Adaptation of the finite element method for the Stieltjes string deformation problem with a nonlinear condition

We study a problem modeling small deformations of a string with features localized in an arbitrary number of points (but not more than a countable number) in the form of elastic supports and concentrated forces. It is assumed that the left end of the string is rigidly fixed and the right end is inside a vertical displacement limiter. Depending on the applied external force, the right end will either remain free or reach the boundary of the limiter. This generates a nonlinear condition at the corresponding point, since the behavior of the solution is not known in advance. The problem under study is described in the form of a variational inequality; the existence and uniqueness theorems of the solution are proved; an algorithm for finding an approximate solution is developed by adapting the finite element method; and an estimate of the deviation of the exact solution from the approximate solution is obtained.