Homogeneous difference schemes on non-uniform nets☆

Abstract Non-uniform nets are widely used in the solution of various differential equations by difference methods. However, little has been done to study the convergence of the difference schemes. The simplest examples show that the most frequently used criteria for judging the quality of difference schemes, a uniform estimate or a mean estimate of the approximation error of the scheme, are unsound for non-uniform nets and can give an untrue idea of the order of accuracy of the scheme.