On a supreme solutions of difference equations enclosing the Laplace operator

The rapid IT development facilitated the math simulation application at various branches of human activity. In present, the huge amount of specialists are dealing with math simulations, that using well known and substantiated numerical algorithms. There were proved some equivalence theorems [1-4] about coupling of numerical solutions convergency with the approximation and stability. So was the theorem of the solution stability conditions [5]. One must remember, however, that all was proved under linearity equations and uniform meshes assumptions. While for nonlinear equations and nonuniform adaptive meshes the approximation errors may be nonconvergency and the supreme solution may differ from exact solution at unlimit growth of number mesh points N. The appearance of nonconverging approximation for the equation of thermal conduction by finite-difference equation at nonuniform meshes was pointed early at [1, 2]. The violation of meshes uniforming leads to entropic trace appearencing at liquid and gas mechanics [6, 7]. In the paper there is treating the problem of difference between exact and supreme solutions of nonlinear equation of thermal conduction or electrostatic equation in the case of strong difference between size of neighbour meshes cells.