On the significance of Galerkin method for the analysis of differential equations

The approach to the construction of approximate solutions of differential equations, suggested in Bubnov and Galerkin, and subsequently developed by Petrov, had a great influence on the development of the theory of partial differential equations derivatives and spawned many methods of quantitative analysis. Galerkin's method stimulated the creation of the concept of generalized solutions and was used to prove the existence of solution of some mathematical physics problems. Modern methods of computational mathematics are widely uses the idea of the Galerkin’s method, which in one or another form is the basis of finite element method, finite volume method, discontinuous Galerkin’s method, dual mixed method and others. The major theoretical problems related with these and other similar methods are proof of convergence to the exact solution and obtaining error estimates. The report provides an overview of the main achievements in this area and a discussing on new unresolved issues related to the quantitative analysis of partial differential equations.