Accelerating the Method of Reflections with Domain Decomposition techniques for Boundary Integral Equations in Multiple Scattering
Abstract
The Method of Reflections was historically introduced to obtain approximate solu-tions as series expansions for the motion of particles in suspension. It can however equally well be used for solving multiple scattering problems numerically. We show for Helmholtz multiple scattering problems that the Method of Reflections, whether applied in its alternating or parallel version, suffers from convergence problems when scatterers are close. We use boundary integral equations to formulate the methods, and then identify them as algebraic Schwarz methods, thereby interpreting them as boundary domain decomposition techniques. This connection allows us to introduce remedies such as overlap (which can be partial, covering only the illuminating region of the obstacles) and coarse spaces from domain decomposition into the Method of Reflections. This leads to substantially accelerated variants, and also naturally makes them suitable preconditioners for GMRES. These new approaches are particularly efficient for closeby obstacles. Moreover, numerical experiments show that the number of iterations remains robust with respect to the wavenumber.
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BibTeX
@article{chaillat2026accelerating,
title = {{Accelerating the Method of Reflections with Domain Decomposition techniques for Boundary Integral Equations in Multiple Scattering}},
author = {Stéphanie Chaillat and Marion Darbas and Martin J Gander and Laurence Halpern},
journal = {BIT Numerical Mathematics},
year = {2026},
month = sep,
doi = {10.1007/s10543-026-01151-7},
url = {https://hal.science/hal-05556432v2},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Accelerating the Method of Reflections with Domain Decomposition techniques for Boundary Integral Equations in Multiple Scattering
AU - Stéphanie Chaillat
AU - Marion Darbas
AU - Martin J Gander
AU - Laurence Halpern
PY - 2026
DA - 2026/09/04
JO - BIT Numerical Mathematics
DO - 10.1007/s10543-026-01151-7
UR - https://hal.science/hal-05556432v2
AB - The Method of Reflections was historically introduced to obtain approximate solu-tions as series expansions for the motion of particles in suspension. It can however equally well be used for solving multiple scattering problems numerically. We show for Helmholtz multiple scattering problems that the Method of Reflections, whether applied in its alternating or parallel version, suffers from convergence problems when scatterers are close. We use boundary integral equations to formulate the methods, and then identify them as algebraic Schwarz methods, thereby interpreting them as boundary domain decomposition techniques. This connection allows us to introduce remedies such as overlap (which can be partial, covering only the illuminating region of the obstacles) and coarse spaces from domain decomposition into the Method of Reflections. This leads to substantially accelerated variants, and also naturally makes them suitable preconditioners for GMRES. These new approaches are particularly efficient for closeby obstacles. Moreover, numerical experiments show that the number of iterations remains robust with respect to the wavenumber.
SN - 0006-3835
SN - 1572-9125
ER -