A study of an implicit difference scheme for the problem of saturated-unsaturated filtration consolidation

An implicit difference scheme for the problem of saturated-unsaturated filtration consolidation under condition of semi-permeability on part of the boundary is constructed and investigated. Using the penalty method the existence of the solution to the difference problem is established. A study of the implicit convergence of the difference scheme is carried out under minimal propositions on the smoothness of the original conditions: the convergence of the piecewise-constant filling of the difference solution to the generalized solution of the problem under consideration is proved.