for the time discretization. Stability and error estimation are analyzed for semi-discrete and full-discrete schemes of tensor ??????2-projection stabilization. Numerical experiments confirm the theoretical results, and three local projection stabilized methods are stable in the convection-dominated case.