The Best Approximation of Plane-Parallel Potential Field on the Set of Fundamental Solutions

Abstract

The problem of constructing the element of the best approximation of the electric or magnetic plane-parallel field potentials on the set of parametrically fundamental solutions of the Laplace equation in a bounded domain is considered. It is assumed that at the boundary of the region the potentials satisfy the Dirichlet boundary condition or the mixed boundary condition. The best approximation element is sought in the form of a linear combination of fundamental solutions whose singular points are located outside the computational domain. The problem is reduced to finding the conditional minimum of the residual function characterizing the error of the potential approximation on the set of fundamental solutions. It is shown that the gradient descent method can be used to solve the optimization problem. The coordinates of special points of basic fundamental solutions and the coordinates of the linear combination formed by them are used as variable variables to minimize the residual. A numerical algorithm is proposed to find the element of the best approximation, the implementation of which does not require variation of the coordinates of special points. The results of the solution of the test problems are presented, which suggest the possibility of approximating the potential of the plane-parallel field on the set of fundamental solutions with high accuracy even with a small number of basis functions.