Discretization of the hypersingular boundary integral operator

Develop a fast and accurate quadrature rule to discretize the hypersingular operator \({\cal T}\) for three-dimensional acoustic scattering by smooth surfaces, extending the paper’s discretization framework for the weakly singular operators \({\cal S}\), \({\cal K}\), and \({\cal K}'\).

Background

The paper develops high-order quadrature schemes for the single-layer operator ${\cal S}$, the double-layer operator ${\cal K}$, and the adjoint double-layer operator ${\cal K}'$, whose kernels are weakly singular. The hypersingular operator ${\cal T}$ arises in the double-layer representation of the interior and exterior Neumann problems and is more difficult to discretize because its integral is interpreted in the Hadamard finite-part sense.

The authors explicitly leave the discretization of ${\cal T}$ unresolved. Completing this task would provide a corresponding high-order, FFT-accelerated boundary integral formulation for the hypersingular representation of Neumann scattering problems.