Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-10-31T22:59:22.380Z Has data issue: false hasContentIssue false

A Simple Method for Computing Singular or Nearly Singular Integrals on Closed Surfaces

Published online by Cambridge University Press:  31 August 2016

J. Thomas Beale*
Affiliation:
Department of Mathematics, Duke University, Durham, NC 27708-0320, USA
Wenjun Ying*
Affiliation:
Department of Mathematics, MOE-LSC and Institute of Natural Sciences, Shanghai Jiao Tong University, Minhang, Shanghai 200240, P.R. China
Jason R. Wilson*
Affiliation:
Mathematics Department, Virginia Tech, Blacksburg, VA 24061-0123, USA
*
*Corresponding author. Email addresses:beale@math.duke.edu (J. T. Beale), wying@sjtu.edu.cn (W. Ying), jasonwil@math.vt.edu (J. R. Wilson)
*Corresponding author. Email addresses:beale@math.duke.edu (J. T. Beale), wying@sjtu.edu.cn (W. Ying), jasonwil@math.vt.edu (J. R. Wilson)
*Corresponding author. Email addresses:beale@math.duke.edu (J. T. Beale), wying@sjtu.edu.cn (W. Ying), jasonwil@math.vt.edu (J. R. Wilson)
Get access

Abstract

We present a simple, accurate method for computing singular or nearly singular integrals on a smooth, closed surface, such as layer potentials for harmonic functions evaluated at points on or near the surface. The integral is computed with a regularized kernel and corrections are added for regularization and discretization, which are found from analysis near the singular point. The surface integrals are computed from a new quadrature rule using surface points which project onto grid points in coordinate planes. The method does not require coordinate charts on the surface or special treatment of the singularity other than the corrections. The accuracy is about O(h3), where h is the spacing in the background grid, uniformly with respect to the point of evaluation, on or near the surface. Improved accuracy is obtained for points on the surface. The treecode of Duan and Krasny for Ewald summation is used to perform sums. Numerical examples are presented with a variety of surfaces.

Type
Research Article
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Atkinson, K. E., The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press, 1997.CrossRefGoogle Scholar
[2] Beale, J. T., A convergent boundary integral method for three-dimensional water waves, Math. Comp., 70 (2001), 9771029.CrossRefGoogle Scholar
[3] Beale, J. T. and Lai, M. C., A method for computing nearly singular integrals, SIAM J. Numer. Anal., 38 (2001), 19021925.CrossRefGoogle Scholar
[4] Beale, J. T., Agrid-based boundary integral method for elliptic problems in three-dimensions, SIAM J. Numer. Anal., 42 (2004), 599620.CrossRefGoogle Scholar
[5] Beale, J. T. and Layton, A. T., On the accuracy of finite difference methods for elliptic problems with interfaces, Commun. Appl. Math. Comput. Sci., 1 (2006), 91119.CrossRefGoogle Scholar
[6] Bremer, J. and Gimbutas, Z., A Nyström method for weakly singular integral operators on surfaces, J. Comput. Phys., 231 (2012), 48854903.CrossRefGoogle Scholar
[7] Bruno, O. and Kunyansky, L., A fast, high-order algorithm for the solution of surface scattering problems: Basic implementation, tests, and applications, J. Comput. Phys., 169 (2001), 80110.CrossRefGoogle Scholar
[8] Chen, M. and Lu, B., TMSmesh: A robust method for molecular surface mesh generation using a trace technique, J. Chem. Theory Comput., 7 (2011), 203–12.CrossRefGoogle ScholarPubMed
[9] Cortez, R., The method of regularized Stokeslets, SIAM J. Sci. Comput., 23 (2001), 1204–25.CrossRefGoogle Scholar
[10] Cortez, R., Fauci, L. and Medovikov, A., The method of regularized Stokeslets in three dimensions: analysis, validation, and application to helical swimming, Phys. Fluids, 17 (2005), 114.CrossRefGoogle Scholar
[11] Duan, Z.-H. and Krasny, R., An Ewald summation based multipole method, J. Chem. Phys. 113, (2000), 3492–5.CrossRefGoogle Scholar
[12] Ganesh, M. and Graham, I. G., A high-order algorithm for obstacle scattering in three dimensions, J. Comput. Phys., 198 (2004), 211–42.CrossRefGoogle Scholar
[13] Graham, I. G. and Sloan, I. H., Fully discrete spectral boundary integral methods for Helmholtz problems on smooth closed surfaces in ℝ3, Numerische Mathematik, 92 (2002), 289323.CrossRefGoogle Scholar
[14] Hackbusch, W., Integral Equations, Theory and Numerical Treatment, Birkhäuser, 1995.CrossRefGoogle Scholar
[15] Helsing, J. and Ojala, R., On the evaluation of layer potentials close to their sources, J. Comput. Phys., 227 (2008), 28992921.CrossRefGoogle Scholar
[16] Helsing, J., A higher-order singularity subtraction technique for the discretization of singular integral operators on curved surfaces, preprint, 2013.Google Scholar
[17] Klöckner, A., Barnett, A., Greengard, L. and O’Neil, M., Quadrature by expansion: A new method for the evaluation of layer potentials, J. Comput. Phys., 252 (2013), 332349.CrossRefGoogle Scholar
[18] Lindsay, K. and Krasny, R., A particle method and adaptive treecode for vortex sheet motion in three-dimensional flow, J. Comput. Phys., 172 (2001), 879907.CrossRefGoogle Scholar
[19] Marin, O., Runborg, O. and Tornberg, A.-K., Corrected trapezoidal rules for a class of singular functions, IMA J. Numer. Anal., 34 (2014), 15091540.CrossRefGoogle Scholar
[20] Mayo, A., Fast high order accurate solution of Laplace's equation on irregular regions, SIAM J. Sc. Statist. Comput., 6 (1985), 144157.CrossRefGoogle Scholar
[21] Nguyen, H.-N. and Cortez, R., Reduction of the regularization error of the method of regularized Stokeslets for a rigid object immersed in a three-dimensional Stokes flow, Commun. Comput. Phys., 15 (2014), 126152.CrossRefGoogle Scholar
[22] Osher, S. and Fedkiw, R., Level Set Methods and Dynamic Implicit Surfaces, Springer, 2003.CrossRefGoogle Scholar
[23] Pozrikidis, C., Interfacial dynamics for Stokes flow, J. Comput. Phys., 169 (2001), 250301.CrossRefGoogle Scholar
[24] Pozrikidis, C., A Practical Guide to Boundary Element Methods with the Software Library BEMLIB, C.R.C., 2002.CrossRefGoogle Scholar
[25] Sauter, S. and C. Schwab, , Boundary Element Methods, Springer, 2010.CrossRefGoogle Scholar
[26] Sethian, J., Level Set Methods and Fast Marching Methods, Cambridge Univ. Press, 1998.Google Scholar
[27] Taus, M., Rodin, G. and Hughes, T. J. R., Isogeometric analysis of boundary integral equations, Math. Models Methods Appl. Sci., 26 (2016), 1447–80.CrossRefGoogle Scholar
[28] Tlupova, S. and Beale, J. T., Nearly singular integrals in 3d Stokes flow, Commun. Comput. Phys., 14 (2013), 1207–27.CrossRefGoogle Scholar
[29] Veerapaneni, S. K., Rahimian, A., Biros, G. and Zorin, D., A fast algorithm for simulating vesicle flows in three dimensions, J. Comput. Phys., 230 (2011), 5610–34.CrossRefGoogle Scholar
[30] Wilson, J. R., On computing smooth, singular and nearly singular integrals on implicitly defined surfaces, Ph.D. thesis, Duke University (2010), http://search.proquest.com/docview/744476497 Google Scholar
[31] Ying, L., Biros, G. and Zorin, D., A kernel-independent adaptive fast multipole algorithm in two and three dimensions, J. Comput. Phys., 196 (2004), 591626.CrossRefGoogle Scholar
[32] Ying, L., Biros, G. and Zorin, D., A high-order 3d boundary integral equation solver for elliptic pdes in smooth domains, J. Comput. Phys., 219 (2006), 247–75.CrossRefGoogle Scholar
[33] Ying, W.-J. and Beale, J. T., A fast accurate boundary integral method for potentials on closely packed cells, Commun. Comput. Phys., 14 (2013), 1073–93.CrossRefGoogle Scholar
[34] Ying, W.-J. and Wang, W.-C., A kernel-free boundary integral method for implicitly defined surfaces, J. Comput. Phys., 252 (2013), 606624.CrossRefGoogle Scholar
[35] Ying, W.-J. and Wang, W.-C., A kernel-free boundary integral method for variable coefficients elliptic PDEs, Commun. Comput. Phys., 15 (2014), 11081140.CrossRefGoogle Scholar
[36] Zinchenko, A. Z., Rother, M. A. and Davis, R.H., Cusping, capture, and breakup of interacting drops by a curvatureless boundary-integral algorithm, J. Fluid Mech., 391 (1999), 249–92.CrossRefGoogle Scholar