In this paper we discuss a fundamental problem in the numerical solution of boundary integral equations. In practice the only function space in which computations are performed is the space of square integrable functions whereas a consistent mathematical formulation of real physical problems based on conservation of energy often leads to more elaborate function spaces. For example, if the acoustic scattering problem is phrased in terms of a velocity potential, then the requirement of locally finite energy implies that the velocity potential and its derivatives both must be square integrable in bounded exterior domains. However, the traces or limiting values of such velocity potentials on the boundary of the scatterer are not only square integrable but have additional smoothness. In contrast, the normal derivative is not necessarily square integrable.
The appropriate function spaces are in fact Sobolev spaces of order + i andi, respectively. Nevertheless, boundary integral equations for either the velocity potential or its normal derivative are usually treated in the space of square integrable functions.
In this paper we present an analysis of both first and second kind integral equations of prototype Dirichlet and Neumann problems. In particular we will treat boundary integral equations for acoustic scattering from bounded impenetrable objects and the corresponding potential (zero frequency) equations.
Specifically, we will show how residual error may be employed to obtain useful error estimates without ever knowing the exact solution. The numerical approximation and residual error calculation will be based on a careful consideration of the appropriate function spaces and we show how the actual computations may be carried out. Moreover, we show that the condition number of the discretized system depends critically on the space in which the elements of the stiffness matrix are approximated. Simple examples will be given which show that systems which are poorly conditioned under one approximation may be well conditioned under another.
2. BOUNDARY INTEGRAL EQUATIONS
The boundary value problem which models the radiation or scattering of acoustic waves from bounded objects involves the solution of the Helmholtz equation, (V2 + k2)u = 0, exterior to the radiating or scattering object, where u may represent either velocity potential or excess pressure. The nature of the body determines the conditions on B, the boundary, and a number of well-known integral equations result from the use of these conditions together with Green's theorem (direct method) or an assumed layer representation (indirect method). With the free