Finite element superconvergence approximation for one-dimensional singularly perturbed problems

Superconvergence approximations of singularly perturbed two-point boundary value problems of reaction-diffusion type and convection-diffusion type are studied. By applying the standard finite element method of any fixed order p on a modified Shishkin mesh, superconvergence error bounds of (N-1 ln(N + 1))(p+1) in a discrete energy norm in approximating problems with the exponential type boundary layers are established. The error bounds are uniformly valid with respect to the singular perturbation parameter. Numerical tests indicate that the error estimates are sharp; in particular, the logarithmic factor is not removable. (C) 2002 Wiley Periodicals, Inc.