Approximate solution for 3D Dirichlet problem in a doubly connected arbitrary finite solid with smooth surface

Abstract

© Published under licence by IOP Publishing Ltd. In this paper we present the spline interpolation method for solving the 3D Laplace equation with Dirichlet boundary conditions for doubly connected solids with smooth surfaces. The solid is divided into N layers and the spline solution construction at each layer for the 3D problem is reduced to the solution of a sequence of 2D Dirichlet problems. The 2D problem solution in each layer is restored via its boundary value with the help of Cauchy integral method. The Cauchy integral method is a boundary element method which reduces the Dirichlet problem to the Fredholm integral equation of the second type. The final spline solution of the 3D problem is continuous with respect to the three variables. Numerical examples are given to verify the efficiency of the method.