On ε-uniform convergence of exponentially fitted methods

Abstract. A class of methods constructed to numerically approximate the solution of two-point singularly perturbed boundary value problems of the form εu + bu + cu = 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 the interpolating function standing behind the method. Because of that, we consider the interpolation error for exponential sums. The main result of the paper is an error bound for interpolation by the exponential sum to the solution of the singularly perturbed problem that does not depend on perturbation parameter ε when ε is small with the respect to mesh width. The numerical experiment implies that the use of a dense mesh in the boundary layer for small meshwidth results in ε-uniform convergence. AMS subject classifications: 65L12, 11E23, 65L10, 65L20