Adaptive Galerkin boundary element methods with panel clustering

Abstract

In this paper, we will propose a boundary element method for solving classical boundary integral equations on complicated surfaces which, possibly, contain a large number of geometric details or even uncertainties in the given data. The (small) size of such details is characterised by a small parameter \(\varepsilon\) and the regularity of the solution is expected to be low in such zones on the surface (which we call the wire-basket zones). We will propose the construction of an initial discretisation for such type of problems. Afterwards standard strategies for boundary element discretisations can be applied such as the h, p, and the adaptive hp-version in a straightforward way.

For the classical boundary integral equations, we will prove the optimal approximation results of our so-called wire-basket boundary element method and discuss the stability aspects. Then, we construct the panel-clustering and \(\mathcal{H}\)-matrix approximations to the corresponding Galerkin BEM stiffness matrix. The method is shown to have an almost linear complexity with respect to the number of degrees of freedom located on the wire basket.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Babuška I., Guo B. (1989) Regularity of the solution of elliptic problems with piecewise analytic data Part II. The trace spaces and application to the boundary value problems with nonhomogeneous boundary conditions. SIAM J. Math. Anal. 20, 763–781

    Article  MathSciNet  MATH  Google Scholar 

  2. 2.

    Bernstein S.N. (1926) Leçons sur les propriétés extrémales et la meilleure approximation des fonctions analytiques d’une variable réelle. Gauthier-Villars, Paris

    Google Scholar 

  3. 3.

    Börm S., Löhndorf M., Melenk J-M. (2005) Approximation of integral operators by variable-order interpolation. Numer. Math. 99, 605–643

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Brückner-Foit, A., Huang, X., Motoyashiki, Y. Mesoscopic simulations of damage accumulation under fatigue loading. ECF 15, Stockholm (2004)

  5. 5.

    Chen G., Zhou J. (1992) Boundary element methods. Academic, New York

    Google Scholar 

  6. 6.

    Costabel M. (1988) Boundary integral operators on Lipschitz domains: elementary results. SIAM J. Math. Anal. 19, 613–626

    Article  MathSciNet  MATH  Google Scholar 

  7. 7.

    Elschner J. (1992) The double layer potential operator over polyhedral domains I: solvability in weighted Sobolev spaces. Appl. Anal. 45, 117–134

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Elschner J. (1995). On the exponential convergence of some boundary element methods for Laplace’s equation in non-smooth domains. In: Costabel M. et al. (eds). Boundary Value Problems and Integral Equations in Nonsmooth Domains. Dekker, New York, pp. 69–80

    Google Scholar 

  9. 9.

    Erichsen S., Sauter S. (1998) Efficient automatic quadrature in 3-d Galerkin BEM. Comp. Meth. Appl. Mech. Eng. 157, 215–224

    Article  MathSciNet  MATH  Google Scholar 

  10. 10.

    Grasedyck L., Hackbusch W. (2003) Construction and arithmetics of \(\mathcal{H}\)-matrices. Computing 70, 295–334

    Article  MathSciNet  MATH  Google Scholar 

  11. 11.

    Grisvard P. (1985) Elliptic problems in non-smooth domains. Pitman, London

    Google Scholar 

  12. 12.

    Guo B., Babuška I. (1997) Regularity of the solutions for problems on nonsmooth domains in \(\mathbb{R}^{3}\), Part 2: regularity in a neighborhood of edges Proc. R. Soc. Edinb., Sect. A127, 517–545

    Google Scholar 

  13. 13.

    Hackbusch W. (1995) Integral equations. Theory and numerical treatment. ISNM 128. Birkhäuser, Basel

    Google Scholar 

  14. 14.

    Hackbusch W. (1999) A sparse matrix arithmetic based on \(\mathcal{H}\)-matrices. Part I: introduction to \(\mathcal{H}\)-matrices. Computing 62, 89–108

    Article  MathSciNet  MATH  Google Scholar 

  15. 15.

    Hackbusch W., Khoromskij B.N. (2000) A sparse \(\mathcal{H}\)-matrix arithmetic. Part II: application to multi-dimensional problems. Computing 64, 21–47

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Hackbusch W., Khoromskij B.N. (2000) \(\mathcal{H}\)-Matrix approximation on graded meshes. In: Whiteman J.R. (eds) The Mathematics of Finite Elements and Applications X, MAFELAP 1999 Chapter 19. Elsevier, Amsterdam, pp. 307–316

    Google Scholar 

  17. 17.

    Hackbusch W., Khoromskij B.N. (2001) Towards \(\mathcal{H}\)-matrix approximation of the linear complexity. Oper. Theory: Adv. Appl. 121, 194–220

    MathSciNet  Google Scholar 

  18. 18.

    Hackbusch W., Khoromskij B.N., Sauter S. (2000) On \(\mathcal{H}^{2}\)-matrices. In: Bungartz H.-J., Hoppe R., Zenger C. (eds). Lectures on Applied Mathematics. Springer, Berlin Heidelberg New York, pp. 9–29

    Google Scholar 

  19. 19.

    Heuer N., Stephan E.P. (1998) Boundary integral operators in countably normed spaces. Math. Nachr. 191, 123–151

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Heuer N., Maischak M., Stephan E.P. (1999) Exponential convergence of the hp-version for the boundary element method on open surfaces. Numer. Math. 83(4): 641–666

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Khoromskij B.N. (2003) Hierarchical matrix approximation to Green’s function via boundary concentrated FEM. J. Numer. Math. 11, 195–223

    Article  MathSciNet  MATH  Google Scholar 

  22. 22.

    Khoromskij B.N., Melenk J.M. (2003) Boundary concentrated finite element methods. SIAM J. Numer. Anal. 41, 1–36

    Article  MathSciNet  MATH  Google Scholar 

  23. 23.

    Khoromskij B.N., Melenk J.M. (2002) Efficient direct solver for the boundary concentrated FEM in 2D. Computing 69, 91–117

    Article  MathSciNet  MATH  Google Scholar 

  24. 24.

    Melenk, J.M. hp-finite element methods for singular perturbations. Springer Lect. Notes Math., 1796 (2002)

  25. 25.

    Nicaise, S., Sauter, S.A. Efficient numerical solution of Neumann problems on compliclated domains. Universität Zürich, Calcolo 43, 95–120 (2006)

  26. 26.

    Petersdorff T.v., Stephan E.P. (1990) Decomposition in edge and corner singularities for the solution of the Dirichlet problem of the Laplacian in a polyhedron. Math. Nachr. 149, 71–104

    MathSciNet  Google Scholar 

  27. 27.

    Sauter, S.A. Über die effiziente Verwendung des Galerkinverfahrens zur Lösung Fredholmscher Integralgleichungen. Ph.D. thesis, Inst. f. Prakt. Math., Universität Kiel (1992)

  28. 28.

    Sauter, S., Schwab, C. Randelementmethoden. B.G. Teubner (2004)

  29. 29.

    Schwab C. (1998) p- and hp-Finite Element Methods. Oxford University Press, Oxford

    Google Scholar 

  30. 30.

    Stephan E.P. (2000) Multilevel methods for the h-, p-, and hp-versions of the boundary element method. J. Comput. Appl. Math. 125, 503–519

    Article  MathSciNet  MATH  Google Scholar 

  31. 31.

    Tadmor E. (1986) The exponential accuracy of Fourier and Chebychev differencing methods. SIAM J. Numer. Anal. 23, 1–23

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Wolfgang Hackbusch.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Hackbusch, W., Khoromskij, B.N. & Sauter, S. Adaptive Galerkin boundary element methods with panel clustering. Numer. Math. 105, 603–631 (2007). https://doi.org/10.1007/s00211-006-0047-9

Download citation

Mathematics Subject Classification

  • 65F50
  • 65F30