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.