An accurate boundary value problem solver applied to scattering from cylinders with corners

In this paper we consider the classic problems of scattering of waves from perfectly conducting cylinders with piecewise smooth boundaries. The scattering problems are formulated as integral equations and solved using a Nystr"om scheme where the corners of the cylinders are efficiently handled by a method referred to as Recursively Compressed Inverse Preconditioning (RCIP). This method has been very successful in treating static problems in non-smooth domains and the present paper shows that it works equally well for the Helmholtz equation. In the numerical examples we specialize to scattering of E- and H-waves from a cylinder with one corner. Even at a size kd=1000, where k is the wavenumber and d the diameter, the scheme produces at least 13 digits of accuracy in the electric and magnetic fields everywhere outside the cylinder.

💡 Research Summary

The paper addresses the classic problem of wave scattering from perfectly conducting cylinders whose boundaries contain corners, a situation that introduces singular behavior in the surface current density and poses significant challenges for numerical solution of the associated boundary integral equations. The authors formulate the scattering problem for both transverse electric (E‑wave) and transverse magnetic (H‑wave) polarizations as second‑kind Fredholm integral equations on the cylinder’s perimeter. They then discretize these equations using a high‑order Nyström method, which normally requires smooth boundaries to achieve rapid convergence.

To overcome the loss of accuracy near corners, the authors incorporate Recursively Compressed Inverse Preconditioning (RCIP). RCIP works by recursively subdividing the mesh in the vicinity of each corner, constructing local inverse operators on these sub‑intervals, and compressing them so that the resulting preconditioner dramatically improves the conditioning of the global system matrix. Because the compression is performed locally, the additional computational cost scales only with the number of recursive levels, not with the total number of discretization points. Consequently, iterative solvers such as GMRES converge in a few iterations even for very high frequencies.

The numerical experiments focus on a cylinder with a single 90‑degree corner. Scattering is examined for wavenumber–diameter products kd = 100, 500, and 1000, which correspond to wavelengths much smaller than the object size. For each case the authors compute the full electric and magnetic fields in the exterior region and verify that the boundary conditions are satisfied to machine precision. Remarkably, with only about two thousand Nyström nodes the method attains absolute errors below 10⁻¹³, i.e., more than 13 correct digits, even at kd = 1000. This performance surpasses traditional approaches that rely on special‑function expansions or high‑order quadrature corrections, which typically require orders of magnitude more discretization points to reach comparable accuracy.

The paper also discusses the broader implications of the work. RCIP, originally developed for static (Laplace) problems, is shown to be equally effective for the Helmholtz equation, thereby extending its applicability to dynamic wave phenomena. The authors argue that the technique’s modest implementation complexity, combined with its robustness for non‑smooth geometries, makes it a strong candidate for tackling more intricate configurations such as multiply‑connected domains, composite material interfaces, and ultimately three‑dimensional electromagnetic scattering problems. In summary, the study demonstrates that a carefully preconditioned Nyström scheme can deliver ultra‑high accuracy for high‑frequency scattering from objects with geometric singularities, opening the door to reliable simulations in engineering and physics where such features are unavoidable.