×

Numerical integration of functions with endpoint singularities and/or complex poles in 3D Galerkin boundary element methods. (English) Zbl 1100.65019

Summary: We propose special strategies to compute 1D integrals of functions having weakly or strong singularities at the endpoints of the interval of integration or complex poles close to the domain of integration. As application of the proposed strategies, we compute a four dimensional integral arising from 3D Galerkin boundary element methods applied to hypersingular boundary integral equations.

MSC:

65D32 Numerical quadrature and cubature formulas
65N38 Boundary element methods for boundary value problems involving PDEs
41A55 Approximate quadratures
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Andrä, H. and Schnack, E., Integration of singular Galerkin-type boundary element integrals for 3D elasticity problems, Numer. Math., 76 (1997), 143-165. · Zbl 0879.73078 · doi:10.1007/s002110050257
[2] Diligenti, M. and Monegato, G., Integral evaluation in the BEM solution of (hy- per)singular integral equations. 2D problems on polygonal domains, J. Comput. Appl. Math., 81 (1997), 29-57. · Zbl 0879.65074 · doi:10.1016/S0377-0427(97)00007-1
[3] Elliott, D. and Prössdorf, S., An algorithm for the approximate solution of integral equations of Mellin type, Numer. Math., 70 (1995), 427-452. · Zbl 0828.65143 · doi:10.1007/s002110050127
[4] Elliott, D. and Venturino, E., Sigmoidal transformations and the Euler-Maclaurin ex- pansion for evaluating certain Hadamard finite-part integrals, Numer. Math., 77 (1997), 453-465. · Zbl 0886.65021 · doi:10.1007/s002110050295
[5] Haas, M. and Kuhn, G., A symmetric Galerkin BEM implementation for 3D elastostatic problems with an extension to curved elements, Comp. Mech., 28 (2002), 250-259. · Zbl 1062.74059 · doi:10.1007/s00466-001-0285-8
[6] Korobov, N. M., Number-Theoretic Methods of Approximate Analysis, GIFL, Moscow, 1963. · Zbl 0115.11703
[7] Monegato, G., Numerical evaluation of hypersingular integrals, J. Comput. Appl. Math., 50 (1994), 9-31. · Zbl 0818.65016 · doi:10.1016/0377-0427(94)90287-9
[8] Monegato, G. and Scuderi, L., High order methods for weakly singular integral equations with non smooth input functions, Math. Comp., 67 (1998), 1493-1515. · Zbl 0907.65139 · doi:10.1090/S0025-5718-98-01005-9
[9] , Numerical integration of functions with boundary singularities, J. Comput. Appl. Math., Special Issue: “Numerical Evaluation of Integrals”, D. Laurie, R. Cools eds., 112 (1999), 201-214. 895 · Zbl 0940.65027 · doi:10.1016/S0377-0427(99)00230-7
[10] Mori, M. and Sugihara, M., The double-exponential transformation in numerical analy- sis, J. Comput. Appl. Math., Numerical Analysis 2000, Vol. V, Quadrature and orthog- onal polynomials, 127 (2001), 287-296. · Zbl 0971.65015 · doi:10.1016/S0377-0427(00)00501-X
[11] Prössdorf, S. and Rathsfeld, A., Quadrature methods for strongly elliptic Cauchy sin- gular integral equations on an interval, in: H. Dym (Ed.), The Gohberg Anniversary Collection, Vol. 2: Topics in Analysis and Operator Theory, Birkhäuser, Basel, (1991), 435-471. · Zbl 0724.65129
[12] Sauter, S. and Lage, C., Transformation of hypersingular integrals and black-box cuba- ture, Math. Comp., 70 (2000), 223-250. · Zbl 0958.65123 · doi:10.1090/S0025-5718-00-01261-8
[13] Sauter, S. and Schwab, C., Quadrature for hp-Galerkin BEM in R I 3, Numer. Math., 78 (1997), 211-258. · Zbl 0901.65069 · doi:10.1007/s002110050311
[14] Sidi, A., A new variable transformation for numerical integration, in: H. Braess, G. Hämmerlin, eds., Numerical Integration IV, Birkhäuser Verlag, Basel, ISNM 112 (1993), 359-373. · Zbl 0791.41027
[15] Sugihara, M., Optimality of the double exponential formula - functional analysis ap- proach, Numer. Math., 75 (1997), 379-395. · Zbl 0868.41019 · doi:10.1007/s002110050244
[16] Takahasi, H. and Mori, M., Double exponential formulas for numerical integration, Publ. RIMS, Kyoto Univ., 9 (1974), 721-741. · Zbl 0293.65011 · doi:10.2977/prims/1195192451
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.