Error estimates of a two‐grid penalty finite element method for the Smagorinsky model

In the paper, we consider the penalty finite element methods (FEMs) for the stationary Smagorinsky model. Firstly, a one‐grid penalty FEM is proposed and analyzed. Since this method is nonlinear, a novel linearized iteration scheme is derived for solving it. We also derived the stability and convergence of numerical solutions for this iteration scheme. Furthermore, a two‐grid penalty FEM is developed for Smagorinsky model. Under , this method consist of solving a nonlinear Smagorinsky model by the one‐grid penalty FEM with the proposed linearized iteration scheme on a coarse mesh with mesh width and then solving a linearized Smagorinsky model based on the Newton iteration on a fine mesh with mesh width , respectively. Stability and error estimates of numerical solutions for two‐grid penalty FEM are presented. Finally, some numerical tests are provided to confirm the theoretical analysis and the effectiveness of the developed methods.