Computing weakly singular and near-singular integrals in high-order boundary elements

Hadrien Montanelli (1, 2) , Matthieu Aussal (3) , Houssem Haddar (2)

1 IP Paris - Institut Polytechnique de Paris
2 IDEFIX - Inversion of Differential Equations For Imaging and physiX
3 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique

We present algorithms for computing weakly singular and near-singular integrals arising when solving the 3D Helmholtz equation with high-order boundary elements. These are based on the computation of the preimage of the singularity in the reference element's space using Newton's method, singularity subtraction, the continuation approach, and transplanted Gauss quadrature. We demonstrate the accuracy of our method for quadratic triangular elements with several numerical experiments, including the scattering by two nearby half-spheres.

Mots clés: Helmholtz equation; integral equations; boundary element method; singular integrals; near-singular integrals; Taylor series; homogeneous functions; continuation approach; Gauss quadrature; 41A55; 41A58; 45E05; 45E99; 65N38; 65R20; 78M15; 35J05

Domaines: Analyse numérique [math.NA]

Hadrien Montanelli, Matthieu Aussal, Houssem Haddar. Computing weakly singular and near-singular integrals in high-order boundary elements. 2022. ⟨hal-03518653v1⟩