Error Estimates on A New Nonlinear Galerkin MethodBased on Two - grid Finite Elements 1

A new nonlinear Galerkin method based on nite element discretization is presented in this paper for a class of second order nonlinear parabolic equations. The new scheme is based on two diierent nite element spaces deened respectively on one coarse grid with grid size H and one ne grid with grid size h H. Nonlinearity and time dependence are both treated on the coarse space and only a xed stationary equation needs to be solved on the ne space at each time. With linear nite element discretizations, it is proved that the diierence between the new nonlinear Galerkin solution and the standard Galerkin solution in H 1 (() norm is of the order of H 3 .