Limited operator semigroups and issues of the convergence of the Bubnov–Galerkin method for one class of shallow shells nonlinear equations

This paper discusses issues related to the rate of convergence of the Bubnov–Galerkin method in numerical calculation of stress-strain state of geometrically nonlinear shells in the dynamic case. To address these issues involved the unit strongly continuous semigroups of limited operators. Methods of functional semigroups of operators was applied effectively in the theory of boundary value problems since the 60s XX-th century. It should be noted author E. Hill, R. Phillips, S. G. Krein, S. Mizohata and others. So, using the methods of strongly continuous semigroups of operators S. G. Krein proved a new theorem on the existence and uniqueness of solutions of linear equations of mechanics in late 60s. In 2000, V. N. Kuznetsov and T. A. Kuznetsova first used the methods limited semigroups of operators to solution of linear equations of shallow shells, which solved the problem of smoothness of solutions of linear systems of equations of shells. At the same time V. N. Kuznetsov and T. A. Kuznetsova have developed a method called a linear approximation in separated parameters, which allow to solve the problem of smoothness of solutions of nonlinear equations of the theory of plates and shells. This made it possible to determine the speed of convergence of the Bubnov–Galerkin method the numerical solution of nonlinear boundary value problems for the geometrically nonlinear shells in the area of sustainability in the parameters.
In this paper, we complete the proof of the result of the rate of convergence of the Bubnov–Galerkin method in the case of an arbitrary configuration shell borders.