Improving the Accuracy of the Exponentially Fitted Scheme on Piecewise Uniform Meshes

A linear one-dimensional singularly perturbed convection-diffusion problem is solved numerically after its solution is decomposed as $u_0+w$, where $u_0$, the corresponding reduced solution, is treated as a function known exactly or approximately. The component $w$ is then calculated using the exponentially fitted Allen-Southwell-Il'in (ASI) scheme on the Shishkin mesh and its asymptotic version. We prove that this numerical method is highly accurate, with errors that diminish when the discretization parameter increases, and, in some cases, even when the perturbation parameter decreases. This is a theoretical confirmation of earlier numerical results showing that the ASI scheme outperforms the general class of Samarskii-type schemes to which it belongs. Even higher accuracy is proved when $u_0$ is linear, in which case, the decomposition is not needed. New numerical experiments are provided to illustrate all this.