The boundary element method is derived through the discretisation of an integral equation that is mathematically equivalent to the original partial differential equation - in the case of this manual, the Helmholtz equation. The essential re-formulation 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 to as the boundary integral equation method or boundary integral method. Over the last twenty years the term boundary element method has become more popular. The other terms are still used in the literature however, particularly when authors wish to refer to the overall derivation and analysis of the methods, rather than their implementation or application.
The advantages in the boundary element method arises from the fact that only the boundary (or boundaries) of the domain of the PDE requires sub-division. (In the finite element method or finite difference method the whole domain of the PDE requires discretisation.) Thus the dimension of the problem is effectively reduced by one, for example an equation governing a three-dimensional region is transformed into one over its surface. In cases where the domain is exterior to the boundary, as it is in exterior problems, the extent of the domain is infinite and hence the advantages of the BEM are even more striking; the equation governing the infinite domain is reduced to an equation over the (finite) boundary.
The application of the BEM in acoustics - considered separately on this website p- involves the solution of the Helmholtz equation. The section on acoustics on this website has been created separately for a number of reasons. One reason is that acoustics is a separate area of physics, mathematics and engineering with its own particular culture . Another reason is that in acoustics, the wavenumber is always real (or clode to being so) and in separating off a generalised Helmholtz problem, we are also allowing for the wavenumber to be complex.