Numerical Algorithms

, Volume 67, Issue 1, pp 1–32 | Cite as

HILBERT — a MATLAB implementation of adaptive 2D-BEM

HILBERT Is a Lovely Boundary Element Research Tool
  • Markus Aurada
  • Michael Ebner
  • Michael Feischl
  • Samuel Ferraz-Leite
  • Thomas Führer
  • Petra Goldenits
  • Michael Karkulik
  • Markus Mayr
  • Dirk Praetorius
Open Access
Original Paper

Abstract

We report on the Matlab program package HILBERT. It provides an easily-accessible implementation of lowest order adaptive Galerkin boundary element methods for the numerical solution of the Poisson equation in 2D. The library was designed to serve several purposes: The stable implementation of the integral operators may be used in research code. The framework of Matlab ensures usability in lectures on boundary element methods or scientific computing. Finally, we emphasize the use of adaptivity as general concept and for boundary element methods in particular. In this work, we summarize recent analytical results on adaptivity in the context of BEM and illustrate the use of HILBERT. Various benchmarks are performed to empirically analyze the performance of the proposed adaptive algorithms and to compare adaptive and uniform mesh-refinements. In particular, we do not only focus on mathematical convergence behavior but also on the usage of critical system resources such as memory consumption and computational time. In any case, the superiority of the proposed adaptive approach is empirically supported.

Keywords

Boundary element methods Adaptive mesh-refinement A posteriori error estimation Matlab implementation 

Mathematics Subject Classifications (2010)

65N38 65Y20 65N50 

References

  1. 1.
    Alberty, J., Carstensen, C., Funken, S.A.: Remarks around 50 lines of Matlab: short finite element implementation. Numer. Algoritm. 20, 117–137 (1999)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Aurada, M., Ebner, M., Feischl, M., Ferraz-Leite, S., Führer, T., Goldenits, P., Karkulik, M., Mayr, M., Praetorius, D.: HILBERT, a MATLAB implementation of adaptive BEM (Release 3). ASC Report 26/2013, Institute for Analysis and Scientific Computing, Vienna University of Technology (2013)Google Scholar
  3. 3.
    Aurada, M., Feischl, M., Führer, T., Karkulik, M., Praetorius, D.: Efficiency and optimality of some weighted-residual error estimator for adaptive 2D boundary element methods. Comput. Methods Appl. Math. 13, 305–332 (2013)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Aurada, M., Feischl, M., Führer, T., Karkulik, M., Melenk, J.M., Praetorius, D.: Inverse estimates for elliptic integral operators and their application to the adaptive coupling of FEM and BEM: ASC Report 07/2012. Institute for Analysis and Scientific Computing, Vienna University of Technology (2012)Google Scholar
  5. 5.
    Aurada, M., Ferraz-Leite, S., Goldenits, P., Karkulik, M., Mayr, M., Praetorius, D.: Convergence of adaptive BEM for some mixed boundary value problem. Appl. Numer. Math. 62, 226–245 (2012)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Aurada, M., Ferraz-Leite, S., Praetorius, D.: Estimator reduction and convergence of adaptive BEM. Appl. Numer. Math. 62, 787–801 (2012)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Cascon, J., Kreuzer, C., Nochetto, R., Siebert, K.: Quasi-optimal convergence rate for an adaptive finite element method. SIAM J. Numer. Anal. 46, 2524–2550 (2008)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Carstensen, C., Praetorius, D.: Averaging techniques for the effective numerical solution of Symm’s integral equation of the first kind. SIAM J. Sci. Comput. 27, 1226–1260 (2006)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Carstensen, C., Praetorius, D.: Averaging techniques for the a posteriori BEM error control for a hypersingular integral equation in two dimensions. SIAM J. Sci. Comput. 29, 782–810 (2007)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Carstensen, C., Praetorius, D.: Convergence of adaptive boundary element methods. J. Integr. Equ. Appl. 24, 1–23 (2012)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Carstensen, C., Stephan, E.P.: A posteriori error estimates for boundary element methods. Math. Comp. 64, 483–500 (1995)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Carstensen, C., Stephan, E.P.: Adaptive boundary element methods for some first kind integral equations. SIAM J. Numer. Anal. 33, 2166–2183 (1996)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Dörfler, W.: A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33, 1106–1124 (1996)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Dörfler, W., Nochetto, R.: Small data oscillation implies the saturation assumption. Numer. Math. 91, 1–12 (2002)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Erath, C., Ferraz-Leite, S., Funken, S., Praetorius, D.: Energy norm based a posteriori error estimation for boundary element methods in two dimensions. Appl. Numer. Math. 59, 2713–2734 (2009)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Erath, C., Funken, S., Goldenits, P., Praetorius, D.: Simple error estimators for the Galerkin BEM for some hypersingular integral equation in 2D. Appl. Anal. 92, 1194–1216 (2013)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Feischl, M., Führer, T., Karkulik, M., Melenk, J.M., Praetorius, D.: Quasi-optimal convergence rates for adaptive boundary element methods with data approximation, part I: Weakly-singular integral equation. Calcolo, in print (2013). doi:10.1007/s10092-013-0100-x
  18. 18.
    Feischl, M., Führer, T., Karkulik, M., Melenk, J.M., Praetorius, D.: Quasi-optimal convergence rates for adaptive boundary element methods with data approximation, part II: Hypersingular integral equation. ASC Report 30/2013, Institute for Analysis and Scientific Computing, Vienna University of Technology (2013)Google Scholar
  19. 19.
    Feischl, M., Karkulik, M., Melenk, J.M., Praetorius, D.: Quasi-optimal convergence rate for an adaptive boundary element method. SIAM J. Numer. Anal. 51, 1327–1348 (2013)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Ferraz-Leite, S., Ortner, C., Praetorius, D.: Convergence of simple adaptive Galerkin schemes based on hh / 2 error estimators. Numer. Math. 116, 291–316 (2010)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Ferraz-Leite, S., Praetorius, D.: Simple a posteriori error estimators for the h-version of the boundary element method. Computing 83, 135–162 (2008)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Funken, S., Praetorius, D., Wissgott, P.: Efficient implementation of adaptive P1-FEM in MATLAB. Comput. Methods Appl. Math. 11, 460–490 (2011)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Karkulik, M., Of, G., Praetorius, D.: Convergence of adaptive 3D BEM for weakly singular integral equations based on isotropic refinement. Numer. Methods Partial Differ. Equ. 29, 2081–2106 (2013)Google Scholar
  24. 24.
    Hackbusch, W.: Hierarchische Matrizen. Algorithmen und Analysis. Springer-Verlag, Berlin (2009)Google Scholar
  25. 25.
    Heuer, N., Mellado, M.E., Stephan, E.P.: hp-adaptive two-level methods for boundary integral equations on curves. Computing 67, 305–334 (2001)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Maischak, M.: The analytical computation of the Galerkin elements for the Laplace, Lamé and Helmholtz equation in 2D-BEM: Preprint. Institute for Applied Mathematics, University of Hannover (2001)Google Scholar
  27. 27.
    Maischak, M., Mund, P., Stephan, E.P.: Adaptive multilevel BEM for acoustic scattering. Comput. Methods Appl. Mech. Eng. 150, 351–367 (1997)CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Maue, A.W.: Zur Formulierung eines allgemeinen Beugungsproblems durch eine Integralgleichung. Z. f. Physik 126, 601–618 (1949)CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Mayr, M.: Stabile Implementierung der Randelementmethode auf stark adaptierten Netzen. Bachelor thesis (in German). Institute for Analysis and Scientific Computing, Vienna University of Technology (2009)Google Scholar
  30. 30.
    McLean, W.: Strongly elliptic systems and boundary integral equations. Cambridge University Press, Cambridge (2000)MATHGoogle Scholar
  31. 31.
    Mund, P., Stephan, E.P.: An adaptive two-level method for hypersingular integral equations in R3. ANZIAM J. 42, C1019–C1033 (2000)MathSciNetGoogle Scholar
  32. 32.
    Mund, P., Stephan, E.P., Weisse, J.: Two-level methods for the single layer potential in \(\mathbb {R}^{3}\). Computing 60, 243–266 (1998)CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Rjasanow, S., Steinbach, O.: The fast solution of boundary integral equations. Springer, New York (2007)MATHGoogle Scholar
  34. 34.
    Sauter, S., Schwab, C.: Boundary element methods, Springer Series in Computational Mathematics, vol. 39. Springer-Verlag, Berlin (2011)Google Scholar
  35. 35.
    Steinbach, O.: Numerical approximation methods for elliptic boundary value problems: finite and boundary elements. Springer, New York (2008)CrossRefGoogle Scholar
  36. 36.
    Tsogtorel, G.: Adaptive boundary element methods with convergence rates. Numer. Math. 124, 471–516 (2013)CrossRefMathSciNetGoogle Scholar
  37. 37.
    Verfürth, R.: A review of a posteriori error estimation and adaptive mesh-refinement techniques. Wiley-Teubner, Chichester (1996)MATHGoogle Scholar
  38. 38.
    Wendland, W.L., Hsiao, G.C.: Boundary integral equations. Springer-Verlag, Berlin (2008)MATHGoogle Scholar

Copyright information

© The Author(s) 2013

Open AccessThis article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Authors and Affiliations

  • Markus Aurada
    • 1
  • Michael Ebner
    • 1
  • Michael Feischl
    • 1
  • Samuel Ferraz-Leite
    • 3
  • Thomas Führer
    • 1
  • Petra Goldenits
    • 1
  • Michael Karkulik
    • 2
  • Markus Mayr
    • 1
  • Dirk Praetorius
    • 1
  1. 1.Vienna University of TechnologyViennaAustria
  2. 2.Pontificia Universidad Católica de ChileSantiagoChile
  3. 3.Max Planck Institute for Mathematics in the SciencesLeipzigGermany

Personalised recommendations