Converging difference schemes of an elliptic equation in a class of summable functions with a network-like carrier

The paper presents the principles of constructing converging difference schemes and the corresponding finite difference method for the analysis of an elliptic equation in a class of summable functions with a network-like carrier. At the same time, the existence of a weak solution of the boundary value problem is not assumed, its weak solvability will be established using the finite difference method. The study is essentially based on some properties of special Sobolev spaces with a finite energy norm. Properties have a general meaning that is not related to the features of a particular boundary value problem and, therefore, determine the general method of analyzing the solution of these problems. Limit transitions are carried out uniformly in all problems: if a uniformly bounded in the norm of a special space is established for approximations of the solution, then the scheme is considered stable in the sense of the metric of this space, and thus the way is opened to obtain sufficient conditions for the weak convergence of the family of approximations, the limiting function is a weak solution to the problem under consideration, and it is possible to establish the conditions of strong convergence. The study is carried out on the example of the Dirichlet problem for an elliptic equation, but not the differential equation itself, but the corresponding integral identity is used, and an approximation of the latter is sought. The described idea can be realized for equations of parabolic and hyperbolic type, although its implementation is somewhat more complicated, however, it is of great help that approximations of their elliptic parts have been established. The proposed finite difference method is applicable with slight modifications in the case of parabolic and other problems for vector functions. An example of the latter is the linearized Navier — Stokes system, widely used in the description of network-like hydrodynamic processes, considered in Sobolev spaces, the elements of which are vector functions with carriers on $n$-dimensional network-like domains ($n>1$). The results obtained can be used in numerical solution of problems of optimal control of thermal and wave processes in structural elements made of composite materials — composites.