A Two-Grid Finite Difference Scheme for Nonlinear Parabolic Equations

We present a two-level finite difference scheme for the approximation of nonlinear parabolic equations. Discrete inner products and the lowest-order Raviart--Thomas approximating space are used in the expanded mixed method in order to develop the finite difference scheme. Analysis of the scheme is given assuming an implicit time discretization. In this two-level scheme, the full nonlinear problem is solved on a "coarse" grid of size H. The nonlinearities areexpanded about the coarse grid solution and an appropriate interpolation operator is used to provide values of the coarse grid solution on the fine grid in terms of superconvergent node points. The resulting linear but nonsymmetric system is solved on a "fine" grid of size h. Some a priori error estimates are derived which show that the discrete L ∞ (L 2 ) and L 2 (H 1 ) errors are O(h² + H4-d/2 + Δ t), where d ≥ 1 is the spatial dimension.