Penalty-Factor-Free Stabilized Nonconforming Finite Elements for Solving Stationary Navier-Stokes Equations (Journal of Computational Mathematics, 2022)
Two nonconforming penalty methods for the two-dimensional stationary Navier-Stokes equations are studied in this paper. These methods are based on the weakly continuous $P_1$ vector fields and the locally divergence-free (LDF) finite elements, which respectively penalize local divergence and are discontinuous across edges. These methods have no penalty factors and avoid solving the saddle-point problems. The existence and uniqueness of the velocity solution are proved, and the optimal error estimates of the energy norms and $L^2$-norms are obtained. Moreover, we propose unified pressure recovery algorithms and prove the optimal error estimates of $L^2$-norm for pressure. We design a unified iterative method for numerical experiments to verify the correctness of the theoretical analysis.
Keywords: Stationary Navier-Stokes equations; Nonconforming finite elements; Penalty stabilization methods; DG methods; Locally divergence-free