Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

BEM with linear complexity for the classical boundary integral operators


Authors: Steffen Börm and Stefan A. Sauter
Journal: Math. Comp. 74 (2005), 1139-1177
MSC (2000): Primary 65N38, 65D05
DOI: https://doi.org/10.1090/S0025-5718-04-01733-8
Published electronically: December 8, 2004
MathSciNet review: 2136997
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Alternative representations of boundary integral operators corresponding to elliptic boundary value problems are developed as a starting point for numerical approximations as, e.g., Galerkin boundary elements including numerical quadrature and panel-clustering. These representations have the advantage that the integrands of the integral operators have a reduced singular behaviour allowing one to choose the order of the numerical approximations much lower than for the classical formulations. Low-order discretisations for the single layer integral equations as well as for the classical double layer potential and the hypersingular integral equation are considered. We will present fully discrete Galerkin boundary element methods where the storage amount and the CPU time grow only linearly with respect to the number of unknowns.


References [Enhancements On Off] (What's this?)

  • 1. R. Bank and J. Xu. An Algorithm for Coarsening Unstructured Meshes. Numer. Math., 73(1):1-36, 1996. MR 1379277 (97c:65055)
  • 2. S. Börm, M. Löhndorf, and M. Melenk. Approximation of integral operators by variable-order interpolation. To appear in Numer. Math.
  • 3. S. Börm, N. Krzebek, and S. A. Sauter. May the singular integrals in BEM be replaced by Zero? Technical Report 86, Max-Planck-Institut, Leipzig, Germany, 2003. To appear in Comp. Meth. Appl. Mech. Eng.
  • 4. P. Clément. Approximation by Finite Element Functions using Local Regularization. RAIRO, Sér. Rouge Anal. Numér., R-2:77-84, 1975. MR 0400739 (53:4569)
  • 5. W. Dahmen, B. Faermann, I. Graham, W. Hackbusch, and S. Sauter. Inverse Inequalities on Non-Quasiuniform Meshes and Application to the Mortar Element Method. Technical Report 24, Max-Planck-Institut, Leipzig, Germany, 2001. Math Comp. 73:1107-1138, 2004. MR 2047080
  • 6. R. DeVore and G. Lorentz. Constructive Approximation. Springer-Verlag, New York, 1993. MR 1261635 (95f:41001)
  • 7. J. Elschner. The Double Layer Potential Operator over Polyhedral Domains II: Spline Galerkin Methods. Math. Meth. Appl. Sci., 15:23-37, 1992. MR 1144458 (94i:65130)
  • 8. S. Erichsen and S. Sauter. Efficient automatic quadrature in 3-d Galerkin BEM. Comp. Meth. Appl. Mech. Eng., 157:215-224, 1998. MR 1634288 (99e:65163)
  • 9. F. T. Johnson. A General Panel Method for the Analysis and Design of Arbitrary Configurations in Incompressible Flows. NASA Report 3079, NASA Ames Research Center, USA, 1980.
  • 10. I. Graham, W. Hackbusch, and S. Sauter. Finite Elements on Degenerate Meshes: Inverse-type Inequalities and Applications. To appear in IMA J. Numer. Anal.
  • 11. L. Grasedyck and W. Hackbusch. Construction and arithmetics of ${\mathcal{H}}$-matrices. Computing, 70(4):295-334, 2003. MR 2011419 (2004i:65035)
  • 12. L. Greengard and V. Rokhlin. A New Version of the Fast Multipole Method for the Laplace Equation in Three Dimensions. Acta Numerica, 6:229-269, 1997. MR 1489257 (99c:65012)
  • 13. M. Griebel and M. Schweitzer. A P article-Partition of Unity Method-Part III: A Multilevel Solver. SIAM J. Sci. Comp., 24(2):377-409, 2002. MR 1951047 (2004e:65125)
  • 14. W. Hackbusch. Integral Equations. ISNM. Birkhäuser, 1995. MR 1350296 (96h:45001)
  • 15. W. Hackbusch, C. Lage, and S. Sauter. On the Efficient Realization of Sparse Matrix Techniques for Integral Equations with Focus on Panel Clustering, Cubature and Software Design Aspects. In W. Wendland, editor, Boundary Element Topics, number 95-4, pages 51-76, Springer-Verlag, Berlin, 1997. MR 1655236 (99h:65204)
  • 16. W. Hackbusch and Z. Nowak. On the Fast Matrix Multiplication in the Boundary Element Method by Panel-Clustering. Num. Math., 54:463-491, 1989. MR 0972420 (89k:65162)
  • 17. 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 1228511 (95c:65204)
  • 18. W. Hackbusch and S. Sauter. Composite Finite Elements for the Approximation of PDEs on Domains with Complicated Micro-Structures. Numer. Math., 75(4):447-472, 1997. MR 1431211 (97k:65251)
  • 19. W. McLean. Strongly Elliptic Systems and Boundary Integral Equations. Cambridge, Univ. Press, 2000. MR 1742312 (2001a:35051)
  • 20. J. Melenk and I.Babuska. The Partition of Unity Finite Element Method: Basic Theory andApplications. Comp. Meth. Appl. Mech. Eng., 139:289-314, 1996. MR 1426012 (97k:65258)
  • 21. J. Nédélec. Integral Equations with Nonintegrable Kernels. Integral Equations Oper. Theory, 5:562-572, 1982. MR 0665149 (84i:45011)
  • 22. J. C. Nédélec. Acoustic and Electromagnetic Equations. Springer, New York, 2001. MR 1822275 (2002c:35003)
  • 23. T. J. Rivlin. The Chebyshev Polynomials. Wiley, New York, 1974. MR 0450850 (56:9142)
  • 24. V. Rokhlin. Rapid solution of integral equations of classical potential theory. Journal of Computational Physics, 60(2):187-207, 1985. MR 0805870 (86k:65120)
  • 25. S. Sauter. Variable order panel clustering. Computing, 64:223-261, 2000. MR 1767055 (2001e:65197)
  • 26. S. Sauter and C. Lage. Transformation of hypersingular integrals and black-box cubature. Math. Comp., 70(97-17):223-250, 2001. MR 1803126 (2001k:65053)
  • 27. S. Sauter and C. Schwab. Randelementmethoden. Teubner, Leipzig, to appear in 2004.
  • 28. S. A. Sauter. Über die effiziente Verwendung des Galerkinverfahrens zur Lösung Fredholmscher Integralgleichungen. Ph.D. thesis, Inst. f. Prakt. Math., Universität Kiel, 1992.
  • 29. S. A. Sauter and C. Schwab. Quadrature for hp-Galerkin BEM in $R^3$. Numer. Math., 78(2):211-258, 1997. MR 1485998 (99f:65207)
  • 30. J. Tausch and J. White. Wavelet-like Bases for Integral Equations on Surfaces with Complex Geometry. In J. Wang, M. Allen, B. Chen, and T. Mathew, editors, IMACS Series in Computational and Applied Mathematics, 1998.
  • 31. T.v.Petersdorff and C. Schwab. Fully Discrete Multiscale Galerkin BEM. In W. Dahmen, P. Kurdila, and P. Oswald, editors, Multiscale Wavelet Methods for Partial Differential Equations, pages 287-346, New York, 1997. Academic Press. MR 1475002 (99a:65158)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65N38, 65D05

Retrieve articles in all journals with MSC (2000): 65N38, 65D05


Additional Information

Steffen Börm
Affiliation: Max-Planck-Institut für Mathematik in den Naturwissenschaften, Inselstraße 22–26, 04103 Leipzig, Germany
Email: sbo@mis.mpg.de

Stefan A. Sauter
Affiliation: Institut für Mathematik, Universität Zürich, Winterthurerstr. 190, CH-8057 Zürich, Switzerland
Email: stas@amath.unizh.ch

DOI: https://doi.org/10.1090/S0025-5718-04-01733-8
Keywords: BEM, data-sparse approximation, ${\mathcal H}^2$-matrices
Received by editor(s): September 9, 2003
Received by editor(s) in revised form: March 29, 2004
Published electronically: December 8, 2004
Additional Notes: This work was supported by the Swiss National Science Foundation, Grant 21-6176400.
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society