Unsaturated Algorithms for the Numerical Solution of Elliptic Boundary Value Problems in Smooth Axisymmetric Domains

Abstract

A fundamentally new, unsaturated, method is constructed for the numerical solution of the Laplace equation in a smooth axisymmetric domain of rather general shape. An essential feature of this method is lack of the leading error term $O(m^{-r})$, where $r$ is a fixed integer with $r>2$. As a result, the method automatically adjusts to the excess (extraordinary) smoothness of solutions of problems. The method provides us with a new computational tool whose discretization inherits both differential and spectral characteristics of the operator of the elliptic problem under consideration. This allows us to take efficiently into account the fact that the domain is axisymmetric which is a stumbling block for numerical methods with the leading error term. Our result is of a fundamental interest because, for $C^{\,\infty }$-smooth solutions, computer constructs a numerical solution (up to a slowly growing multiplier) with an absolutely sharp exponential error estimate. The sharpness is caused by the asymptotics of the Aleksandrov $m$-width of the compact set of $C^{\,\infty }$-smooth functions that contains the exact solution of the problem. This asymptotics is presented by a function that exponentially decays as the integer parameter $m$ grows.