Convergence of adaptive finite element methods in computational mechanics

Applied Numerical Mathematics

Volumn 59, Issue 9, 2009, Pages 2119-2130

  Carstensen, Carsten ^a  

^a HUMBOLDT UNIVERSITY   (Germany)

Author keywords

Adaptive algorithm; Convergence; Convexity; FEM; Hencky elastoplasticity; Linear elasticity

Indexed keywords

CONVERGENCE; CONVEXITY; FEM; HENCKY ELASTOPLASTICITY; LINEAR ELASTICITY;

ADAPTIVE ALGORITHMS; COMPUTATIONAL MECHANICS; COMPUTATIONAL METHODS; ELASTICITY; ELASTOPLASTICITY; FINITE ELEMENT METHOD; PLASTICITY; REFINING; TERNARY SYSTEMS;

CONVERGENCE OF NUMERICAL METHODS;

EID: 67349255264     PISSN: 01689274     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.apnum.2008.12.006     Document Type: Article

References (39)

1. Ainsworth M., and Oden J.T. A Posteriori Error Estimation in Finite Element Analysis (2000), Wiley-Interscience (John Wiley & Sons), New York
2. Alberty J., Carstensen C., Funken S.A., and Klose R. Matlab implementation of the finite element method in elasticity. Computing 69 (2002) 239-263
3. Alberty J., Carstensen C., and Zarrabi D. Adaptive numerical analysis in primal elastoplasticity with hardening. Comput. Methods Appl. Mech. Engrg. 171 (1999) 175-204
4. Babuška I., and Strouboulis T. The Finite Element Method and its Reliability (2001), The Clarendon Press, Oxford University Press xii+802
5. Babuška I., and Vogelius M. Feedback and adaptive finite element solution of one-dimensional boundary value problems. Numer. Math. 44 (1984) 75-102
6. Binev P., Dahmen W., and DeVore R. Adaptive finite element methods with convergence rates. Numer. Math. 97 2 (2004) 219-268
7. Carstensen C. Domain decomposition for a non-smooth convex minimisation problem and its application to plasticity. Numer. Linear Algebra Appl. 4 3 (1997) 1-13
8. Carstensen C. Numerical analysis of primal elastoplasticity with hardening. Numer. Math. 82 (1999) 577-597
9. Carstensen C. Quasi-interpolation and a posteriori error analysis in finite element method. M2AN Math. Model. Numer. Anal. 33 (1999) 1187-1202
10. 2-projection onto Courant finite element spaces. Constr. Approx. 20 4 (2004) 549-564
11. Carstensen C. Convergence of an adaptive FEM for a class of degenerate convex minimization problems. IMA J. Numer. Anal. 28 (2008) 423-439
12. Carstensen C., and Alberty J. Averaging techniques for reliable a posteriori FE-error control in elastoplasticity with hardening. Comput. Meth. Appl. Mech. Engrg. 192 (2003) 1435-1450
13. 2-error control. In: Liu W.-B., Ng M., and Shi Z.-C. (Eds). Recent Progress in Scientific Computing (2007), Science Press 13-39
14. Carstensen C., and Klose R. Elastoviscoplastic finite element analysis in 100 lines of Matlab. J. Numer. Math. 10 3 (2002) 157-192
15. C. Carstensen, Y. Kondratyuk, Optimal adaptive finite element methods for uniformly convex minimisation problems (2009), in preparation
16. Carstensen C., Orlando A., and Hu J. Framework for the a posteriori error analysis of nonconforming finite element. SIAM J. Numer. Anal. 45 1 (2007) 68-82
17. Carstensen C., Orlando A., and Valdman J. A convergent adaptive finite element method for the primal problem of elastoplasticity. Int. J. Numer. Meth. Engrg. 67 13 (2006) 1851-1887
18. Carstensen C., and Verfürth R. Edge residuals dominate a posteriori error estimates for low order finite element methods. SIAM J. Numer. Anal. 36 5 (1999) 1571-1587
19. Cascon J.M., Kreuzer C., Nochetto R.H., and Siebert K.G. Quasi-optimal convergence rate for an adaptive finite element method. SIAM J. Numer. Anal. 46 5 (2008) 2524-2550
20. Dörfler W. A convergent adaptive algorithm for Poisson's equation. SIAM J. Numer. Anal. 33 3 (1996) 1106-1124
21. Eriksson K., Estep D., Hansbo P., and Johnson C. Computational Differential Equations (1995), Cambridge University Press, Cambridge
22. Han W., and Reddy B.D. Plasticity. Mathematical Theory and Numerical Analysis (1999), Springer, New York
23. Johnson C. Existence theorems for plasticity problems. J. Math. Pures Appl. 55 (1976) 431-444
24. Johnson C. A mixed finite element method for plasticity problems with hardening. SIAM J. Numer. Anal. 14 (1977) 575-583
25. Johnson C. On plasticity with hardening. J. Math. Anal. Appl. 62 (1978) 325-336
26. Johnson C., and Hansbo P. Adaptive finite element methods in computational mechanics. Comput. Methods Appl. Mech. Engrg. 101 (1992) 143-181
27. Morin P., Nochetto R.H., and Siebert K.G. Convergence of adaptive finite element method. SIAM Rev. 44 (2003) 631-658
28. Necas J., and Hlavacek I. Mathematical Theory of Elastic and Elasto-Plastic Bodies: An Introduction (1981), Elsevier, New York
29. Neittaanmäki P., and Repin S. Reliable Methods for Mathematical Modelling. Error Control and a Posteriori Estimates (2004), Elsevier, New York
30. Rannacher R., and Suttmeier F.T. A posteriori error control in finite element methods via duality techniques: Application to perfect plasticity. Comput. Mech. 21 2 (1998) 123-133
31. Rannacher R., and Suttmeier F.T. A posteriori error estimation and mesh adaptation for finite element models in elasto-plasticity. Comput. Methods Appl. Mech. Engrg. 176 1-4 (1999) 333-361
32. Simo J.C., and Hughes T.J.R. Computational Inelasticity (1998), Springer-Verlag, New York
33. Stevenson R. Optimality of a standard adaptive finite element method. Found. Comput. Math. 7 2 (2007) 245-269
34. Suquet P.-M. Discontinuities and plasticity. Nonsmooth Mechanics and Applications. CISM Courses vol. 302 (1988), Springer-Verlag, New York 279-341
35. Temam R. Mathematical Problems in Plasticity (1985), Gauthier-Villars, Paris
36. Veeser A. Convergent adaptive finite elements for the nonlinear Laplacian. Numer. Math. 92 4 (2002) 743-770
37. Verfürth R. A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques (1996), Wiley-Teubner, New York, Leipzig
38. Verfürth R. A review of a posteriori error estimation techniques for elasticity problems. Comput. Methods Appl. Mech. Engrg. 176 (1999) 419-440
39. Zeidler E. Nonlinear Functional Analysis and its Applications III and IV (1985), Springer-Verlag, New York and 1988