New preconditioners for Laplace and Helmholtz integral equations on open curves

François Alouges (1) , Martin Averseng (1)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique

The numerical resolution of wave scattering problems by open curves leads to ill-conditioned linear systems which are difficult to precondition due to the geometrical singularities at the edges. We introduce two new preconditioners to tackle this problem respectively for Dirichlet or Neu-mann boundary data, that take the form of square roots of local operators. We describe an adapted analytical setting to analyze them and demonstrate the efficiency of this method on several numerical examples. A complete new pseudo-differential calculus suited to the study of such operators is postponed to the second part of this work.

Mots clés: BEM; Preconditioning; Singularities

Domaines: Mathématiques [math]; Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]

François Alouges, Martin Averseng. New preconditioners for Laplace and Helmholtz integral equations on open curves: I. Theoretical framework and numerical results. 2019. ⟨hal-02144805⟩