The paper presents the results of the numerical study of the stress-strain state on the example of the detail “stand”, which is a bearing element of the pneumatic mechanism design. The numerical modelling is carried out on the basis of difference approximation of the solution of partial differential equations. Some aspects of the theory of difference schemes are considered and approaches to the study of the complex stress state of the real structure are proposed by means of the finite element method and the finite difference method, by means of which approximate solutions can be obtained. The description of the behaviour of the studied element in these methods is carried out in the mesh domain using a set of selected stress and displacement functions. The relations between them are written in one of three main types - stiffness and ductility equations or mixed relations between forces and displacements, satisfying the conditions of continuity of the described properties of the element. It should be noted that both methods require the construction and solution of systems of algebraic equations. Numerical algorithms of approximate solution of differential equations of elasticity theory are implemented in software packages Abaqus (FEM) and Midas MeshFree (meshless method) when modelling the stress-strain state of the detail “stand”. A comparative analysis and accuracy assessment of the results obtained are given.