Error Estimates on A New Nonlinear Galerkin Method
                      Based on Two-grid Finite Elements1

                      Martine Marion and Jinchao Xu

Abstract.  A new nonlinear Galerkin method based on finite element
discretization is presented in this paper for a class of second order
nonlinear parabolic equations.  The new scheme is based on two different
finite element spaces defined respectively on one coarse grid with grid size H
and one fine grid with grid size h << H. Nonlinearity and time dependence are
both treated on the coarse space and only a fixed stationary equation needs to
be solved on the fine space at each time.  With linear finite element
discretizations, it is proved that the difference between the new nonlinear
Galerkin solution and the standard Galerkin solution in H<sup>1</sup>(Omega)
norm is of the order of H<sup>3</sup>.