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Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part I: Low order conforming, nonconforming, and mixed FEM

Authors: Carsten Carstensen and Sören Bartels
Journal: Math. Comp. 71 (2002), 945-969
MSC (2000): Primary 65N30, 65R20, 74B20, 74G99, 74H99
Published electronically: February 4, 2002
MathSciNet review: 1898741
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Abstract: Averaging techniques are popular tools in adaptive finite element methods for the numerical treatment of second order partial differential equations since they provide efficient a posteriori error estimates by a simple postprocessing. In this paper, their reliablility is shown for conforming, nonconforming, and mixed low 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 averaging techniques. Theoretical and numerical evidence supports that the reliability is up to the smoothness of given right-hand sides.

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  • [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 2001:01
  • [A] A. ALONSO: Error estimators for a mixed method. Numer. Math. 74 (1996), 385-395. MR 97g:65212
  • [BaR] I. BABUSSKA, W.C. RHEINBOLDT: Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15 (1978) 736-754. MR 58:3400
  • [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
  • [BC] 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., posted on February 4, 2002, PII S 0025-5718(02)01412-6 (to appear in print).
  • [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
  • [BF] F. BREZZI, M. FORTIN: Mixed and hybrid finite element methods. Springer-Verlag (1991). MR 92d:65187
  • [C1] C. CARSTENSEN: A posteriori error estimate for the mixed finite element method. Math. Comp. 66 (1997) 465-476. MR 98a:65162
  • [C2] C. CARSTENSEN: Quasi interpolation and a posteriori error analysis in finite element method. M2AN Math. Model Numer. Anal. 33 (1999) 1187-1202. MR 2001a:65135
  • [CF1] 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. Math. 8 (3) (2000) 153-175. CMP 2001:07
  • [CF2] 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.
  • [CF3] C. CARSTENSEN, S.A. FUNKEN: Averaging technique for FE--a posteriori error control in elasticity. Part I: conforming FEM, Comp. Meth. Appl. Mech. Engrg. 190 (2001) 2483-2498. MR 2002a:74114
  • [CF4] C. CARSTENSEN, S.A. FUNKEN: Averaging technique for FE--a posteriori error control in elasticity. Part II: $\lambda$-independent estimates, Comput. Methods Appl. Mech. Engrg., 190 (2001), 4663-4675.
  • [CF5] C. CARSTENSEN, S.A. FUNKEN: Averaging technique for FE--a posteriori error control in elasticity. Part III: Locking-free nonconforming FEM, Comput. Methods Appl. Mech. Engrg. (to appear).
  • [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:65115
  • [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
  • [DDPV] E. DARI, R. DURAN, C. PADRA, AND V. VAMPA: A posteriori error estimators for nonconforming finite element methods. RAIRO Model. Math. Anal. Numer. 30 (1996) 385-400. MR 97f:65066
  • [DMR] R. DURAN, M.A. MUSCHIETTI, R. RODRIGUEZ: On the asymptotic exactness of error estimators for linear triangular elements. Numer. Math. 59 (1991) 107-127. MR 92b:65086
  • [EEHJ] K. ERIKSSON, D. ESTEP, P. HANSBO, C. JOHNSON: Introduction to adaptive methods for differential equations. Acta Numer. (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
  • [HSWW] W. HOFFMANN, A.H. SCHATZ, L.B. WAHLBIN, G. WITTUM: Asymptotically exact a posteriori error estimators for the pointwise gradient error on each element in irregular meshes. Part 1: A smooth problem and globally quasi-uniform meshes. Math. Comp. 70 (2001) 897-909.
  • [HW] R.H.W. HOPPE, B. WOHLMUTH: Element-oriented and edge-oriented local error estimators for nonconforming finite element methods. RAIRO Model. Math. Anal. Numer. 30 (1996) 237-263. MR 97e:65124
  • [KS] G. KANSCHAT, F.-T. SUTTMEIER: A posteriori error estimates for nonconforming finite element schemes. Calcolo 36, No.3 (1999) 129-141. MR 2000k:65208
  • [R1] R. RODRIGUEZ: Some remarks on Zienkiewicz-Zhu estimator. Numer. Methods Partial Differential Equations 10 (1994) 625-635. MR 95de:65103
  • [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
  • [V] R. VERFÜRTH: A review of a posteriori error estimation and adaptive mesh-refinement techniques. Wiley-Teubner (1996).
  • [ZZ] O.C. ZIENKIEWICZ, J.Z. ZHU: A simple error estimator and adaptive procedure for practical engineering analysis. Int. J. Numer. Methods Engrg. 24 (1987) 337-357. MR 87m:73055

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Additional Information

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

Sören Bartels
Affiliation: Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel Ludewig-Meyn-Str. 4, D-24098 Kiel, FRG.

Keywords: A~posteriori error estimates, residual based error estimate, adaptive algorithm, reliability, finite element method, mixed finite element method, nonconforming finite element method
Received by editor(s): August 25, 1999
Published electronically: February 4, 2002
Article copyright: © Copyright 2002 American Mathematical Society

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