Uniform convergence analysis of an upwind finite-difference approximation of a convection-diffusion boundary value problem on an adaptive grid

We derive an epsilon-uniform error estimate for the first-order upwind disc retization of a model convection-diffusion boundary value problem on a non- uniform grid. Here, epsilon is the small parameter multiplying the highest derivative term. The grid is constructed by the equidistribution of the pow er of the gradient of the solution of a constant coefficient problem. The u se of equidistribution principles appears in many grid adaption schemes and our analysis indicates the convergence behaviour on such grids. We also sh ow that the condition number of the resulting linear algebraic systems incr eases at the same rate as the singular perturbation parameter is decreased. A simple preconditioning of the original system is shown to lead to system s with condition numbers that are independent of the singular perturbation parameter. Numerical results are given that confirm the epsilon-uniform con vergence rate of the nodal errors and the effect of preconditioning on the linear algebraic systems.