Advertisement

Journal of Scientific Computing

, Volume 79, Issue 2, pp 1161–1181 | Cite as

Optimal Quasi-diagonal Preconditioners for Pseudodifferential Operators of Order Minus Two

  • Thomas FührerEmail author
  • Norbert Heuer
Article
  • 58 Downloads

Abstract

We present quasi-diagonal preconditioners for piecewise polynomial discretizations of pseudodifferential operators of order minus two in any space dimension. Here, quasi-diagonal means diagonal up to a sparse transformation. Considering shape regular simplicial meshes and arbitrary fixed polynomial degrees, we prove, for dimensions larger than one, that our preconditioners are asymptotically optimal. Numerical experiments in two, three and four dimensions confirm our results. For each dimension, we report on condition numbers for piecewise constant and piecewise linear polynomials.

Keywords

Pseudodifferential operator of negative order Diagonal scaling Additive Schwarz method Preconditioner Negative order Sobolev spaces 

Mathematics Subject Classification

65F35 65N30 

Notes

Acknowledgements

This work was supported by CONICYT through FONDECYT Projects 11170050 and 1150056

References

  1. 1.
    Ainsworth, M., Guzmán, J., Sayas, F.-J.: Discrete extension operators for mixed finite element spaces on locally refined meshes. Math. Comput. 85(302), 2639–2650 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bey, J.: Simplicial grid refinement: on Freudenthal’s algorithm and the optimal number of congruence classes. Numer. Math. 85(1), 1–29 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications, Volume 44 of Springer Series in Computational Mathematics. Springer, Heidelberg (2013)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bramble, J.H., Lazarov, R.D., Pasciak, J.E.: A least-squares approach based on a discrete minus one inner product for first order systems. Math. Comput. 66(219), 935–955 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bramble, J.H., Pasciak, J.E.: Least-squares methods for Stokes equations based on a discrete minus one inner product. J. Comput. Appl. Math. 74(1–2), 155–173 (1996). TICAM Symposium (Austin, TX, 1995)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dauge, M.: Elliptic Boundary Value Problems on Corner Domains, Volume 1341 of Lecture Notes in Mathematics. Springer, Berlin (1988). (Smoothness and asymptotics of solutions)Google Scholar
  8. 8.
    Führer, T.: First-order least-squares method for the obstacle problem. arXiv:1801.09622 (2018)
  9. 9.
    Führer, T., Haberl, A., Praetorius, D., Schimanko, S.: Adaptive BEM with inexact PCG solver yields almost optimal computational costs. Numer. Math. (2018). Accepted for publication, preprint:  https://doi.org/10.1007/s00211-018-1011-1
  10. 10.
    Führer, T., Heuer, N., Niemi, A.H.: An ultraweak formulation of the Kirchhoff–Love plate bending model and DPG approximation. Math. Comput. (2018).  https://doi.org/10.1090/mcom/3381
  11. 11.
    Graham, I.G., Hackbusch, W., Sauter, S.A.: Finite elements on degenerate meshes: inverse-type inequalities and applications. IMA J. Numer. Anal. 25(2), 379–407 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Grisvard, P.: Elliptic Problems in Nonsmooth Domains, Volume 24 of Monographs and Studies in Mathematics. Pitman (Advanced Publishing Program), Boston (1985)Google Scholar
  13. 13.
    Heuer, N.: Additive Schwarz method for the \(p\)-version of the boundary element method for the single layer potential operator on a plane screen. Numer. Math. 88(3), 485–511 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Heuer, N., Stephan, E.P., Tran, T.: Multilevel additive Schwarz method for the \(h\)-\(p\) version of the Galerkin boundary element method. Math. Comput. 67(222), 501–518 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hiptmair, R., Mao, S.: Stable multilevel splittings of boundary edge element spaces. BIT 52(3), 661–685 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hsiao, G.C., Wendland, W.L.: A finite element method for some integral equations of the first kind. J. Math. Anal. Appl. 58, 449–481 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hsiao, G.C., Wendland, W.L.: The Aubin–Nitsche lemma for integral equations. J. Integral Equ. 3, 299–315 (1981)MathSciNetzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Langer, U., Pusch, D., Reitzinger, S.: Efficient preconditioners for boundary element matrices based on grey-box algebraic multigrid methods. Int. J. Numer. Methods Eng. 58(13), 1937–1953 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Mund, P., Stephan, E.P., Weiße, J.: Two level methods for the single layer potential in \({{\mathbb{R}}}^3\). Computing 60, 243–266 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Oswald, P.: Multilevel finite element approximation. Teubner Skripten zur Numerik. [Teubner Scripts on Numerical Mathematics]. B. G. Teubner, Stuttgart, Theory and applications (1994)Google Scholar
  22. 22.
    Oswald, P.: Multilevel norms for \(H^{-1/2}\). Computing 61(3), 235–255 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Oswald, P.: Interface preconditioners and multilevel extension operators. In: Eleventh International Conference on Domain Decomposition Methods (London, 1998), pp. 97–104. DDM.org, Augsburg (1999)Google Scholar
  24. 24.
    Steinbach, O., Wendland, W.L.: The construction of some efficient preconditioners in the boundary element method. Adv. Comput. Math. 9(1–2), 191–216 (1998). Numerical treatment of boundary integral equationsMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Stevenson, R.: The completion of locally refined simplicial partitions created by bisection. Math. Comput. 77(261), 227–241 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Stevenson, R., van Venetië, R.: Optimal preconditioning for problems of negative order. arXiv:1803.05226 (2018)
  27. 27.
    Toselli, A., Widlund, O.: Domain Decomposition Methods–Algorithms and Theory, Volume 34 of Springer Series in Computational Mathematics. Springer, Berlin (2005)CrossRefzbMATHGoogle Scholar
  28. 28.
    Tran, T., Stephan, E.P.: Additive Schwarz method for the h-version boundary element method. Appl. Anal. 60, 63–84 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    von Petersdorff, T., Stephan, E.P.: Multigrid solvers and preconditioners for first kind integral equations. Numer. Methods Partial Differ. Equ. 8(5), 443–450 (1992)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Facultad de MatemáticasPontificia Universidad Católica de ChileSantiagoChile

Personalised recommendations