Collocation for Singular Perturbation Problems III: Nonlinear Problems without Turning Points

Abstract

A class of nonlinear singularly perturbed boundary value problems is considered, with restrictions which allow only well-posed problems with possible boundary layers, but no turning points. For the numerical solution of these problems, a close look is taken at a class of general purpose, symmetric finite difference schemes arising from collocation. It is shown that if locally refined meshes, whose practical construction is discussed, are employed, then high order uniform convergence of the numerical solution is obtained. Nontrivial examples are used to demonstrate that highly accurate solutions to problems with extremely thin boundary layers can be obtained in this way at a very reasonable cost.