This paper considers numerical methods for the solution of Poisson's equation in an arbitrary two‐dimensional region with given boundary values. The methods are similar to those discussed by Shortley and Weller for Laplace's equation. It is shown that the entire theory of networks may be applied to Poisson's equation without change. Formulas for the treatment of irregular points and for blocks of points are developed. In addition to the mathematical considerations, the application of the methods to the case of torsion in shafts of any cross section is described. Additional formulas are derived for the determination of the maximum shear at any point in a shaft as well as the torsional stiffness of such a shaft.