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

Abstract

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