A generalized Calderon Formula for open-arc diffraction problems: theoretical considerations

TL;DR

This paper introduces a new second-kind integral equation formulation for open-arc diffraction problems that remains well-conditioned as frequency increases, enabling more efficient numerical solutions.

Contribution

The authors develop a generalized Calderón formula for open-arc scattering problems, providing the first second-kind integral equations with spectra bounded away from zero and infinity.

Findings

Spectral properties show spectra are bounded away from zero and infinity as frequency increases.

Numerical experiments demonstrate improved convergence of Krylov-subspace solvers.

The approach applies to both Dirichlet and Neumann boundary conditions.

Abstract

We deal with the general problem of scattering by open-arcs in two-dimensional space. We show that this problem can be solved by means of certain second-kind integral equations of the form $\tilde{N} \tilde{S}[\varphi] = f$, where $\tilde{N}$ and $\tilde{S}$ are first-kind integral operators whose composition gives rise to a generalized Calder\'on formula of the form $\tilde{N} \tilde{S} = \tilde{J}_0^\tau + \tilde{K}$ in a {\em weighted, periodized} Sobolev space. The $\tilde{N} \tilde{S}$ formulation provides, for the first time, a second-kind integral equation for the open-arc scattering problem with Neumann boundary conditions. Numerical experiments show that, for both the Dirichlet and Neumann boundary conditions, our second-kind integral equations have spectra that are bounded away from zero and infinity as $k\to \infty$; to the authors' knowledge these are the first integral equations for these problems that possess this desirable property. Our proofs rely on three main elements: 1) Algebraic manipulations enabled by the presence of integral weights; 2) Use of the classical result of continuity of the Ces\`aro operator; and 3) Explicit characterization of the point spectrum of $\tilde{J}^\tau_0$, which, interestingly, can be decomposed into the union of a countable set and an open set, both tightly clustered around -1/4. As shown in a separate contribution, the new approach can be used to construct simple spectrally-accurate numerical solvers and, when used in conjunction with Krylov-subspace solvers such as GMRES, gives rise to dramatic reductions of Krylov-subspace iteration numbers vs. those required by other approaches.