Abstract
We present a method for computing nearly singular integrals that occur when single or double layer surface integrals, for harmonic potentials or Stokes flow, are evaluated at points nearby. Such values could be needed in solving an integral equation when one surface is close to another or to obtain values at grid points. We replace the singular kernel with a regularized version having a length parameter \(\delta \) in order to control discretization error. Analysis near the singularity leads to an expression for the error due to regularization which has terms with unknown coefficients multiplying known quantities. By computing the integral with three choices of \(\delta \), we can solve for an extrapolated value that has regularization error reduced to \(O(\delta ^5)\), uniformly for target points on or near the surface. In examples with \(\delta /h\) constant and moderate resolution, we observe total error about \(O(h^5)\) close to the surface. For convergence as \(h \rightarrow 0\), we can choose \(\delta \) proportional to \(h^q\) with \(q < 1\) to ensure the discretization error is dominated by the regularization error. With \(q = 4/5\), we find errors about \(O(h^4)\). For harmonic potentials, we extend the approach to a version with \(O(\delta ^7)\) regularization; it typically has smaller errors, but the order of accuracy is less predictable.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bagge, J., Tornberg, A.K.: Highly accurate special quadrature methods for Stokesian particle suspensions in confined geometries. Int. J. Numer. Methods Fluids 93, 2175–2224 (2021)
Bagge, J., Tornberg, A.K.: Fast Ewald summation for Stokes flow with arbitrary periodicity. J. Comput. Phys. 493, 112473 (2023)
Beale, J.T.: A grid-based boundary integral method for elliptic problems in three dimensions. SIAM J. Numer. Anal. 42(2), 599–620 (2004)
Beale, J.T.: Neglecting discretization corrections in regularized singular or nearly singular integrals. arXiv; Cornell University Library (2020) http://arxiv.org/abs/2004.06686
Beale, J.T., Jones, C., Reale, J., Tlupova S.: A novel regularization for higher accuracy in the solution of the 3-dimensional Stokes flow. Involve, 15, 515–24 (2022)
Beale, J.T., Ying, W., Wilson, J.R.: A simple method for computing singular or nearly singular integrals on closed surfaces. Commun. Comput. Phys. 20(3), 733–753 (2016)
Chwang, A.T., Wu, R.Y.T.: Hydromechanics of low Reynolds number flow. part 2. singularity method for Stokes flows. J. Fluid Mech. 67, 787–815 (1975)
Cortez, R.: The method of regularized Stokeslets. SIAM J. Sci. Comput. 23, 1204 (2001)
Cortez, R., Fauci, L., Medovikov, A.: The method of regularized Stokeslets in three dimensions: analysis, validation, and application to helical swimming. Phys. Fluids 17, 031504 (2005)
Gallagher, M.T., Smith, D.J.: The art of coarse Stokes: Richardson extrapolation improves the accuracy and efficiency of the method of regularized stokeslets. Roy. Soc. Open Sci. 8(5), 210108 (2021)
Greengard, L., Strain, J.: A fast algorithm for the evaluation of heat potentials. Commun. Pure Appl. Math. 43, 949–963 (1990)
Helsing, J.: A higher-order singularity subtraction technique for the discretization of singular integral operators on curved surfaces. arXiv; Cornell University Library (2013) http://arxiv.org/abs/1301.7276
Jiang, S., Greengard, L.: A dual-space multilevel kernel-splitting framework for discrete and continuous convolution. arXiv; Cornell University Library (2023) http://arxiv.org/abs/2308.00292v2
af Klinteberg, L., Shamshirgar, D.S., Tornberg, A.K.: Fast Ewald summation for free-space Stokes potentials. Res. Math. Sci. 4:1:1 (2017)
af Klinteberg, L., Tornberg, A.K.: A fast integral equation method for solid particles in viscous flow using quadrature by expansion. J. Comput. Phys. 326, 420–445 (2016)
Klöckner, A., Barnett, A., Greengard, L., O’Neil, M.: Quadrature by expansion: a new method for the evaluation of layer potentials. J. Comput. Phys. 252, 332–349 (2013)
Krantz, S.G., Parks, H.R.: The implicit function theorem, history, theory, and applications. Birkhauser (2002)
Liron, N., Barta, E.: Motion of a rigid particle in Stokes flow: a new second-kind boundary-integral equation formulation. J. Fluid Mech. 238, 579–598 (1992)
Morse, M., Rahimian, A., Zorin, D.: A robust solver for elliptic PDEs in 3D complex geometries. J. Comput. Phys. 442, 110511 (2021)
Nédélec, J.C.: Acoustic and electromagnetic equations: integral representations for harmonic problems. Springer-Verlag, New York (2001)
Nitsche, M.: Corrected trapezoidal rule for near-singular integrals in axi-symmetric Stokes flow. Adv. Comput. Math. 48, 57 (2022)
Pérez-Arancibia, C., Faria, L.M., Turc, C.: Harmonic density interpolation methods for high-order evaluation of Laplace layer potentials in 2D and 3D. J. Comput. Phys. 376, 411–34 (2019)
Pérez-Arancibia, C., Turc, C., Faria, L.: Planewave density interpolation methods for 3D Helmholtz boundary integral equations. SIAM J. Sci. Comput. 41, A2088–A2116 (2019)
Pozrikidis, C.: Boundary integral and singularity methods for linearized viscous flow. Cambridge Univ, Press (1992)
Shamshirgar, D.S., Bagge, J., Tornberg, A.K.: Fast Ewald summation for electrostatic potentials with arbitrary periodicity. J. Chem. Phys. 154, 164109 (2021)
Shamshirgar, D.S., Tornberg, A.K.: The spectral Ewald method for singly periodic domains. J. Comput. Phys. 347, 341–366 (2017)
Shankar, V., Olson, S.D.: Radial basis function (RBF)-based parametric models for closed and open curves within the method of regularized Stokeslets. Int. J. Numer. Methods Fluids 79, 269–89 (2015)
Siegel, M., Tornberg, A.K.: A local target specific quadrature by expansion method for evaluation of layer potentials in 3D. J. Comput. Phys. 364, 365–392 (2018)
Stein, D.B., Barnett, A.H.: Quadrature by fundamental solutions: kernel-independent layer potential evaluation for large collections of simple objects. Adv. Comput. Math. 48, 60 (2022)
Tlupova, S., Beale, J.T.: Regularized single and double layer integrals in 3D Stokes flow. J. Comput. Phys. 386, 568–584 (2019)
Wang, L., Krasny, R., Tlupova, S.: A kernel-independent treecode algorithm based on barycentric Lagrange interpolation. Commun. Comput. Phys. 28(4), 1415–1436 (2020)
Whittaker, E.T., Watson, G.N.: A course of modern analysis. Cambridge Univ. Press, 4th edition (1927)
Wilson, J.R.: On computing smooth, singular and nearly singular integrals on implicitly defined surfaces. PhD thesis, Duke University (2010) http://search.proquest.com/docview/744476497
Ying, L.: A kernel independent fast multipole algorithm for radial basis functions. J. Comput. Phys. 213, 451–57 (2006)
Ying, L., Biros, G., Zorin, D.: A high-order 3D boundary integral equation solver for elliptic PDEs in smooth domains. J. Comput. Phys. 219, 247–275 (2006)
Funding
The work of ST was supported by the National Science Foundation grant DMS-2012371.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare no competing interests.
Additional information
Communicated by: Anna-Karin Tornberg
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Beale, J.T., Tlupova, S. Extrapolated regularization of nearly singular integrals on surfaces. Adv Comput Math 50, 61 (2024). https://doi.org/10.1007/s10444-024-10161-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10444-024-10161-4