Abstract
The incorporation of analytical kernel information is exploited in the construction of Nyström discretization schemes for integral equations modeling planar Helmholtz boundary value problems. Splittings of kernels and matrices, coarse and fine grids, high-order polynomial interpolation, product integration performed on the fly, and iterative solution are some of the numerical techniques used to seek rapid and stable convergence of computed fields in the entire computational domain.
Similar content being viewed by others
References
Atkinson, K.E.: The numerical Solution of Integral Equations of the Second Kind. Cambridge University Press, Cambridge (1997)
Barnett, A.H.: Evaluation of layer potentials close to the boundary for Laplace and Helmholtz problems on analytic planar domains. SIAM J. Sci. Comput. 36, A427–A451 (2014)
Betcke, T., Chandler-Wilde, S.N., Graham, I.G., Langdon, S., Lindner, M.: Condition number estimates for combined potential integral operators in acoustics and their boundary element discretisation. Numer. Meth. Part. D. E. 27, 31–69 (2011)
Bremer, J., Gillman, A., Martinsson, P.G.: A high-order accurate accelerated direct solver for acoustic scattering from surfaces. arXiv:1308.6643v1 [math.NA] (2013)
Colton, D., Kress, R.: Inverse acoustic and electromagnetic scattering theory. Springer, Berlin (1998)
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, 245–272 (2014)
Helsing, J., Ojala, R.: On the evaluation of layer potentials close to their sources. J. Comput. Phys. 227, 2899–2921 (2008)
Helsing, J.: Integral equation methods for elliptic problems with boundary conditions of mixed type. J. Comput. Phys. 228, 8892–8907 (2009)
Helsing, J.: Solving integral equations on piecewise smooth boundaries using the RCIP method: a tutorial. Abstr. Appl. Anal. 2013 (2013). article ID 938167
Helsing, J., Karlsson, A.: An accurate boundary value problem solver applied to scattering from cylinders with corners. IEEE Trans. Antennas Propag. 61, 3693–3700 (2013)
Helsing, J., Karlsson, A.: An explicit kernel-split panel-based Nyström scheme for integral equations on axially symmetric surfaces. J. Comput. Phys. 272, 686–703 (2014)
Ho, K.L., Greengard, L.: A fast direct solver for structured linear systems by recursive skeletonization. SIAM J. Sci. Comput. 34, A2507–A2532 (2012)
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)
Kolm, P., Rokhlin, V.: Numerical quadratures for singular and hypersingular integrals. Comput. Math. Appl. 41, 327–352 (2001)
Kress, R.: Boundary integral equations in time-harmonic acoustic scattering. Mathl Comput. Modelling 15, 229–243 (1991)
Kress, R.: Linear Integral equations, 2nd ed. Springer, New York (1999)
Mitrea, M.: Boundary value problems and Hardy spaces associated to the Helmholtz equation in Lipschitz domains. J. Math. Anal. Appl. 202, 819–842 (1996)
Saad, Y., Schultz, M.H.: GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comp. 7, 856–869 (1986)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by: Zydrunas Gimbutas
Rights and permissions
About this article
Cite this article
Helsing, J., Holst, A. Variants of an explicit kernel-split panel-based Nyström discretization scheme for Helmholtz boundary value problems. Adv Comput Math 41, 691–708 (2015). https://doi.org/10.1007/s10444-014-9383-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10444-014-9383-y