Approximation of solutions to operator equations in spaces of smooth functions and in spaces of distributions

The Galerkin method, in particular, the Galerkin method with finite elements (called finite element method) is widely used for numerical solution of differential equations. The Galerkin method allows us to obtain approximations of weak solutions only. However, there arises in applications a rich variety of problems where approximations of smooth solutions and solutions in the sense of distributions have to be found. This article is devoted to the employment of the Petrov–Galerkin method for solving such problems. The article contains general results on the Petrov–Galerkin approximations of solutions to linear and nonlinear operator equations. The problem on construction of the subspaces, which ensure the convergence of the approximations, is investigated. We apply the general results to two‐dimensional (2D) and 3D problems of the elasticity, to a parabolic problem, and to a nonlinear problem of the plasticity. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 406–450, 2014