(untitled) (Journal of Computational Mathematics, 2021)
A global finite element nonlinear Galerkin method for the penalized Navier-Stokes equations is presented. This method is based on two finite element spaces X_H and X_h, defined respectively on one coarse grid with grid size H and one fine grid with grid size h＜＜H. Comparison is also made with the finite element Galerkin method. If we choose H=O(ε^-1/4h^1/2), ε＞0 being the penalty parameter, then two methods are of the same order of approximation. However, the global finite element nonlinear Galerkin method is much cheaper than the standard finite element Galerkin method. In fact, in the finite element Galerkin method the nonlinearity is treated on the fine grid finite element space X_h and while in the global finite element nonlinear Galerkin method the similar nonlinearity is treated on the coarse grid finite element space X_H and only the linearity needs to be treated on the fine grid increment finite element space W_h. Finally, we provide numerical test which shows above results stated.