Optimal uniform-convergence results for convection-diffusion problems in one dimension using preconditioning

A linear one-dimensional convection-diffusion problem with a small singular perturbation parameter ε is considered. The problem is discretized using finite-difference schemes on the Shishkin mesh. Generally speaking, such discretizations are not consistent uniformly in ε, so ε-uniform convergence cannot be proved by the classical approach based on ε-uniform stability and ε-uniform consistency. This is why previous proofs of convergence have introduced non-classical techniques (e.g., specially chosen barrier functions). In the present paper, we show for the first time that one can prove optimal convergence inside the classical framework: a suitable preconditioning of the discrete system is shown to yield a method that, uniformly in ε, is both consistent and stable. Using this technique, optimal error bounds are obtained for the upwind and hybrid finite-difference schemes.