Close evaluation of layer potentials

Sponsored by NSF DMS-1819052.

Framework

The close-evaluation problem in boundary integral methods refers to large errors incurred when evaluating layer potentials at points near the boundary of the domain despite being accurate elsewhere in the domain. When using a high order Nyström method to numerically evaluate a layer potential, its high order accuracy will be effective for nearly all points in the domain. However, at close-evaluation points, this quadrature rule will produce an O(1) error. The goal is to address this error using asymptotic analysis of the nearly singular behavior.

Asymptotic method with deferred correction

We developed a numerical method based on asymptotic approximations of the kernel via rational trigonometric functions to capture the peaked behavior for close-evaluation points. Subtracting this approximation allows to use standard Nystrom method, and the remainder can be computed analytically using Fourier convolution sum.

Close evaluation for acoustic scattering. Left: standard Nyström method (O(1) close to the boundary). Right: asymptotic method with deferred correction.

Asymptotic approximations methods

We developed asymptotic approximations methods based matched asymptotic expansions of layer potentials in 2D and 3D to control the error with respect to the distance from the boundary.


Close evaluation for the double-layer potential for Laplace's equation in 3D.

References

  • Codes available on GitHub.
  • Asymptotic approximations for the close evaluation of double-layer potentials,
    C. Carvalho, S. Khatri, A. D. Kim, submitted.
  • Close evaluation for layer potentials in three dimensions,
    C. Carvalho, S. Khatri, A. D. Kim, submitted.
  • Asymptotic analysis for close evaluation for layer potentials,
    C. Carvalho, S. Khatri, A. D. Kim, Journal of Computational Physics, 655, pp 327-341, 2018.
  • Ongoing projects:

  • Applications to Stokes flow
  • Application to transmission problems