Nonlinear Galerkin methods and mixed finite elements: two-grid algorithms for the Navier-Stokes equations

Abstract

A nonlinear Galerkin method using mixed finite elements is presented for the two-dimensional incompressible Navier-Stokes equations. The scheme is based on two finite element spaces $X_H$ and $X_h$ for the approximation of the velocity, defined respectively on one coarse grid with grid size $H$ and one fine grid with grid size $h \ll H$ and one finite element space $M_h$ for the approximation of the pressure. Nonlinearity and time dependence are both treated on the coarse space. We prove that the difference between the new nonlinear Galerkin method and the standard Galerkin solution is of the order of $H^2$, both in velocity ( $H^1 (\Omega)$ and pressure $(L^2(\Omega)$ norm). We also discuss a penalized version of our algorithm which enjoys similar properties.