Computing weakly singular and near-singular integrals over curved boundary elements

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

1 IDEFIX - Inversion of Differential Equations For Imaging and physiX

We present algorithms for computing weakly singular and near-singular integrals arising when solving the 3D Helmholtz equation with curved 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 basis functions and quadratic triangles with several numerical experiments, including the scattering by two half-spheres.

Mots clés: boundary element method; integral equations; Helmholtz equation; continuation approach; Gauss quadrature; homogeneous functions; Taylor series; near-singular integrals; singular integrals

Domaines: Analyse numérique [math.NA]

Hadrien Montanelli, Matthieu Aussal, Houssem Haddar. Computing weakly singular and near-singular integrals over curved boundary elements. SIAM Journal on Scientific Computing, 2022, 44 (6), pp.A3728-A3753. ⟨10.1137/21M1462027⟩. ⟨hal-03518653v4⟩