Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

A posteriori error estimator and error control for contact problems


Authors: Alexander Weiss and Barbara I. Wohlmuth
Journal: Math. Comp. 78 (2009), 1237-1267
MSC (2000): Primary 65N30, 65N15, 65N50
DOI: https://doi.org/10.1090/S0025-5718-09-02235-2
Published electronically: February 20, 2009
MathSciNet review: 2501049
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we consider two error estimators for one-body contact problems. The first error estimator is defined in terms of $ H($div$ )$-conforming stress approximations and equilibrated fluxes while the second is a standard edge-based residual error estimator without any modification with respect to the contact. We show reliability and efficiency for both estimators. Moreover, the error is bounded by the first estimator with a constant one plus a higher order data oscillation term plus a term arising from the contact that is shown numerically to be of higher order. The second estimator is used in a control-based AFEM refinement strategy, and the decay of the error in the energy is shown. Several numerical tests demonstrate the performance of both estimators.


References [Enhancements On Off] (What's this?)

  • 1. M. AINSWORTH, Robust a posteriori error estimation for nonconforming finite element approximation, SIAM J. Numer. Anal., 42 (2005), pp. 2320-234. MR 2139395 (2006j:65331)
  • 2. M. AINSWORTH AND J. ODEN, A posteriori error estimation in finite element analysis, John Wiley and Sons, New York, 2000. MR 1885308 (2003b:65001)
  • 3. M. AINSWORTH, J. ODEN, AND C. LEE, Local a posteriori error estimators for variational inequalities, Numer. Methods Partial Differ. Equations, 9 (1993), pp. 23-33. MR 1193438 (93m:49015)
  • 4. D. ARNOLD AND R. WINTHER, Mixed finite elements for elasticity, Numer. Math., 92 (2002), pp. 401-419. MR 1930384 (2003i:65103)
  • 5. I. BABUSKA AND M. VOGELIUS, Feedback and adaptive finite element solution of one-dimensional boundary value problems, Numer. Math., 44 (1984), pp. 75-102. MR 745088 (85k:65070)
  • 6. I. BABUŠSKA AND T. STROUBOULIS, The finite element method and its reliability, Oxford: Clarendon Press, 2001. MR 1857191 (2002k:65001)
  • 7. F. BEN BELGACEM, Numerical simulation of some variational inequalities arisen from unilateral contact problems by the finite element methods, SIAM J. Numer. Anal., 37 (2000), pp. 1198-1216. MR 1756422 (2002c:74066)
  • 8. F. BEN BELGACEM, P. HILD, AND P. LABORDE, Extension of the mortar finite element method to a variational inequality modeling unilateral contact, Mathematical Models & Methods in Applied Sciences, 9 (1999), pp. 287-303. MR 1674556 (99m:73066)
  • 9. F. BEN BELGACEM AND Y. RENARD, Hybrid finite element methods for the Signorini problem, Mathematics of Computation, 72 (2003), pp. 1117-1145. MR 1972730 (2004d:65139)
  • 10. P. BINEV, W. DAHMEN, AND R. DEVORE, Adaptive finite element methods with convergence rates, Numer. Math, 97 (2004), pp. 219-268. MR 2050077 (2005d:65222)
  • 11. H. BLUM AND F. SUTTMEIER, An adaptive finite element discretisation for a simplified Signorini problem, Calcolo, 37 (2000), pp. 65-77. MR 1819903 (2002a:65171)
  • 12. V. BOSTAN AND W. HAN, A posteriori error analysis for finite element solutions of a frictional contact problem, Comput. Methods Appl. Mech. Engrg., 195 (2006), pp. 1252-1274. MR 2195300 (2008a:74076)
  • 13. V. BOSTAN, W. HAN, AND B. REDDY, A posteriori error estimation and adaptive solution of elliptic variational inequalities of the second kind, Appl. Numer. Math., 52 (2005), pp. 13-38. MR 2102908 (2005i:65167)
  • 14. D. BRAESS, A posteriori error estimators for obstacle problems - another look, Numer. Math., 101 (2005), pp. 523-549. MR 2194822 (2006k:65287)
  • 15. D. BRAESS, C. CARSTENSEN, AND R. HOPPE, Convergence analysis of a conforming adaptive finite element method for an obstacle problem, Numer. Math., 107 (2007), pp. 455-471. MR 2336115 (2008f:65208)
  • 16. D. BRAESS AND W. DAHMEN, The mortar element method revisited-what are the right norms?, in Domain decomposition methods in science and engineering. Papers of the thirteenth international conference on domain decomposition methods, Debit et al., eds., 2002, pp. 27-40. MR 2004064
  • 17. U. BRINK AND E. STEIN, A posteriori error estimation in large-strain elasticity using equilibrated local Neumann problems, Comput. Methods Appl. Mech. Eng., 161 (1998), pp. 77-101. MR 1633995 (99h:73074)
  • 18. G. BUSCAGLIA, R. DURAN, E. FANCELLO, R. FEIJOO, AND C. PADRA, An adaptive finite element approach for frictionless contact problems, Int. J. Numer. Meth. Engng., 50 (2001), pp. 394-418. MR 1807471 (2001j:74085) 3
  • 19. C. CARSTENSEN AND S. BARTELS, Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. I: Low order conforming, nonconforming, and mixed FEM, Math. Comput. 71 (2002), pp. 945-969. MR 1898741 (2003e:65212)
  • 20. C. CARSTENSEN, S. BARTELS, AND S. JANSCHE, A posteriori error estimates for nonconforming finite element methods, Numer. Math., 92 (2002), pp. 233-256. MR 1922920 (2003g:65139)
  • 21. C. CARSTENSEN, P. CAUSIN, AND R. SACCO, A posteriori dual-mixed adaptive finite element error control for Lamé and Stokes equations, Numer. Math., 101 (2005), pp. 309-332. MR 2195346 (2007b:74040)
  • 22. C. CARSTENSEN, O. SCHERF, AND P. WRIGGERS, Adaptive finite elements for elastic bodies in contact, SIAM J. Sci. Comput., 20 (1999), pp. 1605-1626. MR 1694675 (2000j:74065)
  • 23. P. COOREVITS, P. HILD, K. LHALOUANI, AND T. SASSI, Mixed finite element methods for unilateral problems: Convergence analysis and numerical studies, Mathematics of Computation, 71 (2001), pp. 1-25. MR 1862986 (2002g:74043)
  • 24. E. DARI, R. DURAN, C. PADRA, AND V. VAMPA, A posteriori error estimators for nonconforming finite element methods, Math. Modelling Numer. Anal., 30 (1996), pp. 385-400. MR 1399496 (97f:65066)
  • 25. W. D¨ORFLER, A convergent adaptive algorithm for Poisson's equation, SIAM J. Numer. Anal., 33 (1996), pp. 1106-1124. MR 1393904 (97e:65139)
  • 26. B. ERDMANN, M. FREI, R. HOPPE, R. KORNHUBER, AND U. WIEST, Adaptive finite element methods for variational inequalities, East-West J. Numer. Math., 1 (1993), pp. 165-197. MR 1253634 (94k:65154)
  • 27. W. HAN, A posteriori error analysis via duality theory. With applications in modeling and numerical approximations, Springer, New York, 2005. MR 2101057 (2005k:65003)
  • 28. J. HASLINGER, I. HLAVáCEK, AND J. NECAS, Numerical methods for unilateral problems in solid mechanics, in Handbook of Numerical Analysis, P. Ciarlet and J.-L. Lions, eds., vol. IV, North-Holland, Amsterdam, 1996, pp. 313-485. MR 1422506
  • 29. P. HILD, Numerical implementation of two nonconforming finite element methods for unilateral contact, Computer Methods in Applied Mechanics and Engineering, 184 (2000), pp. 99-123. MR 1752624 (2000k:74058)
  • 30. P. HILD AND P. LABORDE, Quadratic finite element methods for unilateral contact problems, Applied Numerical Mathematics, 41 (2002), pp. 410-421. MR 1903172 (2003e:65221)
  • 31. P. HILD AND S. NICAISE, A posteriori error estimations of residual type for Signorini's problem, Numer. Math., 101 (2005), pp. 523-549. MR 2194827 (2006j:65347)
  • 32. R. HOPPE AND R. KORNHUBER, Adaptive multilevel methods for obstacle problems, SIAM J. Numer. Anal., 31 (1994), pp. 301-323. MR 1276702 (95c:65181)
  • 33. S. H¨UEBER AND B. WOHLMUTH, A primal-dual active set strategy for non-linear multibody contact problems, Computer Methods in Applied Mechanics and Engineering, 194 (2005), pp. 3147-3166. MR 2142538 (2005m:74071)
  • 34. C. JOHNSON, Adaptive finite element methods for the obstacle problem, Math. Models Methods Appl. Sci., 2 (1992), pp. 483-487. MR 1189062 (93k:65093)
  • 35. D. KELLY, The self-equilibration of residuals and complementary a posteriori error estimates in the finite element method, Int. J. Numer. Methods Eng., 20 (1984), pp. 1491-1506. MR 759999 (85h:73039)
  • 36. D. KELLY AND J. ISLES, Procedures for residual equilibration and local error estimation in the finite element method, Commun. Appl. Numer. Methods, 5 (1989), pp. 497-505.
  • 37. N. KIKUCHI AND J. ODEN, Contact problems in elasticity: A study of variational inequalities and finite element methods, vol. 8, SIAM Studies in Applied Mathematics, Philadelphia, 1988. MR 961258 (89j:73097)
  • 38. L. SCOTT AND S. ZHANG, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp., 54 (1990), pp. 483-493. MR 1011446 (90j:65021)
  • 39. P. LADEVÈZE AND D. LEGUILLON, Error estimate procedure in the finite element method and applications, SIAM J. Numer. Anal., 20 (1983), pp. 485-509. MR 701093 (84g:65150)
  • 40. P. LADEVÈZE AND E. MAUNDER, A general method for recovering equilibrating element tractions, Comput. Methods Appl. Mech. Engrg., 137 (1996), pp. 111-151.
  • 41. C. LEE AND J. ODEN, A posteriori error estimation of $ h$-$ p$ finite element approximations of frictional contact problems., Comput. Methods Appl. Mech. Eng., 113 (1994), pp. 11-45. MR 1266922 (95f:73070)
  • 42. P. MORIN, R. NOCHETTO, AND K. SIEBERT, Data oscillation and convergence of adaptive FEM, SIAM J. Numer. Anal., 38 (2000), pp. 466-488. MR 1770058 (2001g:65157)
  • 43. -, Convergence of adaptive finite element methods, SIAM Rev., 44 (2002), pp. 631-658. MR 1980447
  • 44. S. NICAISE, K. WITOWSKI, AND B. WOHLMUTH, An a posteriori error estimator for the Lamé equation based on $ {H}(\operatorname{div})$-conforming stress approximations, IMA J. Numer. Anal., 28 (2008), pp. 331-353. MR 2401201
  • 45. R. NOCHETTO AND K. MEKCHAY, Convergence of adaptive finite element methods for general second order linear elliptic PDEs, SIAM J. Numer. Anal., 43 (2005), pp. 1803-1827. MR 2192319 (2006i:65201)
  • 46. R. NOCHETTO, A. SCHMIDT, K. SIEBERT, AND A. VEESER, Pointwise a posteriori error estimates for monotone semi-linear equations, Numer. Math., 104 (2006), pp. 515-538. MR 2249676 (2008a:65201)
  • 47. S. OHNIMUS, E. STEIN, AND E. WALHORN, Local error estimates of FEM for displacements and stresses in linear elasticity by solving local Neumann problems, Int. J. Numer.Meth. Engng., 52 (2001), pp. 727-746. MR 1857633 (2002f:74075)
  • 48. E. STEIN AND S. OHNIMUS, Equilibrium method for postprocessing and error estimation in the finite element method, Comput. Assist. Mech. Eng. Sci., 4 (1997), pp. 645-666.
  • 49. -, Anisotropic discretization- and model-error estimation in solid mechanics by local Neumann problems, Comput. Methods Appl Mech. Engrg., 176 (1999), pp. 363-385. MR 1665353 (2000d:74078)
  • 50. E. STEIN, S. OHNIMUS, AND E. WALHORN, Adaptive finite element discretization in elasticity and elastoplasticity by global and local error estimators using local Neumann-problems, Zeitschrift f. Angewandte Mathematik und Mechanik, 79 (1999), pp. 147-150.
  • 51. R. STEVENSON, An optimal adaptive finite element method, SIAM J. Numer. Anal., 42 (2005), pp. 2188-2217. MR 2139244 (2006e:65226)
  • 52. -, Optimality of a standard adaptive finite element method, Found. Comput. Math., 7 (2007), pp. 245-269. MR 2324418 (2008i:65272)
  • 53. A. VEESER, On a posteriori error estimation for constant obstacle problems, in Numerical methods for viscosity solutions and applications, Falcone, ed., vol. 59 of World Scientific. Ser. Adv. Math. Appl. Sci., Singapore, 2001, pp. 221-234. MR 1886715
  • 54. R. VERFÜRTH, A posteriori error estimation and adaptive mesh-refinement techniques, J. Comp. Appl. Math., 50 (1994), pp. 67-83. MR 1284252 (95c:65171)
  • 55. -, A review of a posteriori error estimation and adaptive mesh-refinement techniques, Wiley-Teubner Series Advances in Numerical Mathematics, Wiley-Teubner, Chichester, Stuttgart, 1996.
  • 56. -, A review of a posteriori error estimation techniques for elasticity problems, Comput. Methods Appl. Mech. Engrg., 176 (1999), pp. 419-440. MR 1665355 (2000f:74075)
  • 57. Y. WANG, Preconditioning for the mixed formulation of linear plane elasticity, Ph.D. thesis, Texas A & M University, 2004.
  • 58. B. WOHLMUTH, Discretization Methods and Iterative Solvers Based on Domain Decomposition, Springer, Berlin, 2001. MR 1820470 (2002c:65231)
  • 59. -, A comparison of dual Lagrange multiplier spaces for mortar finite element discretizations, M$ ^2$AN Math. Model. Numer. Anal., 36 (2002), pp. 995-1012. MR 1958655 (2004b:65193)
  • 60. -, An a posteriori error estimator for two-body contact problems on non-matching meshes, J. Sci Comp., 33 (2007), pp. 25-45. MR 2338331 (2008f:74084)
  • 61. P. WRIGGERS AND O. SCHERF, Different a posteriori error estimators and indicators for contact problems, Math. Comput. Modelling, 28 (1998), pp. 437-447. MR 1648773 (99f:73079)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65N30, 65N15, 65N50

Retrieve articles in all journals with MSC (2000): 65N30, 65N15, 65N50


Additional Information

Alexander Weiss
Affiliation: Institute of Applied Analysis and Numerical Simulations (IANS), Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
Email: weiss@ians.uni-stuttgart.de

Barbara I. Wohlmuth
Affiliation: Institute of Applied Analysis and Numerical Simulations (IANS), Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
Email: wohlmuth@ians.uni-stuttgart.de

DOI: https://doi.org/10.1090/S0025-5718-09-02235-2
Keywords: Equilibrated fluxes, Lagrange multipliers, a posteriori error estimates, contact problems
Received by editor(s): July 17, 2007
Received by editor(s) in revised form: June 2, 2008
Published electronically: February 20, 2009
Additional Notes: This work was supported in part by the Deutsche Forschungsgemeinschaft, SFB 404, B8
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society