A generalized Calderón formula for open-arc diffraction problems: theoretical considerations
نویسندگان
چکیده
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 Ñ S̃[φ] = f , where Ñ and S̃ are first-kind integral operators whose composition gives rise to a generalized Calderón formula of the form ÑS̃ = J̃τ 0 +K̃ in a weighted, periodized Sobolev space. (Here J̃τ 0 is a continuous and continuously invertible operator and K̃ is a compact operator.) The Ñ S̃ 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: 1) Algebraic manipulations enabled by the presence of integral weights; 2) Use of the classical result of continuity of the Cesàro operator; and 3) Explicit characterization of the point spectrum of J̃τ 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 GMRES, it gives rise to dramatic reductions of iteration numbers vs. those required by other approaches.
منابع مشابه
Second-kind integral solvers for TE and TM problems of diffraction by open arcs
[1] We present a novel approach for the numerical solution of problems of diffraction by open arcs in two dimensional space. Our methodology relies on composition of weighted versions of the classical integral operators associated with the Dirichlet and Neumann problems (TE and TM polarizations, respectively) together with a generalization to the open-arc case of the well known closed-surface C...
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