(untitled) (Journal of Computational Mathematics, 2018)
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 subintervals 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.