An efficient high-frequency boundary integral equation

In this paper, we start from the simple observation that scattering processes tend to become local at high-frequencies, as confirmed, for instance, by the successes of the geometrical theory of diffraction. This observation, suggests an efficient integral equation formulation leading to sparse and very well-conditioned matrices. The construction of this integral equation is carried out in two steps. The first step consists in defining a generalization of the well-known combined field integral equation. This generalization inherits the well-posedness, at all frequencies, from the classical one but its regularity is better and allows the use of iterative methods requiring very few iterations. The second step consists in exploiting the asymptotic localization property. We propose a perturbation of the first formulation to define an integral equation accounting for "hidden faces". This means, that a coefficient in the stiffness matrix, relating to basis functions whose supports are not "mutually visible" becomes the more negligible as the wave-number becomes large. Numerical experiments show, that this formulation leads to very well conditioned systems, which can be rapidly solved through iterative methods, maintaining a good accuracy when small coefficients are neglected even with rather high thresholds.