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

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Abstract

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.

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References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ainsworth, M., McLean, W.: Multilevel diagonal scaling preconditioners for boundary element equations on locally refined meshes. Numer. Math. 93(3), 387–413 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cao, T.: Adaptive-additive multilevel methods for hypersingular integral equation. Appl. Anal. 81(3), 539–564 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  13. Griebel, M., Oswald, P.: On additive Schwarz preconditioners for sparse grid discretizations. Numer. Math. 66(4), 449–463 (1994)

    MathSciNet  MATH  Google Scholar 

  14. Greengard, L., Rokhlin, V.: A fast algorithm for particle simulations. J. Comput. Phys. 73(2), 325–348 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hackbusch, W.: A sparse matrix arithmetic based on \({\cal H}\)-matrices. I. Introduction to \({\cal H}\)-matrices. Computing 62(2), 89–108 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hsiao, G.C., Wendland, W.L.: Boundary integral equations In: Applied Mathematical Sciences, vol. 164. Springer, Berlin (2008)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  21. McLean, William: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  22. Mitchell, W.F.: Optimal multilevel iterative methods for adaptive grids. SIAM J. Sci. Stat. Comput. 13(1), 146–167 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  23. McLean, W., Steinbach, O.: Boundary element preconditioners for a hypersingular integral equation on an interval. Adv. Comput. Math. 11(4), 271–286 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  24. Mund, P.: Zwei-Level-Verfahren fr Randintegralgleichungen mit Anwendungen auf die nichtlineare FEM-BEM-Kopplung. PhD thesis, Universitt Hannover (1997)

  25. Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia (2003)

    Book  MATH  Google Scholar 

  26. Śmigaj, W., Betcke, T., Arridge, S., Phillips, J., Schweiger, M.: Solving boundary integral problems with BEM++. 2013. Extended and Revised Preprint. http://www.bempp.org/files/bempp-toms-preprint.pdf

  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)

  28. Stephan, E.P.: Boundary integral equations for screen problems in \({\bf R}^3\). Integral Equ. Oper. Theory 10(2), 236–257 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  29. Steinbach, O.: Stability estimates for hybrid coupled domain decomposition methods. In: Lecture Notes in Mathematics, vol. 1809. Springer, Berlin (2003)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  31. Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54(190), 483–493 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  32. Tran, T., Stephan, E.P.: Additive Schwarz methods for the \(h\)-version boundary element method. Appl. Anal. 60(1–2), 63–84 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  33. Tran, T., Stephan, E.P., Mund, P.: Hierarchical basis preconditioners for first kind integral equations. Appl. Anal. 65(3–4), 353–372 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  34. Tsogtorel, G.: Adaptive boundary element methods with convergence rates. Numer. Math. 124, 471–516 (2013)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  37. Widlund, O.B.: Optimal iterative refinement methods. In: Domain Decomposition Methods. pp. 114–125 (SIAM, Philadelphia, 1989)

  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)

    MathSciNet  MATH  Google Scholar 

  39. Yserentant, H.: On the multilevel splitting of finite element spaces. Numer. Math. 49(4), 379–412 (1986)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

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.

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Authors and Affiliations

  1. School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, 2052, Australia

    Michael Feischl

  2. Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Vickuña Mackenna 4860, 7820436, Santiago, Chile

    Thomas Führer

  3. Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8-10, 1040, Wien, Austria

    Dirk Praetorius

  4. Institute for Applied Mathematics, Leibniz University Hannover, Welfengarten 1, 30167, Hannover, Germany

    Ernst P. Stephan

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  1. Michael Feischl
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  2. Thomas Führer
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  4. Ernst P. Stephan
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Correspondence to Thomas Führer.

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Feischl, M., Führer, T., Praetorius, D. et al. Optimal additive Schwarz preconditioning for hypersingular integral equations on locally refined triangulations. Calcolo 54, 367–399 (2017). https://doi.org/10.1007/s10092-016-0190-3

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  • DOI: https://doi.org/10.1007/s10092-016-0190-3

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