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

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 ÑS[φ] = ƒ, where Ñ and S are first-kind integral operators whose composition gives rise to a generalized Calderón formula of the form ÑS = J^τ_0 + K in a weighted, periodized Sobolev space. (Here J^τ_0 is a continuous and continuously invertible operator and K is a compact operator.) The ÑS 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 the authors' knowledge these are the first integral equations for these problems that possess this desirable property. This situation is in stark contrast with that arising from the related classical open-surface hypersingular and single-layer operators N and S, whose composition NS maps, for example, the function ϕ = 1 into a function that is not even square integrable. Our proofs rely on three main elements: algebraic manipulations enabled by the presence of integral weights; use of the classical result of continuity of the Cesàro operator; and explicit characterization of the point spectrum of J^τ_0, which, interestingly, can be decomposed into the union of a countable set and an open set, both of which are 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 iterative solvers such as the generalized minimal residual method, it gives rise to a dramatic reduction in the number of iterations compared with those required by other approaches.

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