An Introduction to Operator Preconditioning for the Fast Iterative Integral Equation Solution of Time-Harmonic Scattering Problems

Abstract

The aim of this paper is to provide an introduction to the improved iterative Krylov solution of boundary integral equations for time-harmonic scattering problems arising in acoustics, electromagnetism and elasticity. From the point of view of computational methods, considering large frequencies is a challenging issue in engineering since it leads to solving highly indefinite large scale complex linear systems which generally implies a convergence breakdown of iterative methods. More specifically, we explain the problematic and some partial solutions through analytical preconditioning for high-frequency acoustic scattering problems and the introduction of new combined field integral equations. We complete the paper with some recent extensions to the case of electromagnetic and elastic waves equations. Keywords time-harmonic scattering • acoustic • electromagnetism • elasticity • integral equation • Krylov solver • preconditioners • combined field integral equation Contents

A Coupling between Integral Equations and On-Surface Radiation Conditions for Diffraction Problems by Non Convex Scatterers

“…The aim of this section is to provide without any detail the standard integral equation formulations for solving the two-dimensional scattering problem with Dirichlet boundary condition that will be used later. More details can be found in [17,20,28] for the derivation and properties of these integral equations (well-posedness, existence of resonant modes,. .…”

“…where the unknown density ρ is an element of L 2 (Γ). The equation is well-posed and equivalent to the exterior scattering problem (1) as soon as k is not an irregular interior frequency of the associated Dirichlet boundary-value problem [17,20,28]. This integral equation is called Electric Field Integral Equation (EFIE) in electromagnetism.…”

“…One well-known possibility is to use an absorbing boundary condition [1][2][3][4][5][6][7] or a Perfectly Matched Layer [8][9][10][11][12][13] to bound the domain and then solve the resulting problem by the finite element method [14][15][16]. Another widely used alternative is to rewrite equivalently the initial exterior PDE problem as an integral equation over the finite surface Γ of the scatterer Ω − based on the Green's function [17][18][19][20][21][22][23][24][25][26][27][28]. Then, this has the advantage of reducing from one the dimension of the problem.…”

“…When a discretization technique is then applied, as the boundary element method, then the corresponding discrete version of the integral equation leads to the numerical solution of a highly indefinite complex-valued dense linear system which is particularly difficult to tackle in the high-frequency regime. Many technical aspects are then necessary to make it working correctly for some applications, for example to reduce the storage and to accelerate the solution of the linear system [17,18,29]. Various numerical methods were further developed to propose some other ways to solve, at least approximately, the initial scattering problem.…”

Non Uniform Rational B-Splines and Lagrange approximations for time-harmonic acoustic scattering: accuracy and absorbing boundary conditions

“…Broadly speaking, the numerical techniques can be classified as domainand boundary-based approaches. Before focusing on some specific numerical methods related to the paper, we refer to the references [1][2][3][4][5][6][7][8], where the interested reader can find some overviews on general numerical methods for solving acoustic scattering problems.…”