Solving Elliptic Equations by the Generalized Point-Sources Method

Abstract

The aim of this work is to develop a new universal numerical method for solving boundary value problems for linear elliptic equations. This method is based on the transformation of the original mathematical physics equation to a simpler inhomogeneous equation with the known fundamental solution. From this equation the transition is carried out to an inhomogeneous integral equation with a kernel, expressed by the known fundamental solution. Obtained integral equation together with boundary conditions is solved numerically. The result is obtained approximate solution, the field potential, in an analytical form. This allows not only to find approximate value of the field potential at any point in the solutions domain, but also to differentiate this potential and without significant loss of accuracy. This property of the worked out numerical method distinguishes it from traditional numerical methods for solving boundary value problems, such as the finite element method. To confirm the effectiveness of the proposed numerical method the two-dimensional and three-dimensional boundary value problems with the pre known solutions were resolved. The dependences of the numerical solutions error on the number of linear equations in the resulting system were obtained. It is shown that even at a small number of equations in the system, a few hundred, the solutions accuracy being achieved within hundredths of a percent. The results obtained in this work show that a physical field described by any linear elliptic equation, virtually, can be represented as a superposition of point sources fields, satisfying a simpler equation, the solution of which is by the method of point sources of the field. Therefore the numerical method, presented in this paper, can be considered as the generalized point sources method, which allows to greatly extend the scope of the traditional the point sources method in solving applications for modeling fields of different physical nature in various types technical devices.