Global finite element nonlinear galerkin method for the penalized navier-stokes equations

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 much less than H. Comparison is also made with the finite element Galerkin method. If we choose H = O(epsilon (-1/4) h(1/2)), epsilon > 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.