Anisotropic mesh refinement for a singularly perturbed reaction diffusion model problem

The paper is concerned with the finite element resolution of layers appearing in singularly perturbed problems. A special anisotropic grid of Shishkin type is constructed for a reaction diffusion model problem, Estimates of the finite element error in the energy norm are derived for two methods, namely the standard Galerkin method and a stabilized Galerkin method. The estimates are uniformly valid with respect to the (small) diffusion parameter. One ingredient is a pointwise description of derivatives of the continuous solution. A numerical example supports the result, Another key ingredient for the error analysis is a refined estimate for (higher) derivatives of the interpolation error. The assumptions on admissible anisotropic finite elements are formulated in terms of geometrical conditions for triangles and tetrahedra. The application of these estimates is not restricted to the special problem considered in this paper. (C) 1998 IMACS/Elsevier Science B.V.