Regularized integral equations and fast high‐order solvers for sound‐hard acoustic scattering problems

Abstract

This paper presents a theoretical discussion as well as novel solution algorithms for problems of scattering on smooth two-dimensional domains under Zaremba boundary conditions-for which Dirichlet and Neumann conditions are specified on various portions of the domain boundary. The theoretical basis of the proposed numerical methods, which is provided for the first time in the present contribution, concerns detailed information about the singularity structure of solutions of the Helmholtz operator under boundary conditions of Zaremba type. The new numerical method is based on use of Green functions and integral equations, and it relies on the Fourier Continuation method for regularization of all smooth-domain Zaremba singularities as well as newly derived quadrature rules which give rise to high-order convergence even around Zaremba singular points. As demonstrated in this paper, the resulting algorithms enjoy high-order convergence, and they can be used to efficiently solve challenging Helmholtz boundary value problems and Laplace eigenvalue problems with high-order accuracy.

Coercivity of Combined Boundary Integral Equations in High‐Frequency Scattering

“…implies that the numerical range of the associated Galerkin matrix A is bounded away from 0, and thus allows us to obtain k-explicit bounds on the number of GMRES iterations needed to solve Av D f. Indeed, consider the h-version of the Galerkin method; i.e., V N L 2 ./ is the space of piecewise polynomials of degree Ä p for some fixed p 0 on quasiuniform meshes of diameter h, with h decreasing to 0 (thus in the literature; however, results for Nyström discretizations of both the analogous operator for the Neumann problem and modifications of this operator that make it a compact perturbation of the identity on smooth domains can be found in [6,7]. These results show the number of GMRES iterations growing like k a for a range of different 0 < a < 1, depending on the geometry.…”

An accurate boundary element method for the exterior elastic scattering problem in two dimensions

“…Denote by ω > 0 the frequency of propagating elastic waves. The problem to be considered is to determine the elastic displacement field u in the solid provided an incident field u i , and can be formulated as follows: 2) and the Kupradze radiation condition ( [10])…”

“…Among them, the boundary integral equation (BIE) method ( [8]), which has also been widely used in acoustics, electromagnetics and elastostatics ( [2,7,9,14,15]), takes some advantages over domain discretization methods since it is not necessary for the BIE method to impose any artificial boundary condition for the radiation condition, and the reduced BIEs are only discretized on the boundary of the obstacle. To effectively reduce the BIE into a linear system, many different solvers including the boundary element method (BEM) ( [13]), the Nyströme method ( [16]), the fast multipole method [5,17] and the spectral method ( [12]) have been considered.…”

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].…”