Convergence analysis of a high-order Nyström integral-equation method for surface scattering problems

Abstract

In this paper we present a convergence analysis for the Nyström method proposed in [Jour. Comput. Phys. 169 pp. 2921-2934] for the solution of the combined boundary integral equation formulations of sound-soft acoustic scattering problems in three-dimensional space. This fast and efficient scheme combines FFT techniques and a polar change of variables that cancels out the kernel singularity. We establish the stability of the algorithms in the L 2 norm and we derive convergence estimates in both the L 2 and L ∞ norms. In particular, our analysis establishes theoretically the previously observed super-algebraic convergence of the method in cases in which the right-hand side is smooth.

The Exponentially Convergent Trapezoidal Rule

“…Discretizations related to the trapezoidal rule have been used by many authors over the years, including [4,27,43,73,93,98]. We shall not attempt a thorough review of this large subject, but restrict our discussion to an example and a few comments.…”

“…For example, an application in three-dimensional scattering is described in [27]. It is impossible to do justice here to the vast field of integral equations, which encompasses topics including singular integral equations, boundary integral equations, equations of convolution type, and many others.…”

High-Performance Computational Electromagnetics in Frequency-Domain and Time-Domain

“…Meaningful progress concerning well conditioned integral algorithms took place as a result of work sponsored by this contract, including rigorous mathematical theory and powerful numerical algorithms with applicability in a number of fields of science and engineering [12][13][14][15][16][17][18][19][20][21]. A variety of problems and configurations were thus considered, including problems of scattering by open surfaces [12][13][14]; improved integral methods for closed surfaces [15]; problems concerning propagation and scattering by penetrable scatterers [16]; studies of absorption properties of conducting materials containing asperities [17,18]; as well as new methods for evaluation of Laplace eigenfunctions on general domains and under challenging boundary conditions [19,20].…”

“…A variety of problems and configurations were thus considered, including problems of scattering by open surfaces [12][13][14]; improved integral methods for closed surfaces [15]; problems concerning propagation and scattering by penetrable scatterers [16]; studies of absorption properties of conducting materials containing asperities [17,18]; as well as new methods for evaluation of Laplace eigenfunctions on general domains and under challenging boundary conditions [19,20]. A rigorous convergence proof for the original methods [22] was provided in [21]. These contributions are discussed briefly in what follows.…”

“…Complete proofs of the well conditioned character of the aforementioned integral equation formulation for open surfaces was provided in [13,14]. A numerical analysis of the related closed surface solvers [22], in turn, was for the first time put forth in [21]. We believe these two contributions contain a number of innovative theoretical elements which may prove valuable in the general theory of integral equations.…”