Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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



A posteriori error control in low-order finite element discretisations of incompressible stationary flow problems

Authors: Carsten Carstensen and Stefan A. Funken
Journal: Math. Comp. 70 (2001), 1353-1381
MSC (2000): Primary 65N30, 76D07
Published electronically: October 27, 2000
MathSciNet review: 1836908
Full-text PDF

Abstract | References | Similar Articles | Additional Information


Computable a posteriori error bounds and related adaptive mesh-refining algorithms are provided for the numerical treatment of monotone stationary flow problems with a quite general class of conforming and nonconforming finite element methods. A refined residual-based error estimate generalises the works of Verfürth; Dari, Duran and Padra; Bao and Barrett. As a consequence, reliable and efficient averaging estimates can be established on unstructured grids. The symmetric formulation of the incompressible flow problem models certain nonNewtonian flow problems and the Stokes problem with mixed boundary conditions. A Helmholtz decomposition avoids any regularity or saturation assumption in the mathematical error analysis. Numerical experiments for the partly nonconforming method analysed by Kouhia and Stenberg indicate efficiency of related adaptive mesh-refining algorithms.

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. Berichtsreihe des Mathematischen Seminars Kiel, Technical report 98-11 Universität Kiel (1998). Num. Alg. 20 (1999) 117-137. CMP 2000:01
  • [A] A. Alonso: Error estimators for a mixed method. Numer. Math. 74 (1996) 385-395. MR 97g:65212
  • [BB] W. Bao, J.W. Barrett: A priori and a posteriori error bounds for a nonconforming linear finite element approximation of a nonNewtonian flow. RAIRO Modél. Math. Anal. Numér. 32 (1998) 843-858. MR 99i:76086
  • [BW] R.E. Bank, B.D. Welfert: A posteriori error estimates for the Stokes problem. SIAM Numer. Anal. 28 (1991) 591-623. MR 92a:65284
  • [BF] F. Brezzi, M. Fortin: Mixed and hybrid finite element methods. Springer-Verlag 1991. MR 92d:65187
  • [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
  • [Ca1] C. Carstensen: A posteriori error estimate for the mixed finite element method. Math. Comp. 66 (1997) 465-476. MR 98a:65162
  • [Ca2] -: Quasi interpolation and a posteriori error analysis in finite element method. $M^2$AN Math. Model Numer. Anal. 33 (1999) 1187-1202. CMP 2000:07
  • [CB] C. Carstensen, S. Bartels: Averaging techniques yield reliable error control in low order finite element methods on unstructured grids. Berichtsreihe des Mathematischen Seminars Kiel, Technical report 99-11 Universität Kiel (1999)
  • [CD] C. Carstensen, G. Dolzmann: A posteriori error estimates for mixed FEM in Elasticity. Numer. Math. 81 (1998) 187-209. MR 99m:65208
  • [CJ] C. Carstensen, S. Jansche: A posteriori error estimates for finite element discretization of the Stokes problem. Berichtsreihe des Mathematischen Seminars Kiel, Technical report 97-9 Universität Kiel (1997) (unpublished)
  • [CV] C. Carstensen, R. Verfürth: Edge residuals dominate a posteriori error estimates for low order finite element methods. Berichtsreihe des Mathematischen Seminars Kiel, Technical report 97-6 Universität Kiel (1997); SIAM J. Numer. Anal. 36 (1999) 1571-1587. MR 2000g:65115
  • [Ci] P.G. Ciarlet: The finite element method for elliptic problems. North-Holland, Amsterdam 1978. MR 58:25001
  • [Cl] P. Clément: Approximation by finite element functions using local regularization. RAIRO Sér. Rouge Anal. Numér. 9 (1975) no. R-2, 77-84. MR 53:4569
  • [DDP] E. Dari, R. Duran, and C. Padra: Error estimators for nonconforming finite element approximations of the Stokes problem. Math. Comp. 64 (1995) 1017-1033. MR 95j:65136
  • [EHJ] K. Eriksson, D. Estep, P. Hansbo, C. Johnson: Introduction to adaptive methods for differential equations. Acta Numerica (1995) 105-158. MR 96k:65057
  • [FM] R.S. Falk, M. Morley: Equivalence of finite element methods for problems in elasticity. SIAM J. Numer. Anal. 27 (1990) 1486-1505. MR 91i:65177
  • [GR] V. Girault, P.A. Raviart: Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms. Springer, Berlin, 1986. Pitman 1985. MR 88b:65129
  • [Ho] L. Hörmander: Linear Partial Differential Operators. Berlin-Heidelberg-New York: Springer 1963. MR 28:4221
  • [KS] R. Kouhia, R. Stenberg: A linear nonconforming finite element method for nearly incompressible elasticity and Stokes flow. Comput. Meth. Appl. Mech. Engrg. 124 (1995) 195-212. MR 96d:73073
  • [LM] J.L Lions, E. Magenes: Nonhomogeneous boundary value problems and applications, Vol. I. Springer, Berlin, 1972. MR 50:2670
  • [P] C. Padra: A posteriori error estimators for nonconforming approximation of some quasi-Newtonian flows. SIAM J. Numer. Anal. 34 (1997) 1600-1615. MR 98h:65050
  • [QV] A. Quateroni, A. Valli: Numerical Approximation of Partial Differential Equations. Springer, Berlin, 1994.
  • [T] R. Temam: Navier-Stokes Equations. North-Holland, Amsterdam, 1985.
  • [V1] R. Verfürth: A review of a posteriori error estimation and adaptive mesh-refinement techniques. Teubner Skripten zur Numerik. B.G. Willey-Teubner, Stuttgart, 1996.
  • [V2] R. Verfürth: A posteriori error estimators for the Stokes equations. Numer. Math. 55 (1989) 309-325. MR 90d:65187
  • [V3] R. Verfürth: A posteriori error estimators for the Stokes equations. II. Nonconforming discetizations. Numer. Math. 60 (1991) 235-249. MR 92j:65189
  • [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, 76D07

Retrieve articles in all journals with MSC (2000): 65N30, 76D07

Additional Information

Carsten Carstensen
Affiliation: Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, D-24098 Kiel, Germany

Stefan A. Funken
Affiliation: Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, D-24098 Kiel, Germany

Keywords: NonNewtonian flow, Stokes problem, Crouzeix-Raviart element, nonconforming finite element method, a~posteriori error estimates, adaptive algorithm, reliability, efficiency
Received by editor(s): July 24, 1997
Received by editor(s) in revised form: June 2, 1999, and January 6, 2000
Published electronically: October 27, 2000
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society