Nanoindentation is a widely used method to control and determine the elastic properties of materials. For interpretation of the nanoindentation test results, one should solve the problem of constructing an adequate mathematical model. In most cases, analytical formulas obtained using a mathematical model of the foundation in the form of an elastic linearly deformable half-space are used. When formulating the problem, it is assumed that a non-deformable punch is indented in a homogeneous elastic half-space. The numerical formulation of the problem makes it possible to obtain and use a numerical solution obtained taking into account the complete plastic nonlinear behavior of the material. In the present work, we study contact problems on the indentation of a spherical punch into an elastoplastic homogeneous half-space. To verify the numerical solution, a comparison with known analytical solutions was carried out. Issues of convergence and tuning of numerical methods, the influence of plasticity and the applicability of analytical solutions are explored. Problems are solved numerically using the finite element method in the Ansys Mechanical software package.