Advanced Numerical and Semi-Analytical Methods for Differential Equations

8Boundary Element Method

8.1 Introduction

In 1977, for the first time the term boundary element method (BEM) was coined in some publications and the first book on the BEM appeared in 1992 [1]. In the last 20 years the BEM has become one of the important numerical computational techniques along with the FDM (Chapter 5) and FEM (Chapter 6). It is used to solve certain classes of differential equations by formulating as integral equations 2,3]. It is worth mentioning that for certain classes of differential equations, integral equations' reformulation may not always be possible [ 3 ,4]. The advantage in using the BEM is that only the boundaries of the considered domain of the differential equations require subdivision to develop a boundary mesh [5,6]. Whereas, in other methods like FDM or FEM the whole domain of the differential equation requires discretization. In this chapter, we are presenting a brief introduction of the BEM along with a simple example problem for easy understanding of the method.

8.2 Boundary Representation and Background Theory of BEM

Detailed BEM formulation for handling boundary value problem (BVP) in a field or domain may be found in Refs. [ 6 ,7]. By using known data on the boundary S, one can get the rest of information on the boundary S as well as in the domain D in the BEM. Due to various difficulties [ 7 ,8], BVPs do not always have closed‐form solutions or analytic solutions. In general, this may be done by the facility of defining the boundary as ...