Abstract
A high-order accurate quadrature rule for the discretization of boundary integral equations (BIEs) on closed smooth contours in the plane is introduced. This quadrature can be viewed as a hybrid of the spectral quadrature of Kress (Math. Comput. Model. 15(3-5), 229–243 1991) and the locally corrected trapezoidal quadrature of Kapur and Rokhlin (SIAM J. Numer. Anal. 34(4), 1331–1356, 1997). The new technique combines the strengths of both methods, and attains high-order convergence, numerical stability, ease of implementation, and compatibility with the “fast” algorithms (such as the Fast Multipole Method or Fast Direct Solvers). Important connections between the punctured trapezoidal rule and the Riemann zeta function are introduced, which enable a complete convergence analysis and lead to remarkably simple procedures for constructing the quadrature corrections. The paper reports a detailed comparison between the new method and the methods of Kress, of Kapur and Rokhlin, and of Alpert (SIAM J. Sci. Comput. 20(5), 1551–1584, 1999).
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References
Aguilar, J. C., Chen, Y.: High-order corrected trapezoidal quadrature rules for functions with a logarithmic singularity in 2-D. Comput. Math. Appl. 44(8-9), 1031–1039 (2002)
Alpert, B. K.: High-order quadratures for integral operators with singular kernels. J. Comput. Appl. Math. 60(3), 367–378 (1995)
Alpert, B. K.: Hybrid Gauss-trapezoidal quadrature rules. SIAM J. Sci. Comput. 20(5), 1551–1584 (1999)
Borwein, J. M., Glasser, M., McPhedran, R., Wan, J., Zucker, I.: Lattice sums then and now, vol. 150. Cambridge University Press, Cambridge (2013)
Bremer, J., Gillman, A., Martinsson, P. G.: A high-order accurate accelerated direct solver for acoustic scattering from surfaces. BIT Numer. Math. 55(2), 367–397 (2015)
Colton, D., Kress, R.: Inverse acoustic and electromagnetic scattering theory, vol. 93. Springer Nature, Berlin (2019)
Epstein, P.: Zur theorie allgemeiner zetafunctionen. Math. Ann. 56(4), 615–644 (1903)
Greengard, L., Rokhlin, V.: A fast algorithm for particle simulations. J. Comput. Phys. 73(2), 325–348 (1987)
Hao, S., Barnett, A. H., Martinsson, P. G., Young, P.: High-order accurate methods for Nyström discretization of integral equations on smooth curves in the plane. Adv. Comput. Math. 40(1), 245–272 (2014)
Hsiao, G. C., Wendland, W. L.: Boundary integral equations. Springer, Berlin (2008)
Kapur, S., Rokhlin, V.: High-order corrected trapezoidal quadrature rules for singular functions. SIAM J. Numer. Anal. 34(4), 1331–1356 (1997)
Keast, P., Lyness, J. N.: On the structure of fully symmetric multidimensional quadrature rules. SIAM J. Numer. Anal. 16(1), 11–29 (1979)
Kress, R.: Boundary integral equations in time-harmonic acoustic scattering. Math. Comput. Model. 15(3-5), 229–243 (1991)
Kress, R.: Linear integral equations, Applied Mathematical Sciences, 3rd edn., vol. 82. Springer, New York (2014)
Marin, O., Runborg, O., Tornberg, A. K.: Corrected trapezoidal rules for a class of singular functions. IMA J. Numer. Anal. 34(4), 1509–1540 (2014)
Martinsson, P. G.: Fast direct solvers for elliptic PDEs, CBMS-NSF Conference Series, vol. CB96. SIAM, Philadelphia (2019)
Navot, I.: An extension of the Euler-Maclaurin summation formula to functions with a branch singularity. J. Math. Phys. 40(1-4), 271–276 (1961)
Navot, I.: A further extension of the Euler-Maclaurin summation formula. J. Math. Phys. 41(1-4), 155–163 (1962)
Pozrikidis, C., et al.: Boundary integral and singularity methods for linearized viscous flow. Cambridge University Press, Cambridge (1992)
Sidi, A., Israeli, M.: Quadrature methods for periodic singular and weakly singular Fredholm integral equations. J. Sci. Comput. 3(2), 201–231 (1988)
Squire, W., Trapp, G.: Using complex variables to estimate derivatives of real functions. SIAM Rev. 40(1), 110–112 (1998)
Trefethen, L. N., Weideman, J.: The exponentially convergent trapezoidal rule. SIAM Rev. 56(3), 385–458 (2014)
Wu, B., Martinsson, P. G.: Corrected trapezoidal rules for boundary integral equations in three dimensions. arxiv:2007.02512 (2020)
Wu, B., Martinsson, P. G.: Zeta correction: a new approach to constructing corrected trapezoidal quadrature rules for singular integral operators. arxiv:2007.13898 (2020)
Acknowledgements
The authors would like to thank Alex Barnett for sharing valuable perspectives and insights.
Funding
The work reported was supported by the Office of Naval Research (grant N00014-18-1-2354), and by the National Science Foundation (grants DMS-1620472 and DMS-2012606).
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Communicated by: Zydrunas Gimbutas
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This article belongs to the Topical Collection: Advances in Computational Integral Equations Guest Editors: Stephanie Chaillat, Adrianna Gillman, Per-Gunnar Martinsson, Michael O’Neil, Mary-Catherine Kropinski, Timo Betcke, Alex Barnett
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Wu, B., Martinsson, PG. Zeta correction: a new approach to constructing corrected trapezoidal quadrature rules for singular integral operators. Adv Comput Math 47, 45 (2021). https://doi.org/10.1007/s10444-021-09872-9
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DOI: https://doi.org/10.1007/s10444-021-09872-9
Keywords
- Boundary integral equations
- Singular quadrature
- Riemann zeta function
- Nyström discretization
- Kapur-Rokhlin quadrature rule
- Kress quadrature rule
- Alpert quadrature rule
- Corrected trapezoidal rule