An unsaturated numerical method for the exterior axisymmetric Neumann problem for Laplace's equation

Basing on the fundamental ideas of Babenko, we construct a fundamentally new, unsaturated, numerical method for solving the axially symmetric exterior Neumann problem for Laplace's equation. The distinctive feature of this method is the absence of the principal error term enabling us to automatically adjust to every class of smoothness of solutions natural in the problem.
This result is fundamental since in the case of $C^\infty$-smooth solutions the method, up to a slowly increasing factor, realizes an absolutely unimprovable exponential error estimate. The reason is the asymptotics of the Aleksandroff widths of the compact set of $C^\infty$-smooth functions containing the exact solution to the problem. This asymptotics also has the form of an exponential function decaying to zero.