Numerical investigation of a model problem for the Poisson equation with inequality constraints in a domain with a cut

A model problem is considered for the Poisson equation in a two-dimensional domain with a cut. The Dirichlet and Neumann conditions are imposed on the exterior boundary of the domain together with the nonnegativity condition for the jump across the edges of the cut. In addition, the absolute value of the gradient inside the domain must be bounded by some constant. The boundary value problem turns into a variational problem, and the unknown function must yield the minimum of the energy functional on some convex set. After discretization of the problem by the finite element method, an Uzawa-type algorithm is used to find a solution. Some examples are included of solving the discrete problem.