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
Published electronically: February 4, 2002
MathSciNet review: 1898742
Full-text PDF Free Access

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. Babuška, T. Strouboulis, C. S. Upadhyay, S. K. Gangaraj, and K. Copps, Validation of a posteriori error estimators by numerical approach, Internat. J. Numer. Methods Engrg. 37 (1994), no. 7, 1073–1123. MR 1271482, 10.1002/nme.1620370702
  • [BeR] R. Becker and R. Rannacher, A feed-back approach to error control in finite element methods: basic analysis and examples, East-West J. Numer. Math. 4 (1996), no. 4, 237–264. MR 1430239
  • [B] Dietrich Braess, Finite elements, Cambridge University Press, Cambridge, 1997. Theory, fast solvers, and applications in solid mechanics; Translated from the 1992 German original by Larry L. Schumaker. MR 1463151
  • [BV] D. Braess and R. Verfürth, A posteriori error estimators for the Raviart-Thomas element, SIAM J. Numer. Anal. 33 (1996), no. 6, 2431–2444. MR 1427472, 10.1137/S0036142994264079
  • [BS] Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 1994. MR 1278258
  • [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] Houde Han and Weizhu Bao, The discrete artificial boundary condition on a polygonal artificial boundary for the exterior problem of Poisson equation by using the direct method of lines, Comput. Methods Appl. Mech. Engrg. 179 (1999), no. 3-4, 345–360. MR 1716709, 10.1016/S0045-7825(99)00046-8
  • [Cl] Ph. Clément, Approximation by finite element functions using local regularization, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. \jname RAIRO Analyse Numérique 9 (1975), no. R-2, 77–84 (English, with Loose French summary). MR 0400739
  • [Ci] Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. MR 0520174
  • [EEHJ] Kenneth Eriksson, Don Estep, Peter Hansbo, and Claes Johnson, Introduction to adaptive methods for differential equations, Acta numerica, 1995, Acta Numer., Cambridge Univ. Press, Cambridge, 1995, pp. 105–158. MR 1352472, 10.1017/S0962492900002531
  • [GR] Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR 851383
  • [MBa] J. M. Melenk and I. Babuška, The partition of unity finite element method: basic theory and applications, Comput. Methods Appl. Mech. Engrg. 139 (1996), no. 1-4, 289–314. MR 1426012, 10.1016/S0045-7825(96)01087-0
  • [R] Rodolfo Rodríguez, Some remarks on Zienkiewicz-Zhu estimator, Numer. Methods Partial Differential Equations 10 (1994), no. 5, 625–635. MR 1290948, 10.1002/num.1690100509
  • [V] R. VERFÜRTH: A review of a posteriori error estimation and adaptive mesh-refinement techniques. Wiley-Teubner (1996).
  • [Y1] De Hao Yu, Asymptotically exact a posteriori error estimators for elements of bi-odd degree, Chinese J. Numer. Math. Appl. 13 (1991), no. 4, 82–90. MR 1258636
  • [Y2] De Hao Yu, Asymptotically exact a posteriori error estimator for elements of bi-even degree, Chinese J. Numer. Math. Appl. 13 (1991), no. 2, 64–78. MR 1136156
  • [ZZ] O. C. Zienkiewicz and J. Z. Zhu, A simple error estimator and adaptive procedure for practical engineering analysis, Internat. J. Numer. Methods Engrg. 24 (1987), no. 2, 337–357. MR 875306, 10.1002/nme.1620240206

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

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

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