Skip to main content
SpringerLink
Log in

Convergence of natural adaptive least squares finite element methods

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

The first-order div least squares finite element methods provide inherent a posteriori error estimator by the elementwise evaluation of the functional. In this paper we prove Q-linear convergence of the associated adaptive mesh-refining strategy for a sufficiently fine initial mesh with some sufficiently large bulk parameter for piecewise constant right-hand sides in a Poisson model problem. The proof relies on some modification of known supercloseness results to non-convex polygonal domains plus the flux representation formula. The analysis is carried out for the lowest-order case in two-dimensions for the simplicity of the presentation.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Adler, J.H., Manteuffel, T.A., McCormick, S.F., Nolting, J.W., Ruge, J.W., Tang, L.: Efficiency based adaptive local refinement for first-order system least-squares formulations. SIAM J. Sci. Comput. 33(1), 1–24 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arnold, D., Brezzi, F.: Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO Model. Math. Anal. Numer. 19(1), 7–32 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  3. Berndt, M., Manteuffel, T.A., McCormick, S.F.: Local error estimates and adaptive refinement for first-order system least squares (FOSLS). Electron. Trans. Numer. Anal. 6, 35–43 (1997). ((electronic) Special issue on multilevel methods (Copper Mountain, CO, 1997))

    MathSciNet  MATH  Google Scholar 

  4. Binev, P., Dahmen, W., DeVore, R.: Adaptive finite element methods with convergence rates. Numer. Math. 97(2), 219–268 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Binev, P., DeVore, R.: Fast computation in adaptive tree approximation. Numer. Math. 97(2), 193–217 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bochev, P.B., Gunzburger, M.D.: Least-squares Finite Element Methods, volume 166 of Applied Mathematical Sciences. Springer, New York (2009)

    Google Scholar 

  7. Braess, D.: Finite Elements. Cambridge University Press, Cambridge (2007). third edition

    Book  MATH  Google Scholar 

  8. Brandts, J., Chen, Y., Yang, J.: A note on least-squares mixed finite elements in relation to standard and mixed finite elements. IMA J. Numer. Anal. 26(4), 779–789 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Brenner, S.C.: Two-level additive Schwarz preconditioners for nonconforming finite element methods. Math. Comp. 65(215), 897–921 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  10. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics, vol. 15, 3rd edn. Springer, New York (2008)

    Book  MATH  Google Scholar 

  11. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods, volume 15 of Springer Series in Computational Mathematics. Springer, New York (1991)

    Book  MATH  Google Scholar 

  12. Carstensen, C., Feischl, M., Page, M., Praetorius, D.: Axioms of adaptivity. Comput. Math. Appl. 67(6), 1195–1253 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Carstensen, C., Hoppe, R.W.H.: Error reduction and convergence for an adaptive mixed finite element method. Math. Comp. 75, 1033–1042 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Carstensen, C., Hoppe, R.W.H., Eigel, M., Löbhard, C.: A review of unified a posteriori finite element error control. Numer. Math. Theory Methods Appl. 5, 509–558 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Carstensen, C., Park, E.-J.: Convergence and optimality of adaptive least squares finite element methods. SIAM J. Numer. Anal. 53(1), 43–62 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Carstensen, C., Peterseim, D., Schedensack, M.: Comparison results of finite element methods for the Poisson model problem. SIAM J. Numer. Anal. 50(6), 2803–2823 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Carstensen, C., Rabus, H.: An optimal adaptive mixed finite element method. Math. Comp. 80(274), 649–667 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Carstensen, C., Rabus, H.: Axioms of adaptivity for separate marking. (2016). arXiv:1606.02165 [math.NA] (Submitted)

  19. Cascon, J.M., Kreuzer, Ch., Nochetto, R.H., Siebert, K.G.: Quasi-optimal convergence rate for an adaptive finite element method. SIAM J. Numer. Anal. 46, 2524–2550 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Classics in Applied Mathematics, vol. 40. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2002)

    Book  MATH  Google Scholar 

  21. Dörfler, W.: A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33(3), 1106–1124 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ferraz-Leite, C., Ortner, S., Praetorius, D.: Convergence of simple adaptive Galerkin schemes based on \(h-h/2\) error estimators. Numer. Math. 116(2), 291–316 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Girault, V., Raviart, P.-A.: Finite Element Methods for Navier–Stokes Equations, volume 5 of Springer Series in Computational Mathematics. Springer, Berlin (1986)

    Google Scholar 

  24. Marini, L.D.: An inexpensive method for the evaluation of the solution of the lowest order Raviart–Thomas mixed method. SIAM J. Numer. Anal. 22(3), 493–496 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  25. Nochetto, R.H., Siebert, K.G., Veeser, A.: Theory of Adaptive Finite Element Methods: An Introduction Multiscale Nonlinear and Adaptive Approximation. Springer, Berlin (2009). (electronic)

    MATH  Google Scholar 

  26. 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 

  27. Starke, G.: An adaptive least-squares mixed finite element method for elasto-plasticity. SIAM J. Numer. Anal. 45(1), 371–388 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  28. Stevenson, R.: Optimality of a standard adaptive finite element method. Found. Comput. Math. 7(2), 245–269 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. Verfürth, R.: A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley and Teubner, New York (1996)

    MATH  Google Scholar 

Download references

Acknowledgements

The authors thank Eunjung Lee of the Department of Computational Science and Engineering of Yonsei University in Seoul, Korea, and Gerhard Starke from the Faculty of Mathematics of the University of Duisburg-Essen, Germany, for valuable discussions.

Author information

Authors and Affiliations

  1. Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany

    Carsten Carstensen & Philipp Bringmann

  2. Department of Computational Science and Engineering, Yonsei University, Seoul, 03722, Korea

    Eun-Jae Park

Authors
  1. Carsten Carstensen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eun-Jae Park
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Philipp Bringmann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eun-Jae Park.

Additional information

This research was supported by the Deutsche Forschungsgemeinschaft in the Priority Program 1748 “Reliable simulation techniques in solid mechanics. Development of non- standard discretization methods, mechanical and mathematical analysis” under the project “Foundation and application of generalized mixed FEM towards nonlinear problems in solid mechanics” (CA 151/22-1). This research was supported in part by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology NRF 2011-0030934 and NRF-2015R1A5A1009350.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Carstensen, C., Park, EJ. & Bringmann, P. Convergence of natural adaptive least squares finite element methods. Numer. Math. 136, 1097–1115 (2017). https://doi.org/10.1007/s00211-017-0866-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-017-0866-x

Mathematics Subject Classification

Search

Navigation