High-order Nystrom discretizations for the solution of integral equation formulations of two-dimensional Helmholtz transmission problems

摘要

We present and analyse fully discrete Nystrom methods for the solution of three classes of well-conditioned boundary integral equations for the solution of two-dimensional scattering problems by homogeneous dielectric scatterers. Specifically, we perform the stability analysis of Nystrom discretizations of: (1) the classical second-kind integral equations for transmission problems (Kress, R. & Roach, G. F. (1978) Transmission problems for the Helmholtz equation. J. Math. Phys., 19, 1433-1437.), (2) the single integral equation formulations (Kleinman, R. E. & Martin, P. A. (1988) On single integral equations for the transmission problem of acoustics. SIAM J. Appl. Math., 48, 307-325.), and (3) recently introduced Generalized Combined Source Integral Equations (Boubendir et al. (2015) Integral equations requiring small numbers of krylov-subspace iterations for two-dimensional smooth penetrable scattering problems. Appl. Numer. Math., in press.). The Nystrom method that we use for the discretization of the various integral equations under consideration are based on global trigonometric approximations, splitting of the kernels of integral operators into singular and smooth components, and explicit quadratures of products of singular parts (logarithms) and trigonometric polynomials. The discretization of the integral equations (2) and (3) above requires special care, as these formulations feature compositions of boundary integral operators that are pseudodifferential operators of positive and negative orders, respectively. We deal with these compositions through Caldern's calculus, and we establish the convergence of fully discrete Nystrom methods in appropriate Sobolev spaces, which implies pointwise convergence of the discrete solutions. In the case of analytic boundaries, we establish superalgebraic convergence of the method.