On Ɛ-uniform convergence of exponentially fitted methods

Abstract

A class of methods constructed to numerically approximate solution of two-point singularly perturbed boundary value problems of the form $\varepsilon u'' + b u' + c u = f$ use exponentials to mimic exponential behavior of the solution in the boundary layer(s). We refer to them as exponentially fitted methods. Such methods are usually exact on polynomials of certain degree and some exponential functions. Shortly, they are exact on exponential sums. It is often possible that consistency of the method follows from the convergence of interpolating function standing behind the method. Because of that, we consider interpolation error for exponential sums. A main result of the paper is an error bound for interpolation by exponential sum to the solution of singularly perturbed problem that does not depend on perturbation parameter $\varepsilon$ when $\varepsilon$ is small with the respect to mesh width. Numerical experiment implies that the use of dense mesh in the boundary layer for small meshwidth results with $\varepsilon$-uniform convergence.