Finite volume superconvergence approximation for one-dimesional singularly perturbed problems

Abstract

We analyze finite volume schemes of arbitrary order r for the one-dimensional singularly perturbed convection-diffusion problem on the Shishkin mesh. We show that the error under the energy norm decays as (N-1ln(N + 1))r, where 2N is the number of subinter-vals of the primal partition. Furthermore, at the nodal points, the error in function value approximation super-converges with order (N-1ln(N + 1))2r, while at the Gauss points, the derivative error super-converges with order (N-1ln(N + 1))r+1. All the above convergence and superconvergence properties are independent of the perturbation parameter ε. Numerical results are presented to support our theoretical findings. Copyright 2013 by AMSS, Chinese Academy of Sciences.