Two stabilized finite element methods based on local polynomial pressure projection for the steady-state Navier–Stokes–Darcy problem

This study presents two stabilized finite element methods based on local polynomial pressure projections for the mixed steady-state Navier–Stokes–Darcy problem by utilizing the equal order finite element pairs, the $P_1$-$P_1$-$P_1$ and $P_2$-$P_2$-$P_2$ element pairs, for approximating the fluid velocity, kinematic pressure and dynamic pressure, respectively. The presented stabilized methods possess many chief characteristics, for instance, parameter free, simple calculation, element level implementation. The optimal error estimates are established. Finally, some comprehensively numerical tests are reported to examine the efficiency and robustness of the proposed algorithms.