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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
DOI: https://doi.org/10.1090/S0025-5718-00-01264-3
Published electronically: October 27, 2000
MathSciNet review: 1836908
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Abstract:

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.


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Additional Information

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

Stefan A. Funken
Affiliation: Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, D-24098 Kiel, Germany
Email: saf@numerik.uni-kiel.de

DOI: https://doi.org/10.1090/S0025-5718-00-01264-3
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

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