Convergence analysis of a conforming adaptive finite element method for an obstacle problem

Numerische Mathematik

Volumn 107, Issue 3, 2007, Pages 455-471

  Braess, Dietrich ^a   Carstensen, Carsten ^b   Hoppe, Ronald H W ^c,d  

^a RUHR UNIVERSITY BOCHUM   (Germany)

^b HUMBOLDT UNIVERSITY   (Germany)

^c University of Houston   (United States)

^d UNIVERSITY OF AUGSBURG   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 34548069658     PISSN: 0029599X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00211-007-0098-6     Document Type: Article

References (34)

1. Ainsworth M. and Oden J.T. (2000). A Posteriori Error Estimation in Finite Element Analysis. Wiley, Chichester
2. Ainsworth M., Oden J.T. and Lee C.Y. (1993). Local a posteriori error estimators for variational inequalities. Numer. Methods Partial Differ. Equ. 9: 23-33
3. Babuška I. and Strouboulis T. (2001). The Finite Element Method and its Reliability. Clarendon, Oxford
4. Bangerth, W., Rannacher, R.: Adaptive Finite Element Methods for Differential Equations. Lectures in Mathematics. ETH-Zürich. Birkhäuser, Basel (2003)
5. Bartels S. and Carstensen C. (2004). Averaging techniques yield reliable a posteriori finite element error control for obstacle problems. Numer. Math. 99: 225-249
6. Binev P., Dahmen W. and DeVore R. (2004). Adaptive finite element methods with convergence rates. Numer. Math. 97: 219-268
7. Braess D. (2005). A posteriori error estimators for obstacle problems-another look. Numer. Math. 101: 415-421
8. Carstensen C. (2004). Some remarks on the history and future of averaging techniques in finite element error analysis. ZAMM 84: 3-21
9. Carstensen C. (2005). A unifying theory of a posteriori finite element error control. Numer. Math. 100: 617-637
10. Carstensen C. (2006). Reliable and efficient averaging techniques as universal tool for a posteriori finite element error control on unstructured grids. Int. J. Numer. Anal. Model. 3: 333-347
11. Carstensen C. and Hoppe R.H.W. (2005). Convergence analysis of an adaptive edge finite element method for the 2D eddy current equations. J. Numer. Math. 13: 19-32
12. Carstensen C. and Hoppe R.H.W. (2006). Error reduction and convergence for an adaptive mixed finite element method. Math. Comp. 75: 1033-1042
13. Carstensen C. and Hoppe R.H.W. (2006). Convergence analysis of an adaptive nonconforming finite element method. Numer. Math. 103: 251-266
14. Carstensen C. and Verfürth R. (1999). Edge residuals dominate a posteriori error estimates for low order finite element methods. SIAM J. Numer. Anal. 36: 1571-1587
15. Chen Z. and Nochetto R. (2000). Residual type a posteriori error estimates for elliptic obstacle problems. Numer. Math. 84: 527-548
16. Dörfler W. (1996). A convergent adaptive algorithm for Poisson's equation. SIAM J. Numer. Anal. 33: 1106-1124
17. Eriksson K., Estep D., Hansbo P. and Johnson C. (1995). Computational Differential Equations. Cambridge University Press, Cambridge
18. Glowinski R., Lions J.L. and Trémolières R. (1981). Numerical Analysis of Variational Inequalities. North-Holland, Amsterdam
19. Hackbusch W. and Mittelmann H.D. (1983). On multi-grid methods for variational inequalities. Numer. Math. 42: 65-76
20. Hoppe R.H.W. (1987). Multigrid algorithms for variational inequalities. SIAM J. Numer. Anal. 24: 1046-1065
21. Hoppe R.H.W. and Kornhuber R. (1994). Adaptive multilevel methods for obstacle problems. SIAM J. Numer. Anal. 31: 301-323
22. Johnson C. (1992). Adaptive finite element methods for the obstacle problem. Math. Models Methods Appl. Sci. 2: 483-487
23. Kinderlehrer D. and Stampacchia G. (1980). An Introduction to Variational Inequalities. Academic, New York
24. Kornhuber R. (1997). Adaptive Monotone Multigrid Methods for Nonlinear Variational Problems. Teubner, Stuttgart
25. Morin P., Nochetto R.H. and Siebert K.G. (2000). Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal. 38: 466-488
26. Neittaanmäki P. and Repin S. (2004). Reliable Methods for Mathematical Modelling. Error Control and A Posteriori Estimates. Elsevier, New York
27. Nochetto R.H., Siebert K.G. and Veeser A. (2003). Pointwise a posteriori error control for elliptic obstacle problems. Numer. Math. 95: 163-195
28. Nochetto R.H., Siebert K.G. and Veeser A. (2005). Fully localized a posteriori error estimators and barrier sets for contact problems. SIAM J. Numer. Anal. 42: 2118-2135
29. Ortega J.M. and Rheinboldt W.C. (2000). Iterative Solution of Nonlinear Equations in Several Variables. SIAM, Philadelphia
30. Stevenson, R.: Optimality of a standard adaptive finite element method. Preprint. Department of Mathematics, University of Utrecht (2005)
31. Suttmeier F.T. (2005). On a direct approach to adaptive FE-discretisations for elliptic variational inequalities. J. Numer. Math. 13: 73-80
32. Veeser A. (2001). Efficient and reliable a posteriori error estimators for elliptic obstacle problems. SIAM J. Numer. Anal. 39: 146-167
33. Veeser A. (2002). Convergent adaptive finite elements for the nonlinear Laplacian. Numer. Math. 92: 743-770
34. Verfürth R. (1996). A Review of A Posteriori Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, New York