Difference scheme of the second order of accuracy for dirichlet problem in arbitrary area

Difference schemes for two-dimensional Poisson equation in arbitrary domain on standard templates are considered. Tbese schemes also have the second order of local approximation in the nodes near the boundary. The monotonicity of these schemes are proved for a wide class of areas by means of a principle of maximum. Stability of the schemes is proved in the grid norm $W_2^1$ in arbitrary computational domain by the method of energy inequalities.