Asymptotic analysis for close evaluation of layer potentials

• Published in: Journal of computational physics Vol. 355; pp. 327 - 341
• Main Authors: Carvalho, Camille, Khatri, Shilpa, Kim, Arnold D.
• Format: Journal Article
• Language: English
• Published: Cambridge Elsevier Inc 15.02.2018; Elsevier Science Ltd; Elsevier
• Subjects: Analysis of PDEs; Asymptotic series; Boundary integral equations; Boundary layers; Close evaluations; Computational physics; Differential equations; Fluid-structure interaction; Harmonic analysis; Integral equations; Integrals; Laplace's equation; Layer potentials; Mathematical Physics; Mathematics; Nano-optics; Nearly singular integrals; Numerical Analysis; Thickness; Close evaluations; Laplace's equation; Layer potentials; Nearly singular integrals; Boundary integral equations
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2017.11.015

We study the evaluation of layer potentials close to the domain boundary. 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 asymptotic 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. •An accurate method for the close evaluation of layer potentials is developed.•Asymptotic expansions are used to capture the behavior of the peaked kernel.•The method combines these expansions with numerical integration to achieve accuracy.•Results are shown for single- and double-layer potentials for 2D Laplace's equation.