The purpose of this study is to improve the accuracy of the approximate solution of the problem of elasticity theory for incompressible media using the Richardson extrapolation method with respect to a parameter. The Richardson extrapolation method consists in increasing the order of accuracy with respect to the parameter of the approximate solution. Difference schemes for static, dynamic problems of the theory of elasticity for a compressible medium are well studied and investigated and various economical numerical algorithms have been tested for these difference schemes. To solve the problems of an elastic incompressible medium, the solution of problem of an elastic compressible medium is used as the Lame coefficient tends to infinity, since earlier in the works of the authors of this study  asymptotic estimates of the proximity of these solutions were obtained, not only at the differential level, but also at the difference level.