Skip to main content
SpringerLink
Log in

Convergence of goal-oriented adaptive finite element methods for semilinear problems

  • Published:
Computing and Visualization in Science

Abstract

In this article we develop a convergence theory for goal-oriented adaptive finite element algorithms designed for a class of second-order semilinear elliptic equations. We briefly discuss the target problem class, and introduce several related approximate dual problems that are crucial to both the analysis as well as to the development of a practical numerical method. We then review some standard facts concerning conforming finite element discretization and error-estimate-driven adaptive finite element methods (AFEM). We include a brief summary of a priori estimates for this class of semilinear problems, and then describe some goal-oriented variations of the standard approach to AFEM. Following the recent approach of Mommer–Stevenson and Holst–Pollock for increasingly general linear problems, we first establish a quasi-error contraction result for the primal problem. We then develop some additional estimates that make it possible to establish contraction of the combined primal-dual quasi-error, and subsequently show convergence with respect to the quantity of interest. Finally, a sequence of numerical experiments are examined and it is observed that the behavior of the implementation follows the predictions of the theory.

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.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Axelsson, O., Barker, V.A.: Finite Element Solution of Boundary Value Problems: Theory and Computation. Society for Industrial and Applied Mathematics, Philadelphia, PA (2001)

    Book  Google Scholar 

  2. Bangerth, W., Rannacher, R.: Adaptive Finite Element Methods for Differential Equations. Birkhauser, Boston (2003)

    Book  Google Scholar 

  3. Bank, R., Holst, M., Szypowski, R., Zhu, Y.: Finite Element Error Estimates for Critical Growth Semilinear Problems Without Angle Conditions (2011) Available as arXiv:1108.3661 [math.NA]. Submitted

  4. Becker, R., Rannacher, R.: A feed-back approach to error control in finite element methods: basic analysis and examples. East-West J. Numer. Math. 4, 237–264 (1996)

    MathSciNet  Google Scholar 

  5. Becker, R., Rannacher, R.: Weighted a posteriori error control in FE methods. Preprint 96-1, SFB 359, Universitat, pp. 18–22 (1996)

  6. Becker, R., Rannacher, R.: An optimal control approach to a posteriori error estimation in finite element methods. In: Iserles, A. (ed.) Acta Numer, pp. 1–0102. Cambridge University Press, Cambridge (2001)

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

    Article  MathSciNet  Google Scholar 

  8. Brenner, S., Scott, L.: The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, Berlin (2008)

    Book  Google Scholar 

  9. 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)

    Article  MathSciNet  Google Scholar 

  10. Ciarlet, P.G.: Finite Element Method for Elliptic Problems. Society for Industrial and Applied Mathematics, Philadelphia, PA (2002)

    Book  Google Scholar 

  11. Dahmen, W., Kunoth, A., Vorloeper, J.: Convergence of Adaptive Wavelet Methods for Goal-oriented Error Estimation. Sonderforschungsbereich 611, Singuläre Phänomene und Skalierung in Mathematischen Modellen. SFB 611 (2006)

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

    Article  MathSciNet  Google Scholar 

  13. Eriksson, K., Estep, D., Hansbo, P., Johnson, C.: Introduction to adaptive methods for differential equations. In: Iserles, A. (ed.) Acta Numer, pp. 105–158. Cambridge University Press, Cambridge (1995)

  14. Estep, D., Holst, M., Larson, M.: Generalized green’s functions and the effective domain of influence. SIAM J. Sci. Comput. 26, 1314–1339 (2002)

    Article  MathSciNet  Google Scholar 

  15. Estep, D., Holst, M., Mikulencak, D.: Accounting for stability: a posteriori error estimates based on residuals and variational analysis. In: Communications in Numerical Methods in Engineering, pp. 200–202 (2001)

  16. Estep, D., Larson, M.G., Williams, R.D.: Estimating the error of numerical solutions of systems of reaction-diffusion equations. Mem. Am. Math. Soc. 146(696), 101–109 (2000)

    MathSciNet  Google Scholar 

  17. Evans, L.C.: Partial Differential Equations (Graduate Studies in Mathematics, V. 19) GSM/19. American Mathematical Society, New York (1998)

    Google Scholar 

  18. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1977)

    Book  Google Scholar 

  19. Giles, M., Süli, E.: Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality. Acta Numer. 11, 145–236 (2003)

    Google Scholar 

  20. Grätsch, T., Bathe, K.-J.: A posteriori error estimation techniques in practical finite element analysis. Comput. Struct. 83(4–5), 235–265 (2005)

    Article  Google Scholar 

  21. Holst, M.: Adaptive numerical treatment of elliptic systems on manifolds. Adv. Comput. Math. 15(1–4), 139–191 (2001) Available as arXiv:1001.1367 [math.NA]

  22. Holst, M.: Applications of domain decomposition and partition of unity methods in physics and geometry. In: Herrera, I., Keyes, D., Widlund, O., Yates, R. (eds.) Proceedings of the Fourteenth International Conference on Domain Decomposition Methods, pp. 63–78. National Autonomous University of Mexico (UNAM) (2003) Available arXiv:1001.1364 [math.NA]

  23. Holst, M., McCammon, J., Yu, Z., Zhou, Y., Zhu, Y.: Adaptive finite element modeling techniques for the Poisson-Boltzmann equation. Commun. Comput. Phys. 11(1), 179–214 (2012) Available arXiv:1009.6034 [math.NA]

  24. Holst, M., Pollock, S.: Convergence of goal oriented methods for nonsymmetric problems . Numer Methods Partial Differ. Equ. (2015) Available as arXiv:1108.3660 [math.NA]

  25. Holst, M., Szypowski, R., Zhu, Y.: Adaptive finite element methods with inexact solvers for the nonlinear poisson-boltzmann equation. In: Bank, R., Holst, M., Widlund, O., Xu, J. (eds.) Domain Decomposition Methods in Science and Engineering XX, volume 91 of Lecture Notes in Computational Science and Engineering, pp. 167–174. Springer, Berlin (2013)

  26. Holst, M., Szypowski, R., Zhu, Y.: Two-grid methods for semilinear interface problems. Numer Methods Partial Differ. Equ. 29(5), 1729–1748 (2013)

    Article  MathSciNet  Google Scholar 

  27. Holst, M., Tsogtgerel, G., Zhu, Y.: Local and global convergence of adaptive methods for nonlinear partial differential equations (2008) Available as arXiv:1001.1382 [math.NA]

  28. Jüngel, A., Unterreiter, A.: Discrete minimum and maximum principles for finite element approximations of non-monotone elliptic equations. Numer. Math. 99(3), 485–508 (2005)

    Article  MathSciNet  Google Scholar 

  29. Karatson, J., Korotov, S.: Discrete maximum principles for finite element solutions of nonlinear elliptic problems with mixed boundary conditions. Numer. Math. 99, 669–698 (2005)

    Article  MathSciNet  Google Scholar 

  30. Kerkhoven, T., Jerome, J.W.: \(L_{\infty }\) stability of finite element approximations of elliptic gradient equations. Numer. Math. 57, 561–575 (1990)

    Article  MathSciNet  Google Scholar 

  31. Kesavan, S.: Topics in Functional Analysis and Applications. Wiley, New York, NY (1989)

    Google Scholar 

  32. Korotov, S.: A posteriori error estimation of goal-oriented quantities for elliptic type bvps. J. Comput. Appl. Math. 191(2), 216–227 (2006)

    Article  MathSciNet  Google Scholar 

  33. Mekchay, K., Nochetto, R.: Convergence of adaptive finite element methods for general second order linear elliptic PDE. SINUM 43(5), 1803–1827 (2005)

    Article  MathSciNet  Google Scholar 

  34. Mommer, M.S., Stevenson, R.: A goal-oriented adaptive finite element method with convergence rates. SIAM J. Numer. Anal. 47(2), 861–886 (2009)

    Article  MathSciNet  Google Scholar 

  35. Moon, K.-S., von Schwerin, E., Szepessy, A., Tempone, R.: Convergence rates for an adaptive dual weighted residual finite element algorithm. BIT 46(2), 367–407 (2006)

    Article  MathSciNet  Google Scholar 

  36. Nochetto, R.H., Siebert, K.G., Veeser, A.: Theory of Adaptive Finite Element Methods: An Introduction. Springer, Berlin (2009)

    Google Scholar 

  37. Oden, J., Prudhomme, S.: Goal-oriented error estimation and adaptivity for the finite element method. Comput. Math. Appl. 41, 735–756 (2001)

    Article  MathSciNet  Google Scholar 

  38. Sewell, E.G.: Automatic generation of triangulations for piecewise polynomial approximation. Ph. D. dissertation. Purdue Univ., West Lafayette, IN (1972)

  39. Strang, G., Fix, G.J.: An Analysis of the Finite Element Method. Prentice-Hall (Series in Automatic Computation), Englewood Cliffs, NJ (1973)

    Google Scholar 

  40. Struwe, M.: Variational Methods, 3rd edn. Springer, Berlin (2000)

    Book  Google Scholar 

  41. Verfürth, R.: A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations. Math. Comput. 62(206), 445–475 (1994)

    Article  Google Scholar 

  42. Verfürth, R.: A review of a posteriori error estimation and adaptive mesh refinement tecniques. Teubner–Wiley, Stuttgart (1996)

Download references

Acknowledgments

MH was supported in part by NSF Awards 1065972, 1217175, 1262982, 1318480, and by AFOSR Award FA9550-12-1-0046. SP and YZ were supported in part by NSF Awards 1065972 and 1217175. YZ was also supported in part by NSF DMS 1319110, and in part by University Research Committee Grant No. F119 at Idaho State University, Pocatello, Idaho.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California at San Diego, San Diego, CA, 92093, USA

    Michael Holst

  2. Department of Mathematics, Texas A&M University, College Station, TX, 77843, USA

    Sara Pollock

  3. Department of Mathematics, Idaho State University, Pocatello, ID, 83209, USA

    Yunrong Zhu

Authors
  1. Michael Holst
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sara Pollock
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yunrong Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yunrong Zhu.

Additional information

Communicated by Gabriel Wittum.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00791-015-0243-1

Keywords

Mathematics Subject Classification

Search

Navigation