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In this paper, we consider a two-level defect-correction stabilized finite element method (DCS-FEM) for the incompressible Navier-Stokes equation with friction boundary conditions based on local Gauss integration. The main idea is to combine the two-level strategy with the defectcorrection method. Using this technique, the simplified two-level DCSFEM and the Newton two-level DCSFEM are proposed and some error estimates are derived. Finally, the numerical results are displayed to confirm our theoretical findings.

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In this article, we consider a penalty finite element (FE) method for incompressible Navier‐Stokes type variational inequality with nonlinear damping term. First, we establish penalty variational formulation and prove the well‐posedness and convergence of this problem. Then we show the penalty FE scheme and derive some error estimates. Finally, we give some numerical results to verify the theoretical rate of convergence. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 918–940, 2017

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