, Volume 54, Issue 1, pp 367–399 | Cite as

Optimal additive Schwarz preconditioning for hypersingular integral equations on locally refined triangulations

  • Michael Feischl
  • Thomas Führer
  • Dirk Praetorius
  • Ernst P. Stephan


For the non-preconditioned Galerkin matrix of the hypersingular integral operator, the condition number grows with the number of elements as well as the quotient of the maximal and the minimal mesh-size. Therefore, reliable and effective numerical computations, in particular on adaptively refined meshes, require the development of appropriate preconditioners. We propose and analyze a local multilevel preconditioner which is optimal in the sense that the condition number of the corresponding preconditioned system is independent of the number of elements, the local mesh-size, and the number of refinement levels. The theory covers closed boundaries as well as open screens in 2D and 3D. Numerical experiments underline the analytical results and compare the proposed preconditioner to other multilevel schemes as well as techniques based on operator preconditioning.


Preconditioner Multilevel additive Schwarz Hypersingular integral equation 

Mathematics Subject Classification

65N30 65F08 65N38 



The research of the authors is supported by the Austrian Science Fund (FWF) through the research projects Adaptive boundary element method, funded under Grant P21732, and Optimal adaptivity of BEM and FEM-BEM coupling, funded under Grant P27005. In addition, the author TF is supported by the CONICYT project Preconditioned solvers for nonconforming boundary elements, funded under Grant FONDECYT 3150012.


  1. 1.
    Aurada, M., Ebner, M., Feischl, M., Furraz-Leite, S., Führer, T., Goldenits, P., Karkulik, M., Mayr, M., Praetorius, D.: HILBERT—a MATLAB implementation of adaptive 2D-BEM. Numer. Algorit. 67(1), 1–32 (2014)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    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(2013), 305–332 (2013)MathSciNetMATHGoogle Scholar
  3. 3.
    Aurada, M., Feischl, M., Führer, T., Karkulik, M., Melenk, J.M., Praetorius, D.: Local inverse estimates for non-local boundary integral operators. ASC Report, 12/2015, Vienna University of Technology (2015)Google Scholar
  4. 4.
    Aurada, M., Feischl, M., Führer, T., Karkulik, M., Praetorius, D.: Energy norm based error estimators for adaptive BEM for hypersingular integral equations. Appl. Numer. Math. 95, 15–35 (2015)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Ainsworth, M., McLean, W.: Multilevel diagonal scaling preconditioners for boundary element equations on locally refined meshes. Numer. Math. 93(3), 387–413 (2003)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Ainsworth, M., McLean, W., Tran, T.: The conditioning of boundary element equations on locally refined meshes and preconditioning by diagonal scaling. SIAM J. Numer. Anal. 36(6), 1901–1932 (1999)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Cao, T.: Adaptive-additive multilevel methods for hypersingular integral equation. Appl. Anal. 81(3), 539–564 (2002)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    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 51(4), 531–562 (2014)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    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. Electron. Trans. Numer. Anal. 44, 153–176 (2015)MathSciNetMATHGoogle Scholar
  10. 10.
    Feischl, M., Führer, T., Karkulik, M., Praetorius, D.: ZZ-Type a posteriori error estimators for adaptive boundary element methods on a curve. Eng. Anal. Bound. Elem. 38, 49–60 (2014)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Feischl, M., Karkulik, M., Melenk, J.M., Praetorius, D.: Quasi-optimal convergence rate for an adaptive boundary element method. SIAM J. Numer. Anal. 51(2), 1327–1348 (2013)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Graham, I.G., McLean, W.: Anisotropic mesh refinement: the conditioning of Galerkin boundary element matrices and simple preconditioners. SIAM J. Numer. Anal. 44(4), 1487–1513 (2006)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Griebel, M., Oswald, P.: On additive Schwarz preconditioners for sparse grid discretizations. Numer. Math. 66(4), 449–463 (1994)MathSciNetMATHGoogle Scholar
  14. 14.
    Greengard, L., Rokhlin, V.: A fast algorithm for particle simulations. J. Comput. Phys. 73(2), 325–348 (1987)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Hackbusch, W.: A sparse matrix arithmetic based on \({\cal H}\)-matrices. I. Introduction to \({\cal H}\)-matrices. Computing 62(2), 89–108 (1999)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Hiptmair, R., Jerez-Hanckes, C., Urzúa-Torres, C.: Mesh-independent operator preconditioning for boundary elements on open curves. SIAM J. Numer. Anal. 52(5), 2295–2314 (2014)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Hsiao, G.C., Wendland, W.L.: Boundary integral equations In: Applied Mathematical Sciences, vol. 164. Springer, Berlin (2008)Google Scholar
  18. 18.
    Karkulik, M., Pavlicek, D., Praetorius, D.: On 2D newest vertex bisection: optimality of mesh-closure and \(H^1\)-stability of \(L_2\)-projection. Constr. Approx. 38(2), 213–234 (2013)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Lions, P.L.: On the Schwarz alternating method. I. In: First International Symposium on Domain Decomposition Methods for Partial Differential Equations (Paris, 1987). SIAM, Philadelphia, pp. 1–42 (1988)Google Scholar
  20. 20.
    Maischak, M.: A multilevel additive Schwarz method for a hypersingular integral equation on an open curve with graded meshes. Appl. Numer. Math. 59(9), 2195–2202 (2009)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    McLean, William: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)MATHGoogle Scholar
  22. 22.
    Mitchell, W.F.: Optimal multilevel iterative methods for adaptive grids. SIAM J. Sci. Stat. Comput. 13(1), 146–167 (1992)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    McLean, W., Steinbach, O.: Boundary element preconditioners for a hypersingular integral equation on an interval. Adv. Comput. Math. 11(4), 271–286 (1999)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Mund, P.: Zwei-Level-Verfahren fr Randintegralgleichungen mit Anwendungen auf die nichtlineare FEM-BEM-Kopplung. PhD thesis, Universitt Hannover (1997)Google Scholar
  25. 25.
    Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia (2003)CrossRefMATHGoogle Scholar
  26. 26.
    Śmigaj, W., Betcke, T., Arridge, S., Phillips, J., Schweiger, M.: Solving boundary integral problems with BEM++. 2013. Extended and Revised Preprint.
  27. 27.
    Śmigaj, W.., Betcke, T., Arridge, S., Phillips, J., Schweiger, M.: Solving boundary integral problems with BEM++. ACM Trans. Math. Softw. 41(2):Art. 6, 40 (2015)Google Scholar
  28. 28.
    Stephan, E.P.: Boundary integral equations for screen problems in \({\bf R}^3\). Integral Equ. Oper. Theory 10(2), 236–257 (1987)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Steinbach, O.: Stability estimates for hybrid coupled domain decomposition methods. In: Lecture Notes in Mathematics, vol. 1809. Springer, Berlin (2003)Google Scholar
  30. 30.
    Steinbach, Olaf, Wendland, Wolfgang L.: The construction of some efficient preconditioners in the boundary element method. Numerical treatment of boundary integral equations. Adv. Comput. Math. 9(1–2), 191–216 (1998)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54(190), 483–493 (1990)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Tran, T., Stephan, E.P.: Additive Schwarz methods for the \(h\)-version boundary element method. Appl. Anal. 60(1–2), 63–84 (1996)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Tran, T., Stephan, E.P., Mund, P.: Hierarchical basis preconditioners for first kind integral equations. Appl. Anal. 65(3–4), 353–372 (1997)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Tsogtorel, G.: Adaptive boundary element methods with convergence rates. Numer. Math. 124, 471–516 (2013)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Tran, T., Stephan, E.P., Zaprianov, S.: Wavelet-based preconditioners for boundary integral equations. Numerical treatment of boundary integral equations. Adv. Comput. Math. 9(1–2), 233–249 (1998)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Haijun, W., Chen, Z.: Uniform convergence of multigrid V-cycle on adaptively refined finite element meshes for second order elliptic problems. Sci. China Ser. A 49(10), 1405–1429 (2006)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Widlund, O.B.: Optimal iterative refinement methods. In: Domain Decomposition Methods. pp. 114–125 (SIAM, Philadelphia, 1989)Google Scholar
  38. 38.
    Xuejun, X., Chen, H., Hoppe, R.H.W.: Optimality of local multilevel methods on adaptively refined meshes for elliptic boundary value problems. J. Numer. Math. 18(1), 59–90 (2010)MathSciNetMATHGoogle Scholar
  39. 39.
    Yserentant, H.: On the multilevel splitting of finite element spaces. Numer. Math. 49(4), 379–412 (1986)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Italia 2016

Authors and Affiliations

  • Michael Feischl
    • 1
  • Thomas Führer
    • 2
  • Dirk Praetorius
    • 3
  • Ernst P. Stephan
    • 4
  1. 1.School of Mathematics and StatisticsUniversity of New South WalesSydneyAustralia
  2. 2.Facultad de MatemáticasPontificia Universidad Católica de ChileSantiagoChile
  3. 3.Institute for Analysis and Scientific ComputingVienna University of TechnologyWienAustria
  4. 4.Institute for Applied MathematicsLeibniz University HannoverHannoverGermany

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