Error estimates of Galerkin method in studying the dynamics of concrete slab

When calculating the strength of structural elements, one of the steps is to study the dynamics of these elements under various force loads. In this paper, based on the classical model of free vibrations of an elastic plate, in contrast to previous numerical and analytical studies, an analytical method for studying the dynamics of a concrete slab pinned at its edges is developed. According to the Galerkin method, an approximate solution of the partial differential equation used in the model is found as a linear combination of basis functions. This results in a system of ordinary differential equations for determining the coefficients of this combination. Based on the construction of a Lyapunov-type functional for the partial differential equation and on a Lyapunov function for the system of ordinary differential equations, several methods for determining the error of obtained approximate solution are proposed. Numerical calculations demonstrate the accuracy of error estimates. For this purpose, plots of the difference between the approximation under study and the higher-order approximation are constructed. The best estimate was shown by the method of error determination using the following basis function, whose coefficient was found from the equation obtained in study of the Lyapunov-type functional for the original partial differential equation.