Computing and Visualization in Science

, Volume 17, Issue 1, pp 43–63 | Cite as

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

Article

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.

Keywords

Adaptive finite element methods  Goal oriented Semilinear elliptic problems  Quasi-orthogonality Residual-based error estimator Convergence Contraction  A posteriori estimates 

Mathematics Subject Classification

65F08 65F10 65N30 65N50 65N55 

Notes

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.

References

  1. 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)CrossRefGoogle Scholar
  2. 2.
    Bangerth, W., Rannacher, R.: Adaptive Finite Element Methods for Differential Equations. Birkhauser, Boston (2003)CrossRefGoogle Scholar
  3. 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. 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)MathSciNetGoogle Scholar
  5. 5.
    Becker, R., Rannacher, R.: Weighted a posteriori error control in FE methods. Preprint 96-1, SFB 359, Universitat, pp. 18–22 (1996)Google Scholar
  6. 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)Google Scholar
  7. 7.
    Binev, P., Dahmen, W., DeVore, R.: Adaptive finite element methods with convergence rates. Numer. Math. 97(2), 219–268 (2004)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Brenner, S., Scott, L.: The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, Berlin (2008)CrossRefGoogle Scholar
  9. 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)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Ciarlet, P.G.: Finite Element Method for Elliptic Problems. Society for Industrial and Applied Mathematics, Philadelphia, PA (2002)CrossRefGoogle Scholar
  11. 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)Google Scholar
  12. 12.
    Dörfler, W.: A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33, 1106–1124 (1996)MathSciNetCrossRefGoogle Scholar
  13. 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)Google Scholar
  14. 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)MathSciNetCrossRefGoogle Scholar
  15. 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)Google Scholar
  16. 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)MathSciNetGoogle Scholar
  17. 17.
    Evans, L.C.: Partial Differential Equations (Graduate Studies in Mathematics, V. 19) GSM/19. American Mathematical Society, New York (1998)Google Scholar
  18. 18.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1977)CrossRefGoogle Scholar
  19. 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. 20.
    Grätsch, T., Bathe, K.-J.: A posteriori error estimation techniques in practical finite element analysis. Comput. Struct. 83(4–5), 235–265 (2005)CrossRefGoogle Scholar
  21. 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. 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. 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. 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. 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)Google Scholar
  26. 26.
    Holst, M., Szypowski, R., Zhu, Y.: Two-grid methods for semilinear interface problems. Numer Methods Partial Differ. Equ. 29(5), 1729–1748 (2013)MathSciNetCrossRefGoogle Scholar
  27. 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. 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)MathSciNetCrossRefGoogle Scholar
  29. 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)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Kerkhoven, T., Jerome, J.W.: \(L_{\infty }\) stability of finite element approximations of elliptic gradient equations. Numer. Math. 57, 561–575 (1990)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Kesavan, S.: Topics in Functional Analysis and Applications. Wiley, New York, NY (1989)Google Scholar
  32. 32.
    Korotov, S.: A posteriori error estimation of goal-oriented quantities for elliptic type bvps. J. Comput. Appl. Math. 191(2), 216–227 (2006)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Mekchay, K., Nochetto, R.: Convergence of adaptive finite element methods for general second order linear elliptic PDE. SINUM 43(5), 1803–1827 (2005)MathSciNetCrossRefGoogle Scholar
  34. 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)MathSciNetCrossRefGoogle Scholar
  35. 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)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Nochetto, R.H., Siebert, K.G., Veeser, A.: Theory of Adaptive Finite Element Methods: An Introduction. Springer, Berlin (2009)Google Scholar
  37. 37.
    Oden, J., Prudhomme, S.: Goal-oriented error estimation and adaptivity for the finite element method. Comput. Math. Appl. 41, 735–756 (2001)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Sewell, E.G.: Automatic generation of triangulations for piecewise polynomial approximation. Ph. D. dissertation. Purdue Univ., West Lafayette, IN (1972)Google Scholar
  39. 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. 40.
    Struwe, M.: Variational Methods, 3rd edn. Springer, Berlin (2000)CrossRefGoogle Scholar
  41. 41.
    Verfürth, R.: A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations. Math. Comput. 62(206), 445–475 (1994)CrossRefGoogle Scholar
  42. 42.
    Verfürth, R.: A review of a posteriori error estimation and adaptive mesh refinement tecniques. Teubner–Wiley, Stuttgart (1996)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California at San DiegoSan DiegoUSA
  2. 2.Department of MathematicsTexas A&M UniversityCollege StationUSA
  3. 3.Department of MathematicsIdaho State UniversityPocatelloUSA

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