On piecewise-uniform meshes for upwind- and central-difference operators for solving singularly perturbed problems

Numerical methods composed of upwind-difference operators onuniform meshes are shown analytically to be defective for solvingsingularly perturbed differential equations, in the sense that,as the mesh is refined, the error in the numerical approximationincreases until the mesh parameter has decreased to the sameorder of magnitude as the singular perturbation parameter. Itis also shown that the same is true for upwind-difference operatorson piecewise uniform meshes having a transition point that isdependent solely on the singular perturbation parameter. Itis then shown, with specific analytic examples, that both upwind-and central-difference operators on specially designed piecewise-uniformmeshes give numerical methods which do not suffer from thisdefect. Conditions are also given on the structure of a piecewiseuniform mesh that are necessary if the numerical method, composedof this mesh and an upwind-difference operator, is to be convergentuniformly with respect to the singular perturbation parameter.