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A posteriori error estimation
for variational problems
with uniformly convex functionals


Author: Sergey I. Repin
Journal: Math. Comp. 69 (2000), 481-500
MSC (1991): Primary 65N30
DOI: https://doi.org/10.1090/S0025-5718-99-01190-4
Published electronically: August 26, 1999
MathSciNet review: 1681096
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Abstract | References | Similar Articles | Additional Information

Abstract: The objective of this paper is to introduce a general scheme for deriving a posteriori error estimates by using duality theory of the calculus of variations. We consider variational problems of the form

\begin{displaymath}\inf\limits _{v\in V} \{ F(v)+G(\Lambda v) \}, \end{displaymath}

where $F:V\rightarrow \mathbb{R}$ is a convex lower semicontinuous functional, $G: Y\rightarrow \mathbb{R}$ is a uniformly convex functional, $V$ and $Y$ are reflexive Banach spaces, and $\Lambda:V\rightarrow Y$ is a bounded linear operator. We show that the main classes of a posteriori error estimates known in the literature follow from the duality error estimate obtained and, thus, can be justified via the duality theory.


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  • 1. M. Ainsworth and J. T. Oden, A unified approach to a posteriori error estimation using element residual methods, Numer. Math., 65 (1993), 23-50. MR 95a:65185
  • 2. J. P. Aubin and H. G. Burchard, Some aspects of the method of hypercircle applied to elliptic variational problems, In B. Hubbard, editor, Numerical Solutions of Partial Differential Equations - II, SYNSPADE 1970. Academic Press, New York, London, 1971. MR 44:2359
  • 3. I. Babu$\check{\text{s}}$ka and W. C. Rheinboldt, A-posteriori error estimates for the finite element method, Internat. J. Numer. Meth. Engrg., 12 (1978), 1597-1615.
  • 4. I. Babu$\check{\text{s}}$ka and W. C. Rheinboldt, Error estimates for adaptive finite element computations, SIAM J.Numer. Anal., 15(4) (1978), 736-754. MR 58:3400
  • 5. I. Babu$\check{\text{s}}$ka and W. C. Rheinboldt, A posteriori error analysis of finite element solutions for one-dimensional problems, SIAM J.Numer. Anal., 18(3) (1981), 565-589. MR 82j:65082
  • 6. I. Babu$\check{\text{s}}$ka and R. Rodriguez, The problem of the selection of an a posteriori error indicator based on smoothing techniques, Internat. J. Numer. Meth. Engrg., 36 (1993), 539-567.
  • 7. R. E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations, Math. Comp., 44 (1985), 283-301. MR 86g:65207
  • 8. P. Ciarlet, The finite element method for elliptic problems, North-Holland, Amsterdam, 1978. MR 58:25001
  • 9. Ph. Clément, Approximations by finite element functions using local regularization, RIARO Anal. Numer., 2 (1975), 77-84.
  • 10. M. Crouzeix and V. Thomée, The stability in $L_p$ and $W^1_p$ of the $L_2$-projection onto finite element function spaces, Math. Comp., 48 (1987), 521-532. MR 88f:41016
  • 11. J. Douglas, Jr. T. Dupont and L. Wahlbin, The stability in $L^q$ of the $L^2$ projection into finite element function spaces, Numer. Math., 23 (1975), 193-197. MR 52:4669
  • 12. I. Ekeland and R. Temam, Convex analysis and variational problems, North-Holland, Amsterdam, Oxford, New-York, 1976. MR 57:3931b
  • 13. K. Eriksson and C. Johnson, An adaptive finite element method for linear elliptic problems, Math. Comp., 50 (1988), 361-383. MR 89c:65119
  • 14. R. Glowinski, Numerical methods for nonlinear variational problems. Springer-Verlag, New-York, Berlin, Heidelberg, Tokyo, 1984. MR 86c:65004
  • 15. C. Johnson and P. Hansbo, Adaptive finite element methods in computational mechanics, Comput. Methods Appl. Mech. Engrg, 101 (1992), 143-181. MR 93m:65157
  • 16. C. Johnson, Yi-Yong Nie, and V. Thomée, An a posteriori error estimate and adaptive timestep control for a backward Euler discretization of a parabolic problem, SIAM J. Numer. Anal., 27 (1990), 277-291. MR 91g:65199
  • 17. D. W. Kelly, The self-equilibration of residuals and complementary a posteriori error estimates in the finite element method, Internat. J. Numer. Meth. Engrg., 20 (1984), 1491-1506. MR 85h:73039
  • 18. P. Ladevèze and D. Leguillon, Error estimate procedure in the finite element method and applications, SIAM J. Numer. Anal., 20(3) (1983), 485-509. MR 84g:65150
  • 19. P. Ladevèze, J.-P. Pelle, and Ph. Rougeot, Error estimation and mesh optimization for classical finite elements, Engineering Computations, 8 (1991), 69-80. MR 93f:73100
  • 20. S. G. Mikhlin, Variational methods in mathematical physics. Pergamon, Oxford, 1964. MR 30:2712
  • 21. S. G. Mikhlin, Error Analysis in Numerical Processes. Whiley, New York, 1991. MR 92j:65002
  • 22. P. P. Mosolov and V. P. Myasnikov, Mechanics of rigid plastic media. Nauka, Moscow, 1981 (in Russian). MR 84e:73029
  • 23. J. T. Oden, L. Demkowicz, W. Rachowicz, and T. A. Westermann, Towards a universal $h$-$p$ adaptive finite element strategy. Part 2. A posteriori error estimation, Comput. Methods Appl. Mech. Engrg., 77 (1989), 113-180. MR 91m:76073
  • 24. S. I. Repin, A priori error estimates of variational-difference methods for Hencky plasticity problems, Zapiski Nauchnych Seminarov POMI, 221 (1995), 226-234. (in Russian, English translation in the Journal of Mathematical Sciences 1995). MR 97c:73029
  • 25. S. I. Repin, Errors of finite element methods for perfectly elasto-plastic problems, Math. Models Meth. Appl. Sci., 6(5) (1996), 587-604. MR 97f:73049
  • 26. S. I. Repin, A posteriori error estimation for nonlinear variational problems by duality theory, Zapiski Nauchnych Seminarov POMI, 242 (1997), 196-208. MR 99e:49049
  • 27. S. I. Repin, A posteriori error estimates for approximate solutions of variational problems with power growth functionals, Zapiski Nauchnych Seminarov POMI, 249 (1997), 244-255.
  • 28. S. I. Repin, A posteriori error estimates for approximate solutions of variational problems, In the Proceedings of the Second European Conference on Numerical Mathematics and Advanced Applications, Heidelberg 1997. VSP, Utrecht (1999).
  • 29. S. I. Repin, A posteriori error estimates for approximate solutions of variational problems with strongly convex functionals, Problems of Mathematical Analysis, 17 (1997), 199-226. (in Russian, English translation in the Journal of Mathematical Sciences 1998).
  • 30. S. I. Repin, A posteriori estimates of the accuracy of variational methods in problems with nonconvex functionals, Algebra i Analiz, 11 (1999) (in Russian).
  • 31. S. I. Repin, A unified approach to a posteriori error estimation by duality error majorants, Mathematics and Computers in Simulation (1999).
  • 32. S. I. Repin and G. A. Seregin, Error estimates for stresses in the finite element analysis of the two-dimensional elasto-plastic problems, Internat. J. Engrg. Sci., 33(2), (1995), 255-268. MR 96c:73068
  • 33. S. I. Repin and L. S. Xanthis, A posteriori error estimation for elasto-plastic problems based on duality theory, Comput. Methods in Appl. Mech. Engrg., 138 (1996), 317-339. MR 97k:73032
  • 34. S. I. Repin and L. S. Xanthis, A posteriori error estimation for nonlinear variational problems, C. R. Acad. Sci. Paris, Serie I., 324 (1997), 1169-1174. MR 98d:49043
  • 35. S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics. Izdt. Leningrad. Gos. Univ., Leningrad, 1955. in Russian. English translation in Translation of Mathematical Monographs, Volume 90 American Mathematical Society, Providence, Rhode Island. MR 92e:46067
  • 36. R. Verfürth, A review of a posteriori error estimation and adaptive mesh-refinement techniques. Wiley, Teubner, 1996.
  • 37. O. C. Zienkiewicz and J. Z. Zhu, A simple error estimator and adaptive procedure for practical engineering analysis, Internat. J. Numer. Meth. Engrg., 24 (1987), 337-357. MR 87m:73055
  • 38. O. C. Zienkiewicz and J. Z. Zhu, The superconvergent patch recovery (SPR) and adaptive finite element refinement, Comput. Methods Appl. Mech. Engrg., 101 (1992) 207-224. MR 93h:73042

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Additional Information

Sergey I. Repin
Affiliation: Department of Applied Mathematics St. Petersburg State Technical University 195251, St. Petersburg, Russia
Email: repin@mat.amd.stu.neva.ru

DOI: https://doi.org/10.1090/S0025-5718-99-01190-4
Keywords: A posteriori error estimates, duality theory, nonlinear variational problems
Received by editor(s): April 1, 1997
Published electronically: August 26, 1999
Additional Notes: This research was supported by INTAS Grant N 96-0835.
Article copyright: © Copyright 2000 American Mathematical Society

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