A solution of Dirichlet problem using second partial derivatives of boundary function

TL;DR

This paper proposes an alternative boundary element method approach that uses second partial derivatives of the boundary function to solve Dirichlet problems more efficiently, avoiding boundary integral equation solutions.

Contribution

It introduces a novel method leveraging second derivatives of boundary functions to simplify and reduce computational costs in solving Laplace's equation with Dirichlet boundary conditions.

Findings

Applicable to arbitrary shaped domains

Reduces calculation cost significantly

Avoids solving boundary integral equations

Abstract

In Boundary Element Method, Green's function with no boundary conditions is used for solving Laplace's equation with Dirichlet boundary condition. To determine the gradient of solution on the boundary, we need to solve the boundary integral equation numerically in most practical cases. Here we discuss the alternative method to avert solving that boundary integral equation. It is based on the solution for Poisson's equation which has the singularity on the boundary specified by the boundary function and shown it is applicable to arbitrary shaped domain and reduces calculation cost considerably.