We construct the finite-dimensional approximations for mixed variational inequalities with pseudomonotone operators and convex non-differentiable functionals in Banach spaces. Such variational inequalities arise in the mathematical description of the processes of an established filtration and in the problems of determining the equilibrium of soft shells. The convergence of these approximations are investigated.

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This work is devoted to the study of the convergence of an implicit difference scheme for a one-dimensional initial-boundary problem that simulates the process of filtration consolidation with a limiting gradient. From a mathematical point of view, this model is a system of partial differential equations for the displacements of an elastic medium and fluid pressure. In addition, the equation for pressure is degenerate, with nonlinearity in the spatial operator, which generates a non-smooth solution. In this regard, the study of the convergence was carried out under minimal conditions on the smoothness of the initial data. It was based on obtaining a number of a priori estimates that allow, using the monotonicity method, to establish the convergence of piecewise constant completions of the difference solution to a generalized solution of the problem. The spatial operator was approximated using the method of summation identities.