Spline-interpolation solution of 3D Dirichlet problem for one class of solids

TL;DR

This paper introduces a spline-interpolation method for solving the 3D Dirichlet problem in bodies of revolution, cones, and cylinders by reducing it to 2D problems, ensuring boundary continuity even with linear splines.

Contribution

The paper presents a novel spline-interpolation approach that simplifies 3D Dirichlet problems by reducing them to 2D problems, maintaining boundary continuity with linear splines.

Findings

Method effectively reduces 3D problems to 2D problems.

Ensures boundary continuity with linear splines.

Applicable to bodies of revolution, cones, and cylinders.

Abstract

We present the spline-interpolation approximate solution of the Dirichlet problem for the Laplace equation in the bodies of revolution, cones and cylinders. Our method is based on reduction of the 3D problem to the sequence of 2D Dirichlet problems. The main advantage of the spline-interpolation solution of the 3D Dirichlet problem is its continuity in the whole domain up to the boundary even for the case of the linear spline.