Boundary value problems for singularly perturbed semilinear elliptic equations are considered. Special piecewise-uniform meshes are constructed which yield accurate numerical solutions irrespective of the value of the small parameter. Numerical methods composed of standard monotone finite difference operators and these piecewise-uniform meshes are shown theoretically to be uniformly (with respect to the singular perturbation parameter) convergent. Numerical results are also presented, which indicate that in practice the method is first-order accurate.
On the Non-Existence of [Epsilon]-Uniform Finite Difference Methods on Uniform Meshes for Semilinear Two-point Boundary Value Problems04/01/1998
In this paper fitted finite difference methods on a uniform mesh with internodal spacing h, are considered for a singularly perturbed semilinear two-point boundary value problem. It is proved that a scheme of this type with a frozen fitting factor cannot converge epsilon-uniformly in the maximum norm to the solution of the differential equation as the mesh spacing h goes to zero. Numerical experiments are presented which show that the same result is true, for a number of schemes with variable fitting factors.