Graded Galerkin Methods for the High-Order Convection-Diffusion Problem

We develop a Galerkin method using the Hermite spline oil ail admissible graded mesh for solving the high-order singular perturbation problem of the convection-diffusion type. We identify a special function class to which the solution of the convection-diffusion problem belongs and characterize the approximation order of the Hermite spline for such a function class. The approximation order is then used to establish the optimal order of uniform convergence for the Galerkin method. Numerical results are presented to confirm the theoretical estimate. (C) 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 25: 1261-1282,2009