Skip to main content
Springer Nature Link
Log in

On the convergence of adaptive feedback loops

  • Special Issue CS Symposium 2016
  • Published:
Computing and Visualization in Science

Abstract

We present a technique for proving convergence of h and hp adaptive finite element methods through comparison with certain reference refinement schemes based on interpolation error. We then construct a testing environment where properties of different adaptive approaches can be evaluated and improved.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Notes

  1. Finding the maximum will introduce a logarithm into the complexity (e.g., we keep all the elements in a heap based on their errors), but this is not important in this context.

References

  1. Babuška, I., Vogelius, M.: Feedback and adaptive finite element solution of one-dimensional boundary value problems. Numer. Math. 44(1), 75–102 (1984). https://doi.org/10.1007/BF01389757

    Article  MathSciNet  MATH  Google Scholar 

  2. Bank, R.E.: PLTMG: A Software Package for Solving Elliptic Partial Differential Equations, Users’ Guide 12.0. Technical report, Department of Mathematics, University of California at San Diego (2016)

  3. Bank, R.E., Deotte, C.: Adventures in adaptivity. Comput. Vis. Sci. 18(2–3), 79–91 (2017). https://doi.org/10.1007/s00791-016-0272-4

    Article  MathSciNet  MATH  Google Scholar 

  4. Bank, R.E., Nguyen, H.: \(hp\) adaptive finite elements based on derivative recovery and superconvergence. Comput. Vis. Sci. 14(6), 287–299 (2011). https://doi.org/10.1007/s00791-012-0179-7

    Article  MathSciNet  MATH  Google Scholar 

  5. Bank, R.E., Sherman, A.H., Weiser, A.: Refinement algorithms and data structures for regular local mesh refinement. In: Scientific computing (Montreal, Que., 1982), IMACS Trans. Sci. Comput., I, pp. 3–17. IMACS, New Brunswick, NJ (1983)

  6. Bank, R.E., Xu, J., Zheng, B.: Superconvergent derivative recovery for Lagrange triangular elements of degree \(p\) on unstructured grids. SIAM J. Numer. Anal. 45(5), 2032–2046 (2007). https://doi.org/10.1137/060675174

    Article  MathSciNet  MATH  Google Scholar 

  7. Bank, R.E., Yserentant, H.: A note on interpolation, best approximation, and the saturation property. Numer. Math. 131(1), 199–203 (2015). https://doi.org/10.1007/s00211-014-0687-0

    Article  MathSciNet  MATH  Google Scholar 

  8. Bürg, M., Dörfler, W.: Convergence of an adaptive \(hp\) finite element strategy in higher space-dimensions. Appl. Numer. Math. 61(11), 1132–1146 (2011). https://doi.org/10.1016/j.apnum.2011.07.008

    Article  MathSciNet  MATH  Google Scholar 

  9. Carstensen, C., Gallistl, D., Gedicke, J.: Justification of the saturation assumption. Numer. Math. 134(1), 1–25 (2016). https://doi.org/10.1007/s00211-015-0769-7

    Article  MathSciNet  MATH  Google Scholar 

  10. Demlow, A., Stevenson, R.: Convergence and quasi-optimality of an adaptive finite element method for controlling \(L_2\) errors. Numer. Math. 117(2), 185–218 (2011). https://doi.org/10.1007/s00211-010-0349-9

    Article  MathSciNet  MATH  Google Scholar 

  11. Dörfler, W.: A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33(3), 1106–1124 (1996). https://doi.org/10.1137/0733054

    Article  MathSciNet  MATH  Google Scholar 

  12. Dörfler, W., Nochetto, R.H.: Small data oscillation implies the saturation assumption. Numer. Math. 91(1), 1–12 (2002). https://doi.org/10.1007/s002110100321

    Article  MathSciNet  MATH  Google Scholar 

  13. Guo, B., Babuška, I.: The \(hp\) version of the finite element method. Part 1: the basic approximation results. Comput. Mech. 1(1), 21–41 (1986)

    Article  Google Scholar 

  14. Holst, M., Pollock, S.: Convergence of goal-oriented adaptive finite element methods for nonsymmetric problems. Numer. Methods Partial Differ. Equ. 32(2), 479–509 (2016). https://doi.org/10.1002/num.22002

    Article  MathSciNet  MATH  Google Scholar 

  15. Holst, M., Pollock, S., Zhu, Y.: Convergence of goal-oriented adaptive finite element methods for semilinear problems. Comput. Vis. Sci. 17(1), 43–63 (2015). https://doi.org/10.1007/s00791-015-0243-1

    Article  MathSciNet  MATH  Google Scholar 

  16. Mitchell, W.F.: A collection of 2D elliptic problems for testing adaptive grid refinement algorithms. Appl. Math. Comput. 220, 350–364 (2013). https://doi.org/10.1016/j.amc.2013.05.068

    Article  MathSciNet  MATH  Google Scholar 

  17. Mitchell, W.F.: How high a degree is high enough for high order finite elements? Proc. Comput. Sci. 51, 246–255 (2015)

    Article  MathSciNet  Google Scholar 

  18. Mitchell, W.F., McClain, M.A.: A survey of \(hp\)-adaptive strategies for elliptic partial differential equations. In: Recent Advances in Computational and Applied Mathematics. Springer, Dordrecht, pp. 227–258 (2011). https://doi.org/10.1007/978-90-481-9981-5_10

    Chapter  Google Scholar 

  19. Mitchell, W.F., McClain, M.A.: A comparison of \(hp\)-adaptive strategies for elliptic partial differential equations. ACM Trans. Math. Softw. 41(1), Art. 2, 39 (2014). https://doi.org/10.1145/2629459

    Article  Google Scholar 

  20. Nochetto, R.H., Siebert, K.G., Veeser, A.: Theory of adaptive finite element methods: an introduction. In: Multiscale, nonlinear and adaptive approximation. Springer, Berlin, pp. 409–542 (2009). https://doi.org/10.1007/978-3-642-03413-8_12

    Chapter  Google Scholar 

  21. Nochetto, R.H., Veeser, A.: Primer of adaptive finite element methods. In: Multiscale and Adaptivity: Modeling, Numerics and Applications. Lecture Notes in Mathematics, vol. 2040. Springer, Heidelberg, pp. 125–225 (2012). https://doi.org/10.1007/978-3-642-24079-9

    Book  Google Scholar 

  22. Stevenson, R.: Optimality of a standard adaptive finite element method. Found. Comput. Math. 7(2), 245–269 (2007). https://doi.org/10.1007/s10208-005-0183-0

    Article  MathSciNet  MATH  Google Scholar 

  23. Verfürth, R.: A Posteriori Error Estimation Techniques for Finite Element Methods. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2013). https://doi.org/10.1093/acprof:oso/9780199679423.001.0001

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California, San Diego, La Jolla, CA, 92093-0112, USA

    Randolph E. Bank

  2. Institut für Mathematik, Technische Universität Berlin, 10623, Berlin, Germany

    Harry Yserentant

Authors
  1. Randolph E. Bank
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Harry Yserentant
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Randolph E. Bank.

Additional information

Communicated by Volker Schulz.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Randolph E. Bank: The work of this author was supported by the National Science Foundation under Contract DMS-1318480, and the Alexander von Humboldt Foundation through a Humboldt Research Award.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bank, R.E., Yserentant, H. On the convergence of adaptive feedback loops. Comput. Visual Sci. 20, 59–70 (2019). https://doi.org/10.1007/s00791-019-00310-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00791-019-00310-4

Keywords

Mathematics Subject Classification