Accurate discretization for singular perturbations: the one-dimensional case

This paper develops a discretization method for one-dimensional singular perturbation problems based on a Petrov-Galerkin finite element, or an equivalent finite volume, scheme. The method is unique in that: its discretization error has a bound that is second order in the mesh size and uniform in the perturbation parameter; it satisfies a local discrete conservation law; and it exhibits a discrete maximum principle. Numerical results are included for comparison with the theory.