Finite-element solutions of the initial-boundary value problem for a nonlinear parabolic equation in an inhomogeneous domain with the conjugation condition of a non-ideal contact were found. The initial boundary value problem is a mathematical model of an important technical problem of filtration consolidation of inhomogeneous soils. Inhomogeneity is considered in terms of the presence of thin inclusions, physicochemical characteristics of which differ from those of the main soil. The problem of longterm consolidation is especially pronounced in soils with low filtration coefficient. Low permeability of the porous medium causes deviation from the linear relationship between the pressure gradient and the filtration rate. Weak formulation of the problem is suggested, and the accuracy of the approximate finite element solution, its existence and uniqueness are substantiated for the case of Darcy's nonlinear law. A test example and the effect of the nonlinear filtration law for thin inclusion on the dynamics of scattering of excess pressures in the entire area of the problem are considered.

The boundary value problem for the system of quasi-linear parabolic equations in the presence of integral conjugation conditions is considered. The boundary value problem is a mathematical model of the process of non-isothermal filtration consolidation of the soil mass which contains a thin geobarrier. Geobarriers exposed to non-isothermal conditions are a component of waste storage facilities. The change of hydromechanical and thermal properties the geobarriers, as well as the phenomenon of thermal osmosis, require modification of both the equations in the mathematical model and the conjugation conditions. The finite element method is used to find approximate solutions of the corresponding system of quasi-linear parabolic equations. The existence and uniqueness of the approximate generalized solution is proved. The accuracy of finite element solutions in the sense of total approximation are also estimated. The differences in the values of pressure and temperature distributions for the classical case and the case considered in the article were analyzed on the test model example.