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 for convection dominated problems is presented, based on the comparison of the 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 fundamental locus. The fundamental system for a numerical scheme also forms a locus. As a first application, it is shown that a necessary condition for the uniform convergence of a numerical scheme is that the discrete locus should contain the true locus, and it is then shown it is impossible to satisfy this condition with a finite stencil. This shows that results of Shishkin concerning non-uniform convergence at parabolic boundaries are also generic 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 sufficiently close. However, if the loci depart markedly, then the situation is rather more complicated. Under suitable conditions, we develop an explicit numerical lower bound on the attainable relative error in terms of the coefficients in the stencil characterising the scheme and the loci.