Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
Computing and Visualization in Science
Erschienen in: Computing and Visualization in Science 5/2012

01.10.2012

Local multilevel preconditioners for elliptic equations with jump coefficients on bisection grids

verfasst von: Long Chen, Michael Holst, Jinchao Xu, Yunrong Zhu

Erschienen in: Computing and Visualization in Science | Ausgabe 5/2012

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

The goal of this paper is to design optimal multilevel solvers for the finite element approximation of second order linear elliptic problems with piecewise constant coefficients on bisection grids. Local multigrid and BPX preconditioners are constructed based on local smoothing only at the newest vertices and their immediate neighbors. The analysis of eigenvalue distributions for these local multilevel preconditioned systems shows that there are only a fixed number of eigenvalues which are deteriorated by the large jump. The remaining eigenvalues are bounded uniformly with respect to the coefficients and the meshsize. Therefore, the resulting preconditioned conjugate gradient algorithm will converge with an asymptotic rate independent of the coefficients and logarithmically with respect to the meshsize. As a result, the overall computational complexity is nearly optimal.
Vorheriger Artikel Towards practical anisotropic adaptive FEM on triangular meshes: a new refinement paradigm
Nächster Artikel Calculating ellipse overlap areas

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Jetzt informieren
Literatur
1.
Aksoylu, B., Holst, M.: Optimality of multilevel preconditioners for local mesh refinement in three dimensions. SIAM J. Numer. Anal. 44(3), 1005–1025 (2006)CrossRefMATHMathSciNet
2.
Aksoylu, B., Graham, I., Klie, H., Scheichl, R.: Towards a rigorously justified algebraic preconditioner for high-contrast diffusion problems. Comput. Vis. Sci. 11(4), 319–331 (2008)CrossRefMathSciNet
3.
Alcouffe, R.E., Brandt, A., Dendy, J.E., Painter, J.W.: The multi-grid methods for the diffusion equation with strongly discontinuous coefficients. SIAM J. Sci. Stat. Comput. 2, 430–454 (1981)CrossRefMATHMathSciNet
4.
Ashby, S.F., Holst, M.J., Manteuffel, T.A., Saylor, P.E.: The role of the inner product in stopping criteria for conjugate gradient iterations. BIT 41(1), 026–052 (2001)CrossRefMathSciNet
5.
Axelsson, O.: Iteration number for the conjugate gradient method. Math. Comput. Simul. 61(3–6), 421–435 (2003). MODELLING 2001 (Pilsen)
6.
Axelsson, O.: Iterative Solution Methods. Cambridge University Press, Cambridge (1994)CrossRefMATH
7.
8.
9.
10.
Bernardi, C., Verfürth, R.: Adaptive finite element methods for elliptic equations with non-smooth coefficients. Numer. Math. 85(4), 579–608 (2000)CrossRefMATHMathSciNet
11.
Binev, P., Dahmen, W., DeVore, R.: Adaptive finite element methods with convergence rates. Numer. Math. 97(2), 219–268 (2004)
12.
13.
Bramble, J.H., Xu, J.: Some estimates for a weighted \({L}^2\) projection. Math. Comput. 56, 463–476 (1991)MATHMathSciNet
14.
Briggs, W.L., Henson, V.E., McCormick, S.F.: A Multigrid Tutorial, 2nd edn. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2000)CrossRefMATH
15.
Cai, Z., Zhang, S.: Recovery-based error estimator for interface problems: conforming linear elements. SIAM J. Numer. Anal. 47(3), 2132–2156 (2009)CrossRefMATHMathSciNet
16.
Cascon, J.M., Kreuzer, C., Nochetto, R.H., Siebert, K.G.: Quasi-optimal convergence rate for an adaptive finite element method. SIAM J. Numer. Anal. 46(5), 2524–2550 (2008)CrossRefMATHMathSciNet
17.
Chan, T.F., Wan, W.L.: Robust multigrid methods for nonsmooth coefficient elliptic linear systems. J. Comput. Appl. Math. 123(1–2), 323–352 (2000)CrossRefMATHMathSciNet
18.
Chen, L.: iFEM: an integrate finite element methods package in MATLAB. Technical report, University of California at Irvine (2009)
19.
Chen, Z., Dai, S.: On the efficiency of adaptive finite element methods for elliptic problems with discontinuous coefficients. SIAM J. Sci. Comput. 24(2), 443–462 (2002)CrossRefMATHMathSciNet
20.
Chen, L.: Short implementation of bisection in MATLAB. In: Jorgensen, P., Shen, X., Shu, C.-W., Yan, N. (eds.) Recent Advances in Computational Sciences—Selected Papers from the International Workshop on Computational Sciences and Its Education, pp. 318–332. World Scientific Pub Co Inc, Singapore (2007)
21.
Chen, L.: Deriving the X-Z Identity from auxiliary space method. In: Yunqing, Huang, Ralf, Kornhuber, Olof, Widlund, Xu, Jinchao (eds.) Domain Decomposition Methods in Science and Engineering XIX, pp. 309–316. Springer, Berlin (2010)
22.
Chen, L., Zhang, C.-S.: A coarsening algorithm and multilevel methods on adaptive grids by newest vertex bisection. J. Comput. Math. 28(6), 767–789 (2010)MATHMathSciNet
23.
Chen, L., Nochetto, R.H., Xu, J.: Optimal multilevel methods for graded bisection grids. Numer. Math. 120(1), 1–34 (2011)CrossRefMathSciNet
24.
25.
Dryja, M., Sarkis, M.V., Widlund, O.B.: Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions. Numer. Math. 72(3), 313–348 (1996)CrossRefMATHMathSciNet
26.
Golub, G.H., Van Loan, C.F.: Matrix Computations. Johns Hopkins Studies in the Mathematical Sciences, 3rd edn. Johns Hopkins University Press, Baltimore (1996)
27.
Graham, I.G., Hagger, M.J.: Unstructured additive schwarz-conjugate gradient method for elliptic problems with highly discontinuous coefficients. SIAM J. Sci. Comput. 20, 2041–2066 (1999)CrossRefMATHMathSciNet
28.
29.
Heise, B., Kuhn, M.: Parallel solvers for linear and nonlinear exterior magnetic field problems based upon coupled FE/BE formulations. Computing, 56(3), 237–258 (1996). International GAMM-Workshop on Multi-level Methods (Meisdorf, 1994)
30.
31.
Kees, C.E., Miller, C.T., Jenkins, E.W., Kelley, C.T.: Versatile two-level Schwarz preconditioners for multiphase flow. Comput. Geosci. 7(2), 91–114 (2003)CrossRefMATHMathSciNet
32.
Kossaczky, I.: A recursive approach to local mesh refinement in two and three dimensions. J. Comput. Appl. Math. 55, 275–288 (1994)CrossRefMATHMathSciNet
33.
Meza, J., Tuminaro, R.: A multigrid preconditioner for the semiconductor equations. SIAM J. Sci. Comput. 17, 118–132 (1996)CrossRefMATHMathSciNet
34.
Mitchell, W.F.: A comparison of adaptive refinement techniques for elliptic problems. ACM Trans. Math. Softw. (TOMS) Arch. 15(4), 326–347 (1989)CrossRefMATH
35.
Mitchell, W.F.: Optimal multilevel iterative methods for adaptive grids. SIAM J. Sci. Stat. Comput. 13, 146–167 (1992)CrossRefMATH
36.
Nochetto, R., Siebert, K., Veeser, A.: Theory of adaptive finite element methods: an introduction. In: DeVore, R., Kunoth, A. (eds.) Multiscale, Nonlinear and Adaptive Approximation, pp. 409–542. Springer, 2009. Dedicated to Wolfgang Dahmen on the Occasion of His 60th Birthday
37.
Oswald, P.: Multilevel Finite Element Approximation. Theory and Applications. Teubner Skripten zur Numerik. Teubner Verlag, Stuttgart (1994)
38.
Oswald, P.: On the robustness of the BPX-preconditioner with respect to jumps in the coefficients. Math. Comput. 68, 633–650 (1999)CrossRefMATHMathSciNet
39.
Petzoldt, M.: A posteriori error estimators for elliptic equations with discontinuous coefficients. Adv. Comput. Math. 16(1), 47–75 (2002)CrossRefMATHMathSciNet
40.
Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia (2003)CrossRefMATH
41.
Sarkis, M.: Nonstandard coarse spaces and schwarz methods for elliptic problems with discontinuous coefficients using non-conforming elements. Numer. Math. 77(3), 383–406 (1997)CrossRefMATHMathSciNet
42.
Scheichl, R., Vainikko, E.: Additive schwarz with aggregation-based coarsening for elliptic problems with highly variable coefficients. Computing 80(4), 319–343 (2007)CrossRefMATHMathSciNet
43.
Scott, R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54, 483–493 (1990)CrossRefMATHMathSciNet
44.
45.
Stevenson, R.: The completion of locally refined simplicial partitions created by bisection. Math. Comput. 77, 227–241 (2008)
46.
Vohralık, M.: Guaranteed and fully robust a posteriori error estimates for conforming discretizations of diffusion problems with discontinuous coefficients. Technical Report Preprint R08009, Laboratoire Jacques-Louis Lions (2008)
47.
Vuik, C., Segal, A., Meijerink, J.A.: An efficient preconditioned cg method for the solution of a class of layered problems with extreme contrasts in the coefficients. J. Comput. Phys. 152(1), 385–403 (1999)CrossRefMATH
48.
Wang, J., Xie, R.: Domain decomposition for elliptic problems with large jumps in coefficients. In: Proceedings of Conference on Scientific and Engineering Computing, pp. 74–86. National Defense Industry Press (1994)
49.
Wang, J.: New convergence estimates for multilevel algorithms for finite-element approximations. J. Comput. Appl. Math. 50, 593–604 (1994)CrossRefMATHMathSciNet
50.
Wang, Z., Wang, C., Chen, K.: Two-phase flow and transport in the air cathode of proton exchange membrane fuel cells. J. Power Sources 94, 40–50 (2001)CrossRef
51.
Widlund, O.B.: Some Schwarz methods for symmetric and nonsymmetric elliptic problems. In: Keyes, D.E., Chan, T.F., Meurant, G.A., Scroggs, J.S., Voigt, R.G. (eds.) Fifth International Symposium on Domain Decomposition Methods for Partial Differential Equations, pp. 19–36. SIAM, Philadelphia (1992)
52.
Xu, J., Chen, L., Nochetto, R.: Optimal multilevel methods for H (grad), H (curl), and H (div) systems on graded and unstructured grids. In: Multiscale, Nonlinear and Adaptive Approximation, pp. 599–659. Springer, Berlin (2009)
53.
Xu, J.: Counter examples concerning a weighted \({L}^{2}\) projection. Math. Comput. 57, 563–568 (1991)MATH
54.
55.
Xu, J.: A new class of iterative methods for nonselfadjoint or indefinite problems. SIAM J. Numer. Anal. 29, 303–319 (1992)CrossRefMATHMathSciNet
56.
Xu, J.: An introduction to multigrid convergence theory. In: Chan, R., Chan, T., Golub, G. (eds.) Iterative Methods in Scientific Computing. Springer, Berlin (1997)
57.
Xu, J., Zikatanov, L.: The method of alternating projections and the method of subspace corrections in Hilbert space. J. Am. Math. Soc. 15, 573–597 (2002)CrossRefMATHMathSciNet
58.
Xu, J., Zhu, Y.: Uniform convergent multigrid methods for elliptic problems with strongly discontinuous coefficients. Math. Model. Methods Appl. Sci. 18(1), 77–105 (2008)CrossRefMATHMathSciNet
59.
Yserentant, H.: Old and new convergence proofs for multigrid methods. Acta Numer., pp. 285–326 (1993)
60.
Yserentant, H.: Two preconditioners based on the multi-level splitting of finite element spaces. Numer. Math. 58, 163–184 (1990)CrossRefMATHMathSciNet
61.
Zhu, Y.: Domain decomposition preconditioners for elliptic equations with jump coefficients. Numer. Linear Algebra Appl. 15(2–3), 271–289 (2008)CrossRefMATHMathSciNet
Metadaten
Titel
Local multilevel preconditioners for elliptic equations with jump coefficients on bisection grids
verfasst von
Long Chen
Michael Holst
Jinchao Xu
Yunrong Zhu
Publikationsdatum
01.10.2012
Verlag
Springer Berlin Heidelberg
Erschienen in
Computing and Visualization in Science / Ausgabe 5/2012
Print ISSN: 1432-9360
Elektronische ISSN: 1433-0369
DOI
https://doi.org/10.1007/s00791-013-0213-4

Premium Partner

    Bildnachweise