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A method for computing nearly singular integrals

Publication ,  Journal Article
Beale, JT; Lai, MC
Published in: SIAM Journal on Numerical Analysis
December 1, 2001

We develop a method for computing a nearly singular integral, such as a double layer potential due to sources on a curve in the plane, evaluated at a point near the curve. The approach is to regularize the singularity and obtain a preliminary value from a standard quadrature rule. Then we add corrections for the errors due to smoothing and discretization, which are found by asymptotic analysis. We prove an error estimate for the corrected value, uniform with respect to the point of evaluation. One application is a simple method for solving the Dirichlet problem for Laplace's equation on a grid covering an irregular region in the plane, similar to an earlier method of A. Mayo [SIAM J. Sci. Statist. Comput., 6 (1985), pp. 144-157]. This approach could also be used to compute the pressure gradient due to a force on a moving boundary in an incompressible fluid. Computational examples are given for the double layer potential and for the Dirichlet problem.

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Published In

SIAM Journal on Numerical Analysis

DOI

ISSN

0036-1429

Publication Date

December 1, 2001

Volume

38

Issue

6

Start / End Page

1902 / 1925

Related Subject Headings

  • Numerical & Computational Mathematics
  • 4903 Numerical and computational mathematics
  • 4901 Applied mathematics
  • 0103 Numerical and Computational Mathematics
  • 0102 Applied Mathematics
  • 0101 Pure Mathematics
 

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Beale, J. T., & Lai, M. C. (2001). A method for computing nearly singular integrals. SIAM Journal on Numerical Analysis, 38(6), 1902–1925. https://doi.org/10.1137/S0036142999362845
Beale, J. T., and M. C. Lai. “A method for computing nearly singular integrals.” SIAM Journal on Numerical Analysis 38, no. 6 (December 1, 2001): 1902–25. https://doi.org/10.1137/S0036142999362845.
Beale JT, Lai MC. A method for computing nearly singular integrals. SIAM Journal on Numerical Analysis. 2001 Dec 1;38(6):1902–25.
Beale, J. T., and M. C. Lai. “A method for computing nearly singular integrals.” SIAM Journal on Numerical Analysis, vol. 38, no. 6, Dec. 2001, pp. 1902–25. Scopus, doi:10.1137/S0036142999362845.
Beale JT, Lai MC. A method for computing nearly singular integrals. SIAM Journal on Numerical Analysis. 2001 Dec 1;38(6):1902–1925.

Published In

SIAM Journal on Numerical Analysis

DOI

ISSN

0036-1429

Publication Date

December 1, 2001

Volume

38

Issue

6

Start / End Page

1902 / 1925

Related Subject Headings

  • Numerical & Computational Mathematics
  • 4903 Numerical and computational mathematics
  • 4901 Applied mathematics
  • 0103 Numerical and Computational Mathematics
  • 0102 Applied Mathematics
  • 0101 Pure Mathematics