On the Conditioning of an Upwind Finite-Difference Approximation of a Convection-Diffusion Boundary Value Problem on an Adaptive Grid.

On the conditioning of an upwind nite-diierence approximation of a convection-diiusion boundary value problem on an adaptive grid. We derive condition number estimates for the standard rst-order upwind numerical discretisation of a model convection-diiusion boundary value problem on a non-uniform grid. The grid is suggested by the equidistribution of a power of the solution gradient. It is shown that the condition number of the linear algebraic system increases at the same rate as the singular perturbation parameter is decreased. It is also shown that a simple preconditioning of the original system leads to a system with a condition number that is independent of the singular perturbation parameter. Numerical experiments are given to connrm the derived condition estimates.