FINITE-ELEMENT METHODS FOR SINGULARLY PERTURBED HIGH-ORDER ELLIPTIC 2-POINT BOUNDARY-VALUE-PROBLEMS .1. REACTION-DIFFUSION-TYPE PROBLEMS

We consider singularly perturbed high-order elliptic two-point boundary value problems of reaction-diffusion type. It is shown that, on an equidistant mesh, polynomial schemes cannot achieve a high order of convergence that is uniform in the perturbation parameter. Piecewise polynomial Galerkin finite-element methods are then constructed on a Shishkin mesh. Almost optimal convergence results, which are uniform in the perturbation parameter, are obtained in various norms. Numerical results are presented for a fourth-order problem.