Skip to main content
SpringerLink
Log in

Finite-Elemente-Methoden

  • Chapter
  • First Online:
Mehrgittermethoden

  • 2217 Accesses

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 29.95
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 29.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Wir werden auch andere Markierungs-Strategien kennen lernen.

  2. 2.

    Als Träger einer Funktion \(f:A\rightarrow\mathbb{R}^{n}\) bezeichnet man die abgeschlossene Hülle der „Nichtnullstellenmenge“ von \(f\): \(\operatorname{supp}(f):=\overline{\{ x\in A\mid f(x)\ne 0\}}\).

  3. 3.

    Bei der FEM ist eigentlich keine Gitterweite \(h\) definiert. \(h\) kann aber ein Maß für die Element-Größe sein wie die kleinste Höhe oder der größte Inkreis-Durchmesser einer Triangulierung. Auch das ist aber kaum sinnvoll bei adaptiven Netzen. Trotzdem wird allgemein \(h\) als Bezeichnung benutzt. Und ein asymptotisch gegen null gehendes \(h\) ist allemal sinnvoll.

  4. 4.

    Wir verzichten auf die Berücksichtigung von Ansätzen mit mehr als einer Knotenvariablen pro Knotenpunkt, die z. B. bei der Interpolation von Funktions- und Ableitungswerten auftreten.

  5. 5.

    Für andere Relaxationsverfahren und im \(\mathbb{R}^{3}\) ergeben sich andere Faktoren, siehe [15] und [22].

References

  1. Babuška, I.: Error bounds for the finite element method. Numer. Math. 16, 322–333 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  2. Babuška, I., Aziz, A.K.: Survey lectures on the mathematical foundation of the finite element method. In: A.K. Aziz (ed.) Mathematical foundations of the finite element method with Applications to Partial Differential Equations, pp. 1–359. Academic Press, New York (1972)

    Google Scholar 

  3. Babuška, I., Dorr, M.R.: Error estimates for the combined h and p versions of the finite element method. Numer. Math. 37, 257–277 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  4. Babuška, I., Miller, A.: A feedback finite element method with a posteriori error estimation: Part I. Comp. Meth. Appl. Mech. Eng. 61, 1–40 (1987)

    Article  MATH  Google Scholar 

  5. Babuška, I., Rheinboldt, W.C.: Error estimates for adaptive finite element computations. SIAM J, Num. Anal. 15, 736–754 (1978)

    Article  MATH  Google Scholar 

  6. Bank, R.E.: Hierarchical bases and the fem. Acta Numerica 5, 1–47 (1996)

    Article  MathSciNet  Google Scholar 

  7. Bank, R.E.: Pltmg: A software package for solving elliptic partial differential Equations: Users' Guide 8.0. SIAM, Philadelphia (1998)

    Book  Google Scholar 

  8. Bank, R.E.: Pltmg: A software package for solving elliptic partial differential Equations: Users' Guide 10.0. The author, San Diego (2007)

    Google Scholar 

  9. Bank, R.E., Dupont, T., Yserentant, H.: The hierarchical basis multigrid method. Numerische Mathematik 52, 427–458 (1981)

    Article  MathSciNet  Google Scholar 

  10. Bank, R.E. and Weiser, A.: Some a posteriori error estimators for elliptic partial differential equations. Math. Comp. 44, 283–301 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bank, R. E. and Xu, J.: Asymptotically exact a posteriori error estimators, I: Grids with superconvergence. SIAM J. Numer. Anal. 41, 2294–2312 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bank, R. E. and Xu, J.: Asymptotically exact a posteriori error estimators, II: General unstructured grids. SIAM J. Numer. Anal. 41, 2313–2332 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Beck, R. and Erdmann, B. and Roitzsch, R.: Kaskade User's Guide Version 3.x. Tech. rep., Konrad-Zuse-Zentrum für Informationstechnik Berlin (1995). TR-95-11.

    Google Scholar 

  14. Bornemann, F.A., Deuflhard, P.: The cascadic multigrid method for elliptic problems. Numer. Math. 75, 135–152 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  15. Bornemann, F.A., Deuflhard, P.: The cascadic multigrid method for elliptic problems. Numerische Mathematik 75, 135–152 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  16. Bornemann, F.A., Erdmann, B., Kornhuber, R.: A posteriori error estimates for elliptic problems in two and three space dimensions. SIAM J. Numer. Anal. 33, 1188–1204 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  17. Bornemann, F.A., Krause, R.: Classical and cascadic multigrid – a methodical comparison. In: P. Bjôrstad, M. Espedal, D. Keyes (eds.) Proc. 9th Int. Conf. on Domain Decomposition Methods, pp. 64–71. Domain Decomposition Press, Ullenswang, Norway (1996)

    Google Scholar 

  18. Bramble, J.H., Pasciak, J.E., Xu, J.: Parallel multilevel preconditioners. Math. Comp. 55, 1–22 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  19. Deuflhard, P.: Cascadic conjugate gradient methods for elliptic partial differential equation: Algorithm and numerical results. Contemporary Mathematica 180, 29–42 (1994)

    Article  MathSciNet  Google Scholar 

  20. Deuflhard, P., Leinen, P., Yserentant, H.: Concepts of an adaptive hierarchical finite element code. Impact of Computing in Science and Engineering 1, 3–35 (1989)

    Article  MATH  Google Scholar 

  21. Deuflhard, P., P., L., Yserentant, H.: Concepts of an adaptive hierarchical finite element code. Impact Comput. Sci. Eng. 1, 3–35 (1989)

    Article  MATH  Google Scholar 

  22. Deuflhard, P., Weiser, M.: Numerische Mathematik 3. Adaptive Lösung partieller Differentialgleichungen. de Gruyter, Berlin (2011)

    Book  MATH  Google Scholar 

  23. Großmann, C., Roos, H.G.: Numerik partieller Differentialgleichungen. 3.Aufl. Teubner, Wiesbaden (2005)

    Book  Google Scholar 

  24. Hackbusch, W.: Multigrid methods and applications. Springer, Berlin (1985)

    Google Scholar 

  25. Hackbusch, W., Trottenberg, U. (eds.): Multigrid Methods. Springer, Lecture Notes in Mathematics 960, Berlin (1982)

    Google Scholar 

  26. Johnson, C.: Numerical Solution of Partial Differential Equations by the Finite Element Method. Studentlitteratur, Lund, Schweden (1987)

    MATH  Google Scholar 

  27. Johnson, C., Eriksson, K.: Adaptive finite element methods for parabolic problems I: A linear model problem. SIAM J. Numer. Anal, 28 28, 43–77 (1991)

    MathSciNet  MATH  Google Scholar 

  28. Mitchell, W.F.: Unified multilevel adaptive finite element methods for elliptic problems. Ph.D. thesis, University of Illinois at Urbana-Champaign (1988)

    Google Scholar 

  29. Rivara, M. C.: Mesh refinement processes based on the generalized bisection of simplices. SIAM J Numer. Anal. 21(3), 604–613 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  30. Rivara, M. C.: New mathematical tools and techniques for the refinement and/or improvement of unstructured triangulations. In: Proc. 5th Int. Meshing Roundtable. Pittsburgh, PA (1996)

    Google Scholar 

  31. Shaidurov, V.: Multigrid Methods for Finite Elements. Kluwer Academic Publishers, Boston (1995)

    MATH  Google Scholar 

  32. Shaidurov, V.: Some estimates of the rate of convergence for the cascadic conjugate gradient method. Computers Math. Applic. 31(4/5), 161–171 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  33. Shewchuk, J.R.: Lecture notes on delaunay mesh generation. Tech. rep., Dep. EE and CS, University of California at Berkeley (1999). "Uberhaupt alles von Shewchuk, siehe auch http://www.cs.cmu.edu/~jrs/

  34. Stueben, A., Trottenberg, U.: Multigrid methods: fundamental algorithms, model problem analysis and application. In: Hackbusch, W., Trottenberg, U. (eds.): Multigrid Methods, pp. 1–176. Lecture Notes in Mathematics 960. Springer, Berlin (1982)

    Chapter  Google Scholar 

  35. Weiser M. and Schiela A.: (2006). http://www.zib.de/de/numerik/software/kaskade-7.html

  36. Yserentant, H.: Hierachical basis give conjugate gradient type methods a multigrid speed of convergence. Applied Math. Comp. 19, 347–358 (1986)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  38. Yserentant, H.: Preconditioning indefinite discretization matrices. Numer. Math. 54, 719–734 (1989)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Author notes

  1. Norbert Köckler

    Present address: Fakultät für Elektrotechnik, Informatik und Mathemathik, Institut für Mathematik, Universität Paderborn, Paderborn, Deutschland

Authors and Affiliations

    Authors
    1. Norbert Köckler
      View author publications

      You can also search for this author in PubMed Google Scholar

    Corresponding author

    Correspondence to Norbert Köckler .

    Rights and permissions

    Reprints and permissions

    Copyright information

    © 2012 Vieweg+Teubner Verlag | Springer Fachmedien Wiesbaden

    About this chapter

    Cite this chapter

    Köckler, N. (2012). Finite-Elemente-Methoden. In: Mehrgittermethoden. Vieweg+Teubner Verlag, Wiesbaden. https://doi.org/10.1007/978-3-8348-2081-5_8

    Download citation

    Publish with us

    Policies and ethics

    Search

    Navigation