On the numerical evaluation of the singular integrals of scattering theory

In a previous work, the authors introduced a scheme for the numerical evaluation of the singular integrals which arise in the discretization of certain weakly singular integral operators of acoustic and electromagnetic scattering. That scheme is designed to achieve high-order algebraic convergence and high-accuracy when applied to operators given on smoothly parameterized surfaces. This paper generalizes the approach to a wider class of integral operators including many defined via the Cauchy principal value. Operators of this type frequently occur in the course of solving scattering problems involving boundary conditions on tangential derivatives. The resulting scheme achieves high-order algebraic convergence and approximately 12 digits of accuracy. One of the principal observations of integral operator theory is that certain linear elliptic boundary value problems can be reformulated as systems of integral equations whose constituent operators act on spaces of square integrable functions [7, 20]. This observation plays a particularly important role in scattering theory, where such reformulations are standard [14, 11, 15, 13, 12, 7, 8]. Not surprisingly, it also figures prominently in the numerical treatment of scattering problems [16, 17, 1, 10]. But while the integral equation approach to scattering theory is a venerable and well-developed subject, the corresponding numerical analysis — that is, the study of the integral equations of scattering theory using computers and finite precision arithmetic — is rather newer and considerably less developed. As a result, many fundamental problems in numerical scattering theory are as yet unresolved. Examples of this phenomenon can be found in recent contributions like [9] and [2], which offer new integral formulations of certain boundary value problems for Maxwell’s equations that, unlike classical formulations, are amenable to numerical treatment. This article concerns another unresolved problem: the evaluation of the singular integrals of scattering theory. A key difficulty in the discretization of those integral operators which arise from the reformulation of linear elliptic boundary value problems is the efficient and accurate evaluation of integrals of the form