Asymptotic analysis for close evaluation of layer potentials

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 natur...

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Bibliographic Details
Published in:Journal of computational physics 2018-02, Vol.355, p.327-341
Main Authors: Carvalho, Camille, Khatri, Shilpa, Kim, Arnold D.
Format: Article
Language:English
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
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Summary: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.
ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2017.11.015