The aim is to study the efficiency of numerical models of elastic stress fields in deformed solids. The field point-source method (PSM) designated as the method of fundamental solutions (MFS) in the foreign literature is used when creating these models. The PSM system construction under simulating fields of different physical nature is described. We introduced the concept of a point-source elastic displacement field in the deformed solid. The research is resulted in the developed PSM equations system that can be used for solving various problems in the elasticity theory including the classical first and second boundary value problems solution in the elasticity theory (when either voltage or bias is specified at the boundary), as well as a mixed boundary problem (when displacement is given on one part of the boundary, and voltage - on the other). The properties of PSM in solving standard problems and the Dirichlet problem for a circular domain are studied. The dependences of the numerical solution error on the problem parameters, in particular, on the number of charges that simulate the desired field, on the remoteness of the charges from the boundaries of the solution domain are found. Based on these results, it is concluded that in the numerical solution of the elasticity problems, PSM error decreases with the growth of the number of charges exponentially. This numerical solution property allows in certain cases obtaining the extremely accurate for computing solution with a relative error of the order of 10-15 that implies the PSM application perspectiveness under the numerical solution of elasticity problems.