# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Universal quadratures for boundary integral equations on two-dimensional domains with corners. (English) Zbl 1201.65213
Summary: We describe the construction of a collection of quadrature formulae suitable for the efficient discretization of certain boundary integral equations on a very general class of two-dimensional domains with corner points. The resulting quadrature rules allow for the rapid high-accuracy solution of Dirichlet boundary value problems for Laplace’s equation and the Helmholtz equation on such domains under a mild assumption on the boundary data. Our approach can be adapted to other boundary value problems and certain aspects of our scheme generalize to the case of surfaces with singularities in three dimensions. The performance of the quadrature rules is illustrated with several numerical examples.

##### MSC:
 65N38 Boundary element methods (BVP of PDE) 35J05 Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation
Full Text:
##### References:
 [1] , Handbook of mathematical functions (1964) · Zbl 0171.38503 [2] Atkinson, K.: The numerical solution of integral equations of the second kind, (1997) · Zbl 0899.65077 [3] Atkinson, K. E.; Graham, I.: Iterative variants of the Nyström method for second kind boundary integral operators, SIAM journal on scientific computing 13, 694-722 (1990) [4] Björck, A.: Numerical methods for least squares problems, (1996) · Zbl 0847.65023 [5] Bremer, J.; Gimbutas, Z.; Rokhlin, V.: A nonlinear optimization procedure for generalized Gaussian quadratures, SIAM journal of scientific computing 32, 1761-1788 (2010) · Zbl 1215.65045 · doi:10.1137/080737046 [6] Bremer, J.; Rokhlin, V.: Efficient discretization of Laplace boundary integral equations on polygonal domains, Journal of computational physics 229, 2507-2525 (2010) · Zbl 1185.65219 · doi:10.1016/j.jcp.2009.12.001 [7] Bruno, O. P.; Ovall, J. S.; Turc, C.: A high-order integral algorithm for highly singular PDE solutions in Lipschitz domains, Computing 84, 149-181 (2009) · Zbl 1176.65139 · doi:10.1007/s00607-009-0031-1 [8] Coifman, R.; Meyer, Y.: Wavelets: Calderón -- Zygmund and multilinear operators, (1997) · Zbl 0916.42023 [9] Colton, D.; Kress, R.: Inverse acoustic and electromagnetic scattering theory, (1992) · Zbl 0760.35053 [10] Greengard, L.; Gueyffier, D.; Martinsson, P.; Rokhlin, V.: Fast direct solvers for integral equations in complex three-dimensional domains, Acta numerica, 243-275 (2009) · Zbl 1176.65141 · doi:10.1017/S0962492906410011 [11] Gu, M.; Eisenstat, S.: Efficient algorithms for computing a strong rank-revealing QR factorization, SIAM journal on scientific computing 17, 848-869 (1996) · Zbl 0858.65044 · doi:10.1137/0917055 [12] Hackbusch, W.: Integral equations: theory and numerical treatment, (1995) · Zbl 0823.65139 [13] Helsing, J.; Ojala, R.: Corner singularities for elliptic problems: integral equations, graded meshes, quadrature, and compressed inverse preconditioning, Journal of computational physics 227, 8820-8840 (2008) · Zbl 1152.65114 · doi:10.1016/j.jcp.2008.06.022 [14] Koshliakov, N.; Smirnov, M.; Gliner, E. B.: Differential equations of mathematical physics, (1964) · Zbl 0115.30701 [15] Kress, R.: A Nyström method for boundry integral equations in domains with corners, Numerische Mathematik 58 (1991)