Some remarks on the history and future of averaging techniques in a posteriori finite element error analysis

ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik

Volumn 84, Issue 1, 2004, Pages 3-21

  Carstensen, Carsten ^a  

^a VIENNA UNIVERSITY OF TECHNOLOGY   (Austria)

Author keywords

A posteriori error estimates; Averaging techniques; Efficiency; Gradient recovery; Reliability; ZZ error estimator

Indexed keywords

EID: 1642532459     PISSN: 00442267     EISSN: None     Source Type: Journal    
DOI: 10.1002/zamm.200410101     Document Type: Conference Paper

References (48)

1. M. Ainsworth and J. T. Oden, A Posteriori Error Estimation in Finite Element Analysis, Wiley-Interscience (John Wiley & Sons, New York, 2000).
2. J. Alberty and C. Carstensen, Numerical analysis of time-depending primal elastoplasticity with hardening, SIAM J. Numer. Anal. 37(4), 1271-1294 (2000) (electronic).
3. J. Alberty and C. Carstensen, Averaging techniques for reliable a posteriori FE-error control in elastoplasticity with hardening, Comput. Methods Appl. Mech. Eng. 192, 1435-1450 (2003).
4. I. Babuška and A. Miller, A feedback finite element method with a posteriori error estimation, Part I: The finite element method and some properties of the a posteriori estimator, Comput. Methods Appl. Mech. Eng. 61, 1-40 (1987).
5. I. Babuška and W. C. Rheinboldt, Error estimates for adaptive finite element computations, SIAM J. Numer. Anal. 15, 736-754 (1978).
6. I. Babuška and T. Strouboulis, The Finite Element Method and its Reliability (The Clarendon Press, Oxford University Press, 2001).
7. R. E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations, Math. Comput. 44(170), 283-301 (1985).
8. R. E. Bank and J. Xu, Asymptotically exact a posteriori error estimators, Part I: Grids with superconvergence; Part II: General unstructured grids, Preprint.
9. W. Bao and J. W. Barrett, A priori and a posteriori error bounds for a non-conforming linear finite element approximation of a non-Newtonian flow, RAIRO Model. Math. Anal. Numer. 32(7), 843-858 (1998).
10. S. Bartels and C. Carstensen, Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids, Part II: Higher order FEM, Math. Comput. 71(239), 971-994 (2002).
11. S. Bartels, C. Carstensen, and G. Dolzmann, Inhomogeneous Dirichlet conditions in a priori and a posteriori finite element error analysis, Kieler Berichtsreihe 1 (2002).
12. R. Becker, R. Rannacher, A feed-back approach to error control in finite element methods: basic analysis and examples, East-West J. Numer. Math. 4(4), 237-264 (1996).
13. R. Becker and R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods, Acta Numer., 1-102 (2001).
14. P. Binev, W. Dahmen, and R. DeVore, Adaptive Finite Element Methods with Convergence Rates, IGPM Report, RWTH Aachen, (2002), preprint available at http://www.igpm.rwth-aachen.de/dahmen/rep_02.html.
15. D. Braess, Finite Elements (Cambridge University Press, Cambridge, 2001).
16. S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods (Springer Verlag, New York, 2002).
17. C. Carstensen, Numerical analysis of the primal problem of elastoplasticity with hardening, Numer. Math. 82(4), 577-597 (1999).
18. C. Carstensen, Quasi-interpolation and a posteriori error analysis in finite element method, M2AN 33, 1187-1202 (1999).
19. C. Carstensen, A unifying theory of a posteriori finite element error control, Preprint (2002).
20. C. Carstensen, All first-order averaging techniques for a posteriori finite element error control on unstructured grids are efficient and reliable, preprint (2002).
21. C. Carstensen, Estimation of higher Sobolev norm from lower order approximation, preprint (2002).
22. 2-projection onto finite element spaces, Math. Comput. 71(237), 157-163 (2002).
23. C. Carstensen and S. Bartels, Averaging techniques yield reliable a posteriori finite element error control for obstacle problems, Publ. Max-Planck-Inst. Math. Sci. (Leipzig) 2 (2001), preprint available at http://www.mis.mpg.de/preprints/2001/prepr0201-abstr.html
24. 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(239), 945-969 (2002).
25. C. Carstensen, S. Bartels, and R. Klose, An experimental survey of a posteriori Courant finite element error control for the Poisson equation, Adv. Comput. Math. 15, 1-4, 79-106 (2001).
26. C. Carstensen and S. A. Funken, Coupling of nonconforming finite elements and boundary elements I: A priori estimates, Computing 62(3), 229-241 (1999).
27. C. Carstensen and S. A. Funken, A posteriori error control in low-order finite element discretisations of incompressible stationary flow problems, Math. Comput. 70(236), 1353-1381 (2001).
28. C. Carstensen and S. A. Funken, Averaging technique for FE-a posteriori error control in elasticity, Part II: λ-independent estimates, Comput. Methods Appl. Mech. Eng. 190, 4663-4675 (2001).
29. C. Carstensen and S. A. Funken, Averaging technique for a posteriori error control in elasticity. III. Locking-free nonconforming FEM, Comput. Methods Appl. Mech. Eng. 191, 8-10, 861-877 (2001).
30. C. Carstensen and K. Jochimsen, Adaptive finite element error control for non-convex minimization problems: Numerical two-well model example allowing microstructures, Computing (to appear).
31. C. Carstensen, W. Liu, and N. Yan, A Posteriori Error Estimators Based on Gradient Recovery for Finite Element Approximation of p-Laplacian, Report UKC/IMS/01/44 (2001).
32. C. Carstensen and S. Müller, Local stress regularity in scalar non-convex variational problems, SIAM J. Math. Anal. (to appear), preprint 49 of DFG Priority Program Multiscale Problems available at http://www.mathematik. uni-stuttgart.de/mathA/lst1/mehrskalen/.
33. C. Carstensen and R. Verfürth, Edge residuals dominate a posteriori error estimates for low order finite element methods, SIAM J. Numer. Anal. 36(5), 1571-1587 (1999).
34. P. G. Ciarlet, The Finite Element Method for Elliptic Problems (North-Holland Publishing Co., Amsterdam, 1978).
35. P. Clément, Approximation by finite element functions using local regularization, RAIRO, Sér. Rouge Anal. Numer. R-2, 77-84 (1975).
36. W. Dörfler, A convergent adaptive algorithm for Poisson's equation, SIAM J. Numer. Anal. 33, 1106-1124 (1996).
37. W. Dörfler and R. H. Nochetto, Small data oscillation implies the saturation assumption, Numer. Math. 91(1), 1-12 (2002).
38. K. Eriksson and C. Johnson, Error estimates and automatic time step control for nonlinear parabolic problems I, SIAM J. Numer. Anal. 24(1), 12-23 (1978).
39. K. Eriksson, D. Estep, P. Hansbo, and C. Johnson, Introduction to adaptive methods for differential equations, Acta Numer., 105-158 (1995).
40. W. Hoffmann, A. H. Schatz, L. B. Wahlbin, and G. Wittum, Asymptotically exact a posteriori estimators for the pointwise gradient error on each element in irregular meshes. I. A smooth problem and globally quasi-uniform meshes, Math. Comput. 70, 235, 897-909 (2001).
41. R. Kouhia and R. Stenberg, A linear nonconforming finite element method for nearly incompressible elasticity and Stokes flow, Comput. Methods Appl. Mech. Eng. 124(3), 195-212 (1995).
42. P. Morin and R. H. Nochetto, and K. G. Siebert, Local problems on stars: a posteriori error estimation, convergence, and performance, Math. Comput., to appear.
43. R. H. Nochetto, Removing the saturation assumption in a posteriori error analysis, Ist. Lomb. Accad. Sci. Lett. Rend. A 127, 67-82 (1993).
44. R. Rodriguez, Some remarks on Zienkiewicz-Zhu estimator, Int. J. Numer. Methods Partial Diff. Equ. 10, 625-635 (1994).
45. R. Rodriguez, A posteriori error analysis in the finite element method, in: Finite Element Methods. 50 Years of the Courant Element, Conference held at the Univ. of Jyväskylä, Finland, 1993, edited by M. Krizek (Marcel Dekker, New York, NY); Inc. Lect. Notes Pure Appl. Math. 164, 389-397 (1994).
46. R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques (Wiley-Teubner, 1996).
47. L. B. Wahlbin, Superconvergence in Galerkin Finite Element Methods (Springer-Verlag, Berlin, 1995).
48. M. Wheeler and J. Whiteman, Superconvergent recovery of gradients on subdomains from piecewise linear finite-element approximations, Numer. Methods Partial Differ. Equ. 3(4), 357-374 (1987).