1 Introduction

The boundary element method (BEM) has been developed over recent decades as an alternative to more traditional methods, such as finite differences or finite elements, for solving partial differential equations. Although the BEM is not as widely applicable as the other methods, it has the important advantage that only the boundary requires discretization rather than the full domain. It is particularly attractive for exterior problems where the domain extends to infinity; the two other methods mentioned are clearly difficult to apply in such cases. For further details on the method see references [1], [2], [3].

In this paper the processes which give rise to errors in the implementation of the BEM are considered, and the results of the numerical experiments are given. The setting for the experiments is the boundary S of a region D together with the surrounding region E. In general the BEM is derived via a method which involves introducing an approximation to the boundary, [S'], say (and consequently gives rise to and approximate exterior [E'] and and approximate interior [D']), and an approximation to the boundary functions.

The derivation of the BEM from the boundary integral equation (BIE) formulation of the partial differential equation is well established [1], and is briefly reviewed later in this paper. The implementation of the BEM may be divided into two stages:

The Primary Stage - the computation of the unknown boundary function, that is the potential j(p) in the direct approach (to Neumann problems) and the source density function in the indirect approach, s(p).

The Secondary Stage - the computation of the potential (usually for several values of p) in the problem domain.

There are four discernible sources of error in the approximation to the boundary function obtained as a result of carrying out the primary stage of the BEM, they are the

(Ia) boundary approximation error,

(Ib) approximation of the boundary functions,

(Ic) numerical integration error, and

(Id) error in the solution of the linear system of equations.

The secondary stage of the BEM will also incur an error, some part of which will be inherited from the primary stage;

(IIa) boundary approximation error,

(IIb) approximation of the boundary functions, and

(IIc) numerical integration error.

The source of error (Id) can usually be disregarded if the linear systems of equations that arise in the method are well-conditioned (as they are in the examples considered) and a Gaussian elimination type method is used for their solution. In some cases a significant error may arise if an iterative method is used. In this paper the numerical integrations have been carried out to sufficient accuracy in order to ensure that their contribution to the error, (Ic) and (IIc) is negligible in comparison with the overall error.

In section 2 the general Laplace problem which is being considered, and its integral formulations, are stated. In section 3 the derivation of the BEM via collocation is reviewed and the approximations used to represent the geometry and the boundary functions are listed. Details of the test examples, which involve a square and a circle, are given in section 4, followed by results and conclusions in section 5 and 6.