A novel stabilized P1 × P0 finite element formulation for the time-dependent incompressible Navier-Stokes equations

In this paper, we present a derivation of a new stabilized finite element formulation for the time-dependent incompressible Navier-Stokes equations when the P1 × P0 element pair is used. Unlike the traditional choice in the literature, we motivate the expression of the stabilization from the inconsistency caused by the P1 × P0 element pair in the procedure of integration by parts and also suggest adding a grad-div term to the stabilization. We show that for large γ, the conventional approach may lead to locking and result in a less accurate numerical velocity, while the addition of grad-div stabilization may help to improve performance as demonstrated through numerical experiments. Numerical experiments with the Taylor-Green vortex show the effectiveness of the dissipation provided by the stabilization in our and the conventional formulations for both large and small viscosities. A brief discussion on the interpretation of simulation results from both the perspectives of numerical partial differential equations and physics is presented, and a slightly different new view is proposed within the finite element framework.