Optimal Uniform Convergence Analysis for a Two-dimensional Parabolic Problem with Two Small Parameters

In this paper, we consider a two-dimensional parabolic equation with two small parameters. These small parameters make the underlying problem containing multiple scales over the whole problem domain. By using the maximum principle with carefully chosen barrier functions, we obtain the pointwise derivative estimates of arbitrary order, from which an anisotropic mesh is constructed. This mesh uses very finer mesh inside the small scale regions (where the boundary layers are located) than elsewhere (large scale regions). A fully discrete backward difference Galerkin scheme based on this mesh with arbitrary k-th (k ≥ 1) order conforming rectangular elements is discussed. Note that the standard finite element analysis technique can not be used directly for such highly nonuniform anisotropic meshes because of the violation of the quasi-uniformity assumption. Then we use the integral identity superconvergence technique to prove the optimal uniform convergence O(N−(k+1) +M−1) in the discrete L2-norm, where N and M are the number of partitions in the spatial (same in both the xand y-directions) and time directions, respectively.