Generalized combined field integral equations for the iterative solution of the three-dimensional Helmholtz equation

Abstract

Abstract. This paper addresses the derivation of new second-kind Fredholm combined field integral equations for the Krylov iterative solution of tridimensional acoustic scattering problems by a smooth closed surface. These integral equations need the introduction of suitable tangential square-root operators to regularize the formulations. Existence and uniqueness occur for these formulations.

The Wave-Matching Boundary Integral Equation — An energy approach to Galerkin BEM for acoustic wave propagation problems

“…Another research area where this effectively is done is in the group of algorithms that apply a Dirichlet-to-Neumann map (or vice-versa) to regularise the BIE and ensure a unique solution [10,[34][35][36][37]. These are typically stated as being motivated by the work on On-Surface Radiation Conditions (OSRCs) [38], though appear to most commonly be applied to generate matrix pre-conditioners…”

A new accurate residual-based a posteriori error indicator for the BEM in 2D-acoustics

“…The operator Λ is consequently of order −s: it is an ''approximation of the inverse of the square root of A 0 ''. In a way, this problem is really close to analytical preconditioning techniques [20,21] where one is looking for an approximate inverse of A.…”

A well-conditioned integral equation for iterative solution of scattering problems with a variable Leontovitch boundary condition

“…corresponds to the Dirichlet-To-Neumann operator for the acoustic scattering problem [3]. For an efficient approximation of this one, we introduce a damping parameter ε in order to perturb the wavenumber k by k ε = k + iε and so we regularize the singularity of the square root at the level of the transition region of glancing rays.…”