Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

 
 

 

Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part II: Higher order FEM


Authors: Sören Bartels and Carsten Carstensen
Journal: Math. Comp. 71 (2002), 971-994
MSC (2000): Primary 65N30, 65R20, 74B20, 74G99, 74H99
DOI: https://doi.org/10.1090/S0025-5718-02-01412-6
Published electronically: February 4, 2002
MathSciNet review: 1898742
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Averaging techniques are popular tools in adaptive finite element methods since they provide efficient a posteriori error estimates by a simple postprocessing. In the second paper of our analysis of their reliability, we consider conforming $h$-FEM of higher (i.e., not of lowest) order in two or three space dimensions. In this paper, reliablility is shown for conforming higher order finite element methods in a model situation, the Laplace equation with mixed boundary conditions. Emphasis is on possibly unstructured grids, nonsmoothness of exact solutions, and a wide class of local averaging techniques. Theoretical and numerical evidence supports that the reliability is up to the smoothness of given right-hand sides.


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

  • [ACF] J. ALBERTY, C. CARSTENSEN, S.A. FUNKEN: Remarks around $50$ lines of Matlab: short finite element implementation. Numer. Algorithms 20 (1999) 117-137. CMP 2000:01
  • [Baetal] I. BABUSSKA, T. STROUBOULIS, C.S. UPADHYAY, S.K. GANGARAJ, K. COPPS: Validation of a posteriori error estimators by numerical approach. Int. J. Numer. Meth. Engrg. 37 (1994) 1073-1123. MR 95e:65096
  • [BeR] 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, No. 4 (1996) 237-264. MR 98m:65185
  • [B] D. BRAESS: Finite Elements. Cambridge University Press (1997). MR 98f:65002
  • [BV] D. BRAESS, R. VERFÜRTH: A posteriori error estimators for the Raviart-Thomas element. SIAM J. Numer. Anal. 33 (1996) 2431-2444. MR 97m:65201
  • [BS] S.C. BRENNER, L.R. SCOTT: The Mathematical Theory of Finite Element Methods. Texts Appl. Math. 15, Springer, New-York (1994). MR 95f:65001
  • [Ca] C. CARSTENSEN: Quasi interpolation and a posteriori error analysis in finite element method, M2AN 33 (1999) 1187-1202.
  • [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., posted on February 4, 2002, PII S 0025-5718(02)01402-3 (to appear in print).
  • [CF] C. CARSTENSEN, S.A. FUNKEN: Constants in Clément-interpolation error and residual based a posteriori error estimates in Finite Element Methods. East-West J. Numer. Anal. 8 (2000), 153-175. CMP 2001:07
  • [CMS] C. CARSTENSEN, S. MAISCHAK, E.P. STEPHAN: A posteriori error estimate and $h$-adaptive algorithm on surfaces for Symm's integral equation. Numer. Math. (2001) Published online May 30, 2001, DOI 10.1007/s002110100287.
  • [CV] C. CARSTENSEN, R. VERFÜRTH: Edge residuals dominate a posteriori error estimates for low order finite element methods. SIAM J. Numer. Anal. 36 No. 5 (1999) 1571-1587. MR 2000g:65125
  • [Cl] P. CL´EMENT: Approximation by finite element functions using local regularization. RAIRO Sér. Rouge Anal. Numér. R-2 (1975) 77-84. MR 53:4569
  • [Ci] P.G. CIARLET: The finite element method for elliptic problems. North-Holland, Amsterdam (1978). MR 58:25001
  • [EEHJ] K. ERIKSSON, D. ESTEP, P. HANSBO, C. JOHNSON: Introduction to adaptive methods for differential equations. Acta Numerica (1995) 105-158. MR 96k:65057
  • [GR] V. GIRAULT, P.A. RAVIART: Finite Element Methods for Navier-Stokes Equations. Springer, Berlin (1986). MR 88b:65129
  • [MBa] J.M. MELENK, I. BABUSSKA: The partition of unity finite element method: Basic theory and applications. Comput. Methods Appl. Mech. Eng. 139, No.1-4, 289-314 (1996). MR 97k:65258
  • [R] R. RODRIGUEZ: Some remarks on Zienkiewicz-Zhu estimator. Int. J. Numer. Meth. in PDE 10 (1994) 625-635. MR 95e:65103
  • [V] R. VERFÜRTH: A review of a posteriori error estimation and adaptive mesh-refinement techniques. Wiley-Teubner (1996).
  • [Y1] D. YU: Asymptotically exact a posteriori error estimators for elements of bi-odd degree. Chinese J. Num. Math. and Appl. 13, no. 4, 82-90 (1991). MR 94h:65103
  • [Y2] D. YU: Asymptotically exact a posteriori error estimators for elements of bi-even degree. Chinese J. Num. Math. and Appl. 13, no. 2, 64-78 (1991). MR 92i:65173
  • [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

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65N30, 65R20, 74B20, 74G99, 74H99

Retrieve articles in all journals with MSC (2000): 65N30, 65R20, 74B20, 74G99, 74H99


Additional Information

Sören Bartels
Affiliation: Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel Ludewig-Meyn-Str. 4, D-24098 Kiel, Germany
Email: sba@numerik.uni-kiel.de

Carsten Carstensen
Affiliation: Institute for Applied Mathematics and Numerical Analysis, Vienna University of Technology, Wiedner Hauptstrasse 8-10, A-1040 Vienna, Austria
Email: Carsten.Carstensen@tuwien.ac.at

DOI: https://doi.org/10.1090/S0025-5718-02-01412-6
Keywords: A posteriori error estimates, residual based error estimate, adaptive algorithm, reliability, finite element method, higher order finite element method
Received by editor(s): February 17, 2000
Published electronically: February 4, 2002
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society