Tangent space correction method for the Galerkin approximation based on two-grid finite element

A novel fully discrete two-grid finite element method for certain semi-linear parabolic equations is presented in this paper. The new scheme is based on two different finite element subspaces defined respectively on one coarse grid with grid size H and one fine grid with grid size h ≪ H. Nonlinearity is treated in the coarse grid subspace by solving the standard Galerkin equation, while in the fine grid subspace, only a linear equation has to be solved. Differing from the usual two-grid method, the splitting of the coarse grid subspace and its related fine grid incremental subspace is based on a new projection, in which sense the incremental subspace is closely identified with the tangent space of certain operator on each time step. With linear finite element discretization, the stability and error estimate results for such new scheme are derived. The results show that the difference between this new approximation and the fine grid standard Galerkin approximation in H1(Ω) norm is of the order H4-d2, where d = 1, 2, 3 is the space dimension.