All first-order averaging techniques for a posteriori finite element error control on unstructured grids are efficient and reliable

Mathematics of Computation

Volumn 73, Issue 247, 2004, Pages 1153-1165

  Carstensen, C ^a  

^a VIENNA UNIVERSITY OF TECHNOLOGY   (Austria)

Author keywords

A posteriori error estimate; Averaging operator; Efficiency; Finite element method; Gradient recovery; Mixed finite element method; Nonconforming finite element method

Indexed keywords

EID: 2942651067     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0025-5718-03-01580-1     Document Type: Article

References (22)

1. [AO] M. AINSWORTH, J.T ODEN: A posteriori error estimation in finite element analysis, John Wiley & Sons, New York, 2001. MR 2003b:65001
2. [BS] I. BABUŠKA, T. STROUBOULIS: The Finite Element Method and its Reliability. Oxford University Press, 2001. MR 2002k:65001
3. [BC1] S. BARTELS, C. CARSTENSEN: Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part II: Higher order FEM. Math. Comp. 71 (2002) 971-994. MR 2003e:65207
4. [BC2] S. BARTELS, C. CARSTENSEN: Averaging techniques yield reliable a posteriori finite element error control for obstacle problems. Numer. Math. (2003) to appear.
5. [BR] R. BECKER, R. RANNACHER: A feed-back approach to error control in finite element methods: basic analysis and examples. East-West Journal of Numerical Mathematics 4 Number 4 (1996) 237-264. MR 98m:65185
6. [B] D. BRAESS: Enhanced assumed strain elements and locking in membrane problems, Comp. Meths. Appl. Mech. Engrg. 165 (1998) 155-174. MR 2000j:74084
7. [C] C. CARSTENSEN: Quasi-interpolation and a posteriori error analysis in finite element method. M2AN 33 (1999) 1187-1202. MR 2001a:65135
8. [CA] C. CARSTENSEN, J. ALBERTY: Averaging techniques for reliable a posteriori FE-error control in elastoplasticity with hardening. Comput. Methods Appl. Mech. Engrg. 192 (2003) 1435-1450.
9. [CB] C. CARSTENSEN, S. BARTELS: Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids, part I: Low order conforming, nonconforming, and mixed FEM. Math. Comp. 71 (2002) 945-969. MR 2003e:65212
10. [CBJ] C. CARSTENSEN, S. BARTELS, S. JANSCHE: A posteriori error estimates for nonconforming finite element methods. Numer. Math. 92 (2002) 233-256. MR
11. [CF1] C. CARSTENSEN, S.A. FUNKEN: Constants in Clément-interpolation error and residual based a posteriori estimates in finite element methods, East-West Journal of Numerical Analysis, 8, 3, 153-256. MR 2002a:65173
12. [CF2] C. CARSTENSEN, S.A. FUNKEN: Fully reliable localised error control in the FEM, SIAM J. Sci. Comp., 21, 4, 1465-1484. MR 2000k:65205
13. [CF3] C. CARSTENSEN, S.A. FUNKEN: Averaging technique for FE - a posteriori error control in elasticity. Part I: Conforming FEM. Comput. Methods Appl. Mech. Engrg. 190 (2001), pp. 2483-2498, Part II: λ-independent estimates. Comput. Methods Appl. Mech. Engrg. 190 (2001) 4663-4675. Part III: Locking-free nonconforming FEM. Comput. Methods Appl. Mech. Engrg. 191 (2001), no. 8-10, 861-877. MR 2002a:74114, MR 2002d:65140, MR 2002j:65106
14. [CF3] C. CARSTENSEN, S.A. FUNKEN: Averaging technique for FE - a posteriori error control in elasticity. Part I: Conforming FEM. Comput. Methods Appl. Mech. Engrg. 190 (2001), pp. 2483-2498, Part II: λ-independent estimates. Comput. Methods Appl. Mech. Engrg. 190 (2001) 4663-4675. Part III: Locking-free nonconforming FEM. Comput. Methods Appl. Mech. Engrg. 191 (2001), no. 8-10, 861-877. MR 2002a:74114, MR 2002d:65140, MR 2002j:65106
15. [CF3] C. CARSTENSEN, S.A. FUNKEN: Averaging technique for FE - a posteriori error control in elasticity. Part I: Conforming FEM. Comput. Methods Appl. Mech. Engrg. 190 (2001), pp. 2483-2498, Part II: λ-independent estimates. Comput. Methods Appl. Mech. Engrg. 190 (2001) 4663-4675. Part III: Locking-free nonconforming FEM. Comput. Methods Appl. Mech. Engrg. 191 (2001), no. 8-10, 861-877. MR 2002a:74114, MR 2002d:65140, MR 2002j:65106
16. [CF4] C. CARSTENSEN, S.A. FUNKEN: A posteriori error control in low-order finite element discretisations of incompressible stationary flow problems. Math. Comp. 70 (2001) 1353-1381. MR 2002f:65157
17. [CV] C. CARSTENSEN, R. VERFÜRTH: Edge residuals dominate a posteriori error estimates for low order finite element methods, SIAM J. Numer. Anal. 36, 5,(1999) 1571-1587. MR 2000g:65115
18. [N] R. NOCHETTO: Removing the saturation assumption in a posteriori error analysis. Rend., Sci. Mat. Appl., A 127, 67-82 (1994). MR 95c:65187
19. [R1] R. RODRIGUEZ: Some remarks on Zienkiewicz-Zhu estimator. Int. J. Numer. Meth. in PDE 10 (1994) 625-635. MR 95e:65103
20. [R2] R. RODRIGUEZ: A posteriori error analysis in the finite element method. Finite element methods. 50 years of the Courant element. Conference held at the University of Jyvaeskylae, Finland, 1993. Inc. Lect. Notes Pure Appl. Math. 164, 389-397 (1994). MR 95g:65158
21. [V] R. VERFÜRTH: A review of a posteriori error estimation and adaptive mesh-refinement techniques, 1996, Wiley-Teubner.
22. [ZZ] O.C. ZIENKIEWICZ, J.Z. ZHU: A simple error estimator and adaptive procedure for practical engineering analysis, Int. J. Numer. Meth. Engrg., 24 (1987) 337-357. MR 87m:73055