Asymptotic analysis for close evaluation of layer potentials
This work provides a method to improve accuracy in boundary integral evaluations for applications like fluid-structure interactions and near-field scattering, but is incremental as it extends existing asymptotic techniques to a specific numerical issue.
The paper addresses the problem of large errors when evaluating layer potentials near domain boundaries using standard quadrature rules. By employing an asymptotic expansion of the kernel, they eliminate the O(1) error in a boundary layer of thickness O(1/N), demonstrating effectiveness for Laplace's equation in 2D.
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.