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

Numerical analysis of a modified finite element nonlinear Galerkin method

“…The case of finite element approximations for general nonlinear evolution equations are considered in [18][19]. The case of mixed finite elements approximations for the Navier-Stokes equations is treated in [1,[10][11][12].…”

“…For the standard finite element Galerkin method ( SG method), the discrete solution Uh(', t) is determined in finite element spaces denoted by X h • As in [1], for h < H, we introduce the decomposition…”

“…where, as in [1,19], the intermediate space Xf! is the L 2 -orthogonal supplementary of X H in X h .…”

“…In the equation for w h , the simplification is different from ones given in [1,[10][11], but the equation for w h is still linear.…”

“…Our main results are the following: (1.1) where to > a is a fixed time, while the finite element nonlinear Galerkin method (NG method I) given in [1] has the following convergence rates: (…”