The authors consider the nonlinear parabolic equation
with the initial condition in and Neumann boundary condition on , where is a rectangular domain in , is its boundary, is the outward unit normal vector on and is a symmetric, positive definite second-order diagonal tensor.
Using the variational formulation of the above problem they obtain a two-level nonlinear cell-centered finite difference scheme on a “coarse” grid of mesh size . For belonging to on each time-level and for being of class the uniqueness and the existence of the solution to the discrete problem is proved, when is small enough, where , . An a priori error estimation of order is also given. (Here ; in general, the order of the error estimation is . Next, the authors construct a linear difference scheme on a “fine” grid of mesh size , using the nonlinear solution on the coarse grid. The a priori error estimation is now of order .
No computational results are given, but they are announced to appear in later papers.