A uniformly accurate finite-element method for a singularly perturbed one-dimensional reaction-diffusion problem

Abstract

A finite-element method with exponential basis elements is applied to a selfadjoint, singularly perturbed, two-point boundary value problem. The tridiagonal difference scheme generated is shown to be uniformly second-order accurate for this problem (i.e., the nodal errors are bounded by C h 2 C{h^2} , where C is independent of the mesh size h and the perturbation parameter). With a certain choice of trial functions, uniform first-order accuracy is obtained in L ∞ [ 0 , 1 ] {L^\infty }[0,1] .