Fundamental systems of numerical schemes for linear convection-diffusion equations and their relationship to accuracy

A new approach towards the assessment and derivation of numerical methods f or convection dominated problems is presented. based on the comparison or t he fundamental systems of the continuous and discrete operators. In two or more space dimensions, the dimension of the fundamental system is infinite, and may be identified with a ball. This set is referred to as the true fun damental locus. The fundamental system for a numerical scheme also forms a locus. As a first application, it is shown that a necessary condition for t he uniform convergence of a numerical scheme is that the discrete locus sho uld contain the true locus, and it is then shown it is impossible to satisf y this condition with a finite stencil. This shows that results of Shishkin concerning non-uniform convergence at parabolic boundaries are also generi c for outflow boundaries. It is shown that the distance between the loci is related to the accuracy of the schemes provided that the loci are sufficie ntly close. However, if the loci depart markedly, then the situation is rat her more complicated. Under suitable conditions, we develop an explicit num erical lower bound on the attainable relative error in terms of the coeffic ients in the stencil characterising the scheme and the loci.