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

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A linear one-dimensional convection–diffusion problem with a small singular perturbation parameter �� is considered. The problem is discretized using finite-difference schemes on the Shishkin mesh. Generally speaking, such discretizations are not consistent uniformly in ��, so ��-uniform convergence cannot be proved by the classical approach based on ��-uniform stability and ��-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 ��, is both consistent and stable. Using this technique, optimal error bounds are obtained for the upwind and hybrid finite-difference schemes.