A new local projection stabilization virtual element method for the Oseen problem on polygonal meshes

For the Oseen problem, we present a new stabilized virtual element method on polygonal meshes that allows us to employ “equal-order” virtual element pairs to approximate both velocity and pressure. By introducing the local projection type stabilization terms to the virtual element method, the method can not only circumvent the discrete Babuška-Brezzi condition, but also maintain the favorable stability and approximation properties of residual-based stabilization methods. In particular, it does not need to calculate complex high-order derivative terms and avoids the strong coupling terms of velocity and pressure. Error estimates are obtained without depending on the inverse of the viscosity, which means that the method is effective in the convective-dominated regime. Some numerical experiments are performed to verify the method has good behaviors.