Boundary Integral Solutions of Laplace's Equation

Abstract

Although Laplace's equation is simple, the region over which it is to be solved is often complicated. Both the shape of the region and the boundary conditions can induce solutions Φ which are singular at isolated points on the boundary of the region. Boundary integral equation methods are well-suited to the problem, reducing a two-dimensional partial differential equation to a one-dimensional integral equation. Unfortunately, the standard boundary integral equation methods lead to an ill-conditioned set of linear equations, restricting the achievable accuracy in the approximate solution. This paper describes an improved boundary integral method. A new integral equation is derived. Laplace's equation is reduced to solving two coupled, one-dimensional integral equations. The resulting linear equations are well-conditioned. A program package for solving Laplace's equation has been developed. The package solves Laplace's equation in two dimensions or in three dimensions with axial symmetry. The region may extend to infinity, and may be multiply-connected. In addition to smooth basis functions, the program automatically includes appropriate singular basis functions, greatly improving the achievable accuracy for regions with corners.