Robust Error Estimates for Second‐Order Stabilization Finite Element Method for Navier–Stokes Equations With Small Viscosity and Nonsmooth Initial Data

ABSTRACT In this article, we study the fully discrete finite element approximation to the two‐dimensional Navier–Stokes equations with a small viscosity coefficient and initial data by using semi‐implicit two‐step backward differentiation formulae (BDF2) method with variable step‐sizes and grad‐div stabilization. The variable step sizes allow us to take special viscosity‐dependent, locally refined time step sizes and thus prove that the variable step‐size BDF2 method can achieve second‐order convergence in time. By adding the grad‐div stabilization, we obtain a robust, fully discrete error bound in which the constants reduce dependence on inverse powers of the viscosity. Numerical results illustrate the effectiveness of the proposed method for such types of Navier–Stokes equations.