Optimal uniform convergence analysis for a two-dimensional parabolic problem with two small parameters
Jichun Li 11 Department of Mathematical Sciences, University of Nevada, Las Vegas, NV 89154, USA
Received by the editors June 14, 2004 and, in revised form, July 19, 2004
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 point-wise 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 L-2-norm, where N and M are the number of partitions in the spatial (same in both the x- and y-directions) and time directions, respectively.
AMS subject classifications: 65N30, 65N15
Key words: singular perturbation; anisotropic mesh and uniform convergence