Combined Adaptive Meshfree Method for Modeling Potential Physical Fields

Abstract

A procedure for numerical solution of the Dirichlet problem for the Laplace equation, which is often re-duced to modeling potential physical fields in homogeneous media, is described. The approximate solution is proposed to be found using a combined meshfree Monte Carlo method and fundamental solutions, which is implemented in two stages. At the first stage, the element of the best approximation in the linear shell of the fundamental solutions of the Laplace equation is determined. At the second stage, the solution is refined using the potential values found by the Monte Carlo method at individual points in the computational domain. Algorithms are given for finding the defining parameters of both methods used to reduce the error. The procedure for evaluating the accuracy of the found approximate solution of the problem is described. An example of calculating the potential distribution in the angular zone under specified boundary conditions using the combined meshfree method is given. The accuracy of the approximate solution is estimated by comparing it with the exact solution. It is shown that the use of the meshfree method leads to a decrease in the error without a significant increase in the computational resources required for its implementation