Numerical Modeling

**Chapter 1**

**Abstract**

**1. Introduction**

coupled [2].

**3**

A Pilot Fortran Software Library

The boundary element method (BEM) is developed from the standpoint of software design. The Fortran language is used to produce a structured library for solving Laplace's equation in various domain topologies and dimensions with generalised boundary conditions. Subroutines that compute the discrete Laplace operators, which are the core components for populating the matrices in the BEM, are developed. The main subroutines for solving Laplace's equation in 2D, 3D and axisymmetric cases for open and closed boundaries are introduced. The methods

The boundary element method (BEM) has established itself as an important numerical technique for solving partial differential equations (PDEs) over the last half century [1, 2]. It distinguishes itself from competing methods, such as the finite element method (FEM) [3] in that the latter method requires a mesh of the domain, whereas the BEM only requires a mesh of the boundary (of the domain). The BEM is not as widely applicable as the FEM, particularly in that it is much more of a struggle to apply the BEM to non-linear problems. However, for problems to which the boundary element method is viable, the advantage of only requiring a boundary mesh is a significant one; the BEM is likely to be more efficient but also the relative simplicity of meshing, and the method is easier to use and is more accessible. This advantage is more notable for exterior problems; the domain is infinite, and 'domain methods'such as the FEM require special treatment, but for the BEM, only a (finite) boundary mesh is required. Computational methods may be combined or

The boundary element method is derived through the discretisation of an integral equation that is mathematically equivalent to the original partial differential equation. The essential reformulation of the PDE that underlies the BEM consists of an integral equation that is defined on the boundary of the domain and an integral that relates the boundary solution to the solution at points in the domain. The former is termed a boundary integral equation (BIE), and the BEM is often referred

**Keywords:** boundary element method, Laplace's equation, Fortran

s

for the Solution of Laplace'

Equation by the Boundary

Element Method

are demonstrated on test problems.

*Stephen Kirkup and Javad Yazdani*
