ABOUT PRACTICAL APPLICATIONS FOR METHODS OF POINT SOURCES

Abstract

The method of point sources (or the method of fundamental solutions in the Englishlanguage literature), which was introduced in 1963 by Georgian mathematicians M.A. Aleksidze and V.D. Kupradze, in typical use, has a solution of various types of boundary problems for differential equations for which fundamental solutions are known. During the lifetime of the method, its convergence was theoretically proved under a series of restrictions in domains of arbitrary geometric shape. However, a review of literary sources shows that the practical application of the method was carried out by its researchers exclusively in problems with domains of simple geometric shape. This is mainly due to the need to have an analytical solution to the problem for assessing the accuracy of various modifications of the point source method. This paper compares the computational capabilities of the point source method and the finite element method. The domain of simulation of the stationary temperature field in the first problem is the traditional rectangle. In the second problem, the rectangle on two opposite sides has cuts of different geometric shapes. The boundary conditions are kept the same. Convective heat exchange is observed at first boundary of the domain, the temperature of the second boundary is kept constant, and adiabaticity conditions are satisfied at the other boundaries of the domain. In solving the basic system of linear algebraic equations in the method of point sources, regularization according to A.N. Tikhonov and the analysis of the differential properties of the L-curve to find the value of the regularization parameter are used. Computational experiments have shown that even for a rectangular domain, the condition number of this system of equations varies non-monotonously depending on the radius of the circle on which fictitious sources are placed. It is shown that the point source method is significantly inferior in accuracy to the finite element method in domains of complex geometric shape. The continuation of research is connected with the construction of approximate conformal mappings of a given domain onto domain which is bounded by the contour of fictitious sources.