On the convergence of adaptive feedback loops

  • Randolph E. BankEmail author
  • Harry Yserentant
Special Issue CS Symposium 2016


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.


Adaptive feedback loop Finite elements Convergence proof 

Mathematics Subject Classification

65N30 65N15 65N50 



  1. 1.
    Babuška, I., Vogelius, M.: Feedback and adaptive finite element solution of one-dimensional boundary value problems. Numer. Math. 44(1), 75–102 (1984). MathSciNetzbMATHGoogle Scholar
  2. 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)Google Scholar
  3. 3.
    Bank, R.E., Deotte, C.: Adventures in adaptivity. Comput. Vis. Sci. 18(2–3), 79–91 (2017). MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bank, R.E., Nguyen, H.: \(hp\) adaptive finite elements based on derivative recovery and superconvergence. Comput. Vis. Sci. 14(6), 287–299 (2011). MathSciNetzbMATHGoogle Scholar
  5. 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)Google Scholar
  6. 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). MathSciNetzbMATHGoogle Scholar
  7. 7.
    Bank, R.E., Yserentant, H.: A note on interpolation, best approximation, and the saturation property. Numer. Math. 131(1), 199–203 (2015). MathSciNetzbMATHGoogle Scholar
  8. 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). MathSciNetzbMATHGoogle Scholar
  9. 9.
    Carstensen, C., Gallistl, D., Gedicke, J.: Justification of the saturation assumption. Numer. Math. 134(1), 1–25 (2016). MathSciNetzbMATHGoogle Scholar
  10. 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). MathSciNetzbMATHGoogle Scholar
  11. 11.
    Dörfler, W.: A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33(3), 1106–1124 (1996). MathSciNetzbMATHGoogle Scholar
  12. 12.
    Dörfler, W., Nochetto, R.H.: Small data oscillation implies the saturation assumption. Numer. Math. 91(1), 1–12 (2002). MathSciNetzbMATHGoogle Scholar
  13. 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)zbMATHGoogle Scholar
  14. 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). MathSciNetzbMATHGoogle Scholar
  15. 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). MathSciNetzbMATHGoogle Scholar
  16. 16.
    Mitchell, W.F.: A collection of 2D elliptic problems for testing adaptive grid refinement algorithms. Appl. Math. Comput. 220, 350–364 (2013). MathSciNetzbMATHGoogle Scholar
  17. 17.
    Mitchell, W.F.: How high a degree is high enough for high order finite elements? Proc. Comput. Sci. 51, 246–255 (2015)MathSciNetGoogle Scholar
  18. 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).
  19. 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).
  20. 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).
  21. 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).
  22. 22.
    Stevenson, R.: Optimality of a standard adaptive finite element method. Found. Comput. Math. 7(2), 245–269 (2007). MathSciNetzbMATHGoogle Scholar
  23. 23.
    Verfürth, R.: A Posteriori Error Estimation Techniques for Finite Element Methods. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2013).

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California, San DiegoLa JollaUSA
  2. 2.Institut für MathematikTechnische Universität BerlinBerlinGermany

Personalised recommendations