Improved finite difference method for equilibrium problems based on differentiation of the partial differential equations and the boundary conditions

A numerical algorithm for producing high‐order solutions for equilibrium problems is presented. The approximated solutions are improved by differentiating both the governing partial differential equations and their boundary conditions. The advantages of the proposed method over standard finite difference methods are: the possibility of using arbitrary meshes; the possibility of using simultaneously approximations with different (distinct) orders of accuracy at different locations in the problem domain; an improvement in approximating the boundary conditions; the elimination of the need for ‘fictitious’ or ‘external’ nodal points in treating the boundary conditions. Furthermore, the proposed method is capable of reaching approximate solutions which are more accurate than other finite difference methods, when the same number of nodal points participate in the local scheme. A computer program was written for solving two‐dimensional problems in elasticity. The solutions of a few examples clearly illustrate these advantages.