Asymptotic analysis for close evaluation of layer potentials

Camille Carvalho (1) , Shilpa Khatri (1) , Arnold D Kim (1)

1 UC Merced - University of California [Merced]

Accurate evaluation of layer potentials near boundaries is needed in many applications, including fluid-structure interactions and near-field scattering in nano-optics. When numerically evaluating layer potentials, it is natural to use the same quadrature rule as the one used in the Nyström method to solve the underlying boundary integral equation. However, this method is problematic for evaluation points close to boundaries. For a fixed number of quadrature points, N , this method incurs O(1) errors in a boundary layer of thickness O(1/N). Using an asymp-totic expansion for the kernel of the layer potential, we remove this O(1) error. We demonstrate the effectiveness of this method for interior and exterior problems for Laplace's equation in two dimensions.

Mots clés: Laplace's equation; Boundary integral equations; Layer potentials; Nearly singular integrals; Close evaluations

Domaines: Analyse numérique [math.NA]; Physique mathématique [math-ph]; Equations aux dérivées partielles [math.AP]

Camille Carvalho, Shilpa Khatri, Arnold D Kim. Asymptotic analysis for close evaluation of layer potentials. Journal of Computational Physics, 2018. ⟨hal-01530928v2⟩