Petrov-Galerkin Method for Approximation of Solutions to Operator equations in Positive and Negative Banach Spaces

The Galerkin method, in particular, the Galerkin method with finite elements (called finite element method) are widely used in numerical solving of differential equations. The Galerkin method allows us to obtain approximations of weak solutions only. However, a rich variety of problems arise in applications, where approximations of smooth solutions and solutions in negative spaces have to be found. This paper is devoted to the employment of the Petrov-Galerkin method for solving such problems. General results on convergence of the Petrov-Galerkin approximations of solutions to operator equations are obtained. The problem on construction of the subspaces, which ensure the convergence of the approximations, is investigated. By way of example, we consider two--and--three-dimensional problems of the elasticity, a parabolic problem, and a nonlinear problem of the plasticity