Transformation of hypersingular integrals and black-box cubature

Sauter, S.; Lage, C.

doi:10.1090/S0025-5718-00-01261-8

Transformation of hypersingular integrals and black-box cubature

Authors: S. A. Sauter and C. Lage
Journal: Math. Comp. 70 (2001), 223-250
MSC (2000): Primary 65N38, 65R10, 65R20
DOI: https://doi.org/10.1090/S0025-5718-00-01261-8
Published electronically: June 12, 2000
MathSciNet review: 1803126
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Abstract:

In this paper, we will consider hypersingular integrals as they arise by transforming elliptic boundary value problems into boundary integral equations. First, local representations of these integrals will be derived. These representations contain so-called finite-part integrals. In the second step, these integrals are reformulated as improper integrals. We will show that these integrals can be treated by cubature methods for weakly singular integrals as they exist in the literature.

1. D. G. Anderson.
Gaussian quadrature formula for $\int -\ln \left( x\right) f\left( x\right) dx$ .
Math. Comp., 19:477-481, 1965. MR 31:2826
2. K. Atkinson.
Solving Integral Equations on Surfaces in Space.
In G. Hämmerlin and K. Hoffmann, editors, Constructive Methods for the Practical Treatment of Integral Equations, pages 20-43. Birkhäuser: ISNM, 1985. CMP 19:10
3. M. Costabel.
Boundary integral operators on Lipschitz domains: Elementary results.
SIAM, J. Math. Anal., 19:613-626, 1988. MR 89h:35090
4. S. Erichsen and S. Sauter.
Efficient automatic quadrature in 3-d Galerkin BEM.
Comp. Meth. Appl. Mech. Eng., 157:215-224, 1998. MR 99e:65163
5. I. Graham, W. Hackbusch, and S. Sauter.
Discrete boundary element methods on general meshes in 3d.
Technical Report 97/19, University of Bath, U.K., 1997.
Mathematics Preprint, to appear in Numer. Math.
6. M. Guiggiani.
Direct Evaluation of Hypersingular Integrals in 2D BEM.
In W. Hackbusch, editor, Proc. Of 7th GAMM Seminar on Numerical Techniques for BEM, Kiel 1991, pages 23-34, Braunschweig, 1991. Vieweg.
7. M. Guiggiani and A. Gigante.
A general algorithm for multidimensional Cauchy principal value integrals in the boundary element method.
ASME J. Appl. Mech., 57:907-915. MR 93b:73037
8. W. Hackbusch.
Integral Equations.
ISNM. Birkhäuser, 1995. MR 96h:45001
9. W. Hackbusch and S. Sauter.
On the Efficient Use of the Galerkin Method to Solve Fredholm Integral Equations.
Applications of Mathematics, 38(4-5):301-322, 1993. MR 95c:65204
10. H. Han.
A Boundary Element Method for Signorini Problems in Three Dimensions.
Numer. Math., 60:63-76, 1991. MR 93a:65166
11. H. Han.
The Boundary Integro-Differential Equations of Three-Dimensional Neumann Problem in Linear Elasticity.
Numer. Math., 68(2):269-281, 1994. MR 95e:65109
12. R. Kieser.
Über einseitige Sprungrelationen und hypersinguläre Operatoren in der Methode der Randelemente.
PhD thesis, Mathematisches Institut A, Universität Stuttgart, Germany, 1990.
13. R. Kieser, C. Schwab, and W. L. Wendland.
Numerical Evaluation of Singular and Finite-Part Integrals on Curved Surfaces Using Symbolic Manipulation.
Computing, 49:279-301, 1992. MR 93e:65037
14. C. Lage.
Software Development for Boundary Element Mehtods: Analysis and Design of Efficient Techniques (in German).
PhD thesis, Lehrstuhl Prakt. Math., Universität Kiel, 1995.
15. J. Lyness.
Applications of extrapolation techniques to multidimensional quadrature of some integrand functions with a singularity.
J. Comp. Phys., 20:346-364, 1976. MR 52:15972
16. J. Lyness.
An error functional expression for N-dimensional quadrature with an integrand function singular at a point.
Math. Comp., 30:1-23, 1976. MR 53:11976
17. J. Lyness and G. Monegato.
Quadrature error functional expansions for the simplex when the integrand function has singularities at vertices.
Math. Comp., 34:213-225, 1980. MR 80m:65017
18. S. Mikhlin and S. Prößdorf.
Singular Integral Operators.
Springer-Verlag, Heidelberg, 1986. MR 88e:47097
19. J. Nédélec.
Curved Finite Element Methods for the Solution of Singular Integral Equations on Surfaces in $\mathbf{R}^3$ .
Comput. Meth. Appl. Mech. Engrg., 8:61-80, 1976. MR 58:19275
20. J. Nédélec.
Integral Equations with Non Integrable Kernels.
Integral Equations Oper. Theory, 5:562-572, 1982. MR 84i:45011
21. O. Oleinik, A. Shamaev, and G. Yosifian.
Mathematical Problems in Elasticity and Homogenization.
North-Holland, Amsterdam, 1992. MR 93k:35025
22. S. Sauter and C. Lage.
Transformation of hypersingular integrals and black-box cubature (extended version).
Technical Report 97-17, Universität Kiel, 1997.
(Available via WWW-address: http://www.numerik.uni-kiel.de/reports/1997/).
23. S. A. Sauter.
Über die effiziente Verwendung des Galerkinverfahrens zur Lösung Fredholmscher Integralgleichungen.
PhD thesis, Inst. f. Prakt. Math., Universität Kiel, 1992.
24. S. A. Sauter and A. Krapp.
On the Effect of Numerical Integration in the Galerkin Boundary Element Method.
Numer. Math., 74(3):337-360, 1996. MR 97j:65189
25. C. Schwab and W. Wendland.
Kernel Properties and Representations of Boundary Integral Operators.
Math. Nachr., 156:187-218, 1992. MR 94g:65135
26. C. Schwab and W. Wendland.
On Numerical Cubatures of Singular Surface Integrals in Boundary Element Methods.
Numer. Math., 62:343-369, 1992. MR 93h:65035
27. T.von Petersdorff and C. Schwab.
Fully Discrete Multiscale Galerkin BEM.
In W. Dahmen, P. Kurdila, and P. Oswald, editors, Multiresolution Analysis and Partial Differential Equations, pages 287-346, New York, 1997. Academic Press. MR 99a:65158
28. W. Wendland.
Strongly elliptic boundary integral equations.
In A. Iserles and M. Powell, editors, The State of the Art in Numerical Analysis, pages 511-561, Oxford, 1987. Clarendon Press. MR 88m:65209