Superconvergent algorithms for the numerical solution of the Laplace equation in smooth axisymmetric domains

A fundamentally new–nonsaturable–method is constructed for the numerical solution of elliptic boundary value problems for the Laplace equation in ${{C}^{\infty}}$-smooth axisymmetric domains of fairly arbitrary shape. A distinctive feature of the method is that it has a zero leading error term. As a result, the method is automatically adjusted to any redundant (extraordinary) smoothness of the solutions to be found. The method enriches practice with a new computational tool capable of inheriting, in discretized form, both differential and spectral characteristics of the operator of the problem under study. This underlies the construction of a numerical solution of guaranteed quality (accuracy) if the elliptic problem under study has a sufficiently smooth (e.g., ${{C}^{\infty }}$-smooth) solution. The result obtained is of fundamental importance, since, in the case of ${{C}^{\infty }}$-smooth solutions, the solution is constructed with an absolutely sharp exponential error estimate. The sharpness of the estimate is caused by the fact that the Aleksandrov $m$-width of the compact set of ${{C}^{\infty }}$-smooth functions, which contains the exact solution of the problem, is asymptotically represented in the form of an exponential function decaying to zero (with growing integer parameter $m$).