New preconditioners for the Laplace and Helmholtz integral equations on open curves: analytical framework and numerical results

François Alouges (1, 2) , Martin Averseng (1)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 CB - CB - Centre Borelli - UMR 9010

Helmholtz wave scattering by open screens in 2D can be formulated as first-kind integral equations which lead to ill-conditioned linear systems after discretization. We introduce two new preconditioners in the form of square-roots of on-curve differential operators both for the Dirichlet and Neumann boundary conditions on the screen. They generalize the so-called “analytical” preconditioners available for Lipschitz scatterers. We introduce a functional setting adapted to the singularity of the problem and enabling the analysis of those preconditioners. The efficiency of the method is demonstrated on several numerical examples.

Domaines: Mathématiques [math]

François Alouges, Martin Averseng. New preconditioners for the Laplace and Helmholtz integral equations on open curves: analytical framework and numerical results. Numerische Mathematik, 2021, 148 (2), pp.255-292. ⟨10.1007/s00211-021-01189-5⟩. ⟨hal-04450564⟩