Kropinski, Mary Catherine A.; Nigam, Nilima Fast integral equation methods for the Laplace-Beltrami equation on the sphere. (English) Zbl 1302.65262 Adv. Comput. Math. 40, No. 2, 577-596 (2014). Summary: Integral equation methods for solving the Laplace-Beltrami equation on the unit sphere in the presence of multiple “islands” are presented. The surface of the sphere is first mapped to a multiply-connected region in the complex plane via a stereographic projection. After discretizing the integral equation, the resulting dense linear system is solved iteratively using the fast multipole method for the 2D Coulomb potential in order to calculate the matrix-vector products. This numerical scheme requires only \(O(N)\) operations, where \(N\) is the number of nodes in the discretization of the boundary. The performance of the method is demonstrated on several examples. Cited in 7 Documents MSC: 65N38 Boundary element methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations 58J05 Elliptic equations on manifolds, general theory Keywords:Laplace-Beltrami boundary value problems; integral equation method; fast multipole methods; numerical example; stereographic projection; Coulomb potential PDFBibTeX XMLCite \textit{M. C. A. Kropinski} and \textit{N. Nigam}, Adv. Comput. Math. 40, No. 2, 577--596 (2014; Zbl 1302.65262) Full Text: DOI arXiv References: [1] Bertalmio, M., Cheng, L.T., Osher, S., Sapiro, G.: Variational problems and partial differential equations on implicit surfaces. J. Comput. Phys. 174(2), 759-780 (2001). doi:10.1006/jcph.2001.6937. http://www.sciencedirect.com/science/article/pii/S0021999101969372 · Zbl 0991.65055 [2] Bonner, B.D., Graham, I.G., Smyshlyaev, V.P.: The computation of conical diffraction coefficients in high-frequency acoustic wave scattering. SIAM J. Numer. 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