ABSTRACT
The method of improving the convergence for trigonometric series due to Academician A. N. Krylov is well known ([1], see also [2]). This method consists in splitting the given slowly converging series into two series, one of which also converges slowly but can be explicitly summed, while the second one converges rapidly. Here we show that the same principle, selection of an elementary part which contains a singularity, is a general approach to improving convergence for all methods of approximate solution to boundary value problems and, as we shall see below, the improvement of convergence of trigonometric series is a rather particular case of the application of this method. (Some other applications of this method are given in my paper [3].)
So we consider the following boundary value problem: find the solution of the equation of elliptic type
(for example, the Poisson equation Au = f ( x , y ) ) under the boundary con dition
u(x, y) — cf)(s) on the contour L. To be definite we confine ourselves here to the Dirichlet problem and
second order equations. The domain bounded by the contour L will be denoted by D.
