Unsaturated algorithms for the numerical solution of elliptic boundary value problems in smooth axisymmetric domains

A fundamentally new —unsaturated — method for the numerical solution of the Laplace equation in smooth axisymmetric domains of sufficiently arbitrary shape is constructed based on the boundary integral equation. A distinctive feature of the method is — the absence of the principal value of error $O(m^{-r})$ ($r>2$ — fixed integer), and as a result — the ability to automatically adjust to any excess (extraordinary) reserves of smoothness of the sought solutions to problems. The method endows computer practice with a new computing tool capable in a discretized form to inherit both differential and spectral characteristics of the operator of the elliptic problem under study. This makes it possible to effectively take into account the axisymmetric specifics of the problem, which is a “stumbling block” for any numerical methods with a major error term. The result obtained is of fundamental interest, because in the case of $C^{ \infty}$-smooth solutions, a computer numerical answer is constructed (up to a slowly growing multiplier) with an absolutely unimproved exponential error estimate. The unimprovability of the estimate is due to the asymptotics of the Alexandrovsky $m$-diameter of the compact $C^{ \infty}$-smooth functions containing the exact solution of the problem. This asymptotic also has the form of exponential decreasing to zero (with the growth of the whole parameter $m$).