Error reduction and convergence for an adaptive mixed finite element method
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- by Carsten Carstensen and R. H. W. Hoppe PDF
- Math. Comp. 75 (2006), 1033-1042 Request permission
Abstract:
An adaptive mixed finite element method (AMFEM) is designed to guarantee an error reduction, also known as saturation property: after each refinement step, the error for the fine mesh is strictly smaller than the error for the coarse mesh up to oscillation terms. This error reduction property is established here for the Raviart–Thomas finite element method with a reduction factor $\rho <1$ uniformly for the $L^2$ norm of the flux errors. Our result allows for linear convergence of a proper adaptive mixed finite element algorithm with respect to the number of refinement levels. The adaptive algorithm surprisingly does not require any particular mesh design, unlike the conforming finite element method. The new arguments are a discrete local efficiency and a quasi-orthogonality estimate. The proof does not rely on duality or on regularity.References
- Mark Ainsworth and J. Tinsley Oden, A posteriori error estimation in finite element analysis, Pure and Applied Mathematics (New York), Wiley-Interscience [John Wiley & Sons], New York, 2000. MR 1885308, DOI 10.1002/9781118032824
- A. Alonso, Error estimators for a mixed method, Numer. Math. 74 (1996), no. 4, 385–395. MR 1414415, DOI 10.1007/s002110050222
- Ivo Babuška and Theofanis Strouboulis, The finite element method and its reliability, Numerical Mathematics and Scientific Computation, The Clarendon Press, Oxford University Press, New York, 2001. MR 1857191
- Eberhard Bänsch, Pedro Morin, and Ricardo H. Nochetto, An adaptive Uzawa FEM for the Stokes problem: convergence without the inf-sup condition, SIAM J. Numer. Anal. 40 (2002), no. 4, 1207–1229. MR 1951892, DOI 10.1137/S0036142901392134
- Bahriawati, C. and Carstensen, C. Three Matlab implementations of the lowest-order Raviart-Thomas MFEM with a posteriori error control. Newton Institute Preprint NI03069-CPD available at http://www.newton .cam.ac.uk/preprints/NI03069.pdf
- Wolfgang Bangerth and Rolf Rannacher, Adaptive finite element methods for differential equations, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2003. MR 1960405, DOI 10.1007/978-3-0348-7605-6
- Peter Binev, Wolfgang Dahmen, and Ron DeVore, Adaptive finite element methods with convergence rates, Numer. Math. 97 (2004), no. 2, 219–268. MR 2050077, DOI 10.1007/s00211-003-0492-7
- Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205, DOI 10.1007/978-1-4612-3172-1
- Carsten Carstensen, A posteriori error estimate for the mixed finite element method, Math. Comp. 66 (1997), no. 218, 465–476. MR 1408371, DOI 10.1090/S0025-5718-97-00837-5
- Carsten Carstensen and Sören Bartels, Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. I. Low order conforming, nonconforming, and mixed FEM, Math. Comp. 71 (2002), no. 239, 945–969. MR 1898741, DOI 10.1090/S0025-5718-02-01402-3
- Willy Dörfler, A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal. 33 (1996), no. 3, 1106–1124. MR 1393904, DOI 10.1137/0733054
- K. Eriksson, D. Estep, P. Hansbo, and C. Johnson, Computational differential equations, Cambridge University Press, Cambridge, 1996. MR 1414897
- Ronald H. W. Hoppe and Barbara Wohlmuth, Adaptive multilevel techniques for mixed finite element discretizations of elliptic boundary value problems, SIAM J. Numer. Anal. 34 (1997), no. 4, 1658–1681. MR 1461801, DOI 10.1137/S0036142994276992
- Luisa Donatella Marini, An inexpensive method for the evaluation of the solution of the lowest order Raviart-Thomas mixed method, SIAM J. Numer. Anal. 22 (1985), no. 3, 493–496. MR 787572, DOI 10.1137/0722029
- Pedro Morin, Ricardo H. Nochetto, and Kunibert G. Siebert, Data oscillation and convergence of adaptive FEM, SIAM J. Numer. Anal. 38 (2000), no. 2, 466–488. MR 1770058, DOI 10.1137/S0036142999360044
- Pedro Morin, Ricardo H. Nochetto, and Kunibert G. Siebert, Local problems on stars: a posteriori error estimators, convergence, and performance, Math. Comp. 72 (2003), no. 243, 1067–1097. MR 1972728, DOI 10.1090/S0025-5718-02-01463-1
- Verfürth, R. (1996). A Review of A Posteriori Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, New York, Stuttgart.
- Wohlmuth, B. and Hoppe, R.H.W. (1999). A comparison of a posteriori error estimators for mixed finite element discretizations. Math. Comp., 82, 253–279.
Additional Information
- Carsten Carstensen
- Affiliation: Department of Mathematics, Humboldt-Universität zu Berlin, D-10099 Berlin, Germany
- R. H. W. Hoppe
- Affiliation: Institute of Mathematics, Universität Augsburg, D-86159 Augsburg, Germany; and Department of Mathematics, University of Houston, Houston, Texas 77204-3008
- Received by editor(s): April 11, 2004
- Published electronically: March 13, 2006
- © Copyright 2006 American Mathematical Society
- Journal: Math. Comp. 75 (2006), 1033-1042
- MSC (2000): Primary 65N30, 65N50
- DOI: https://doi.org/10.1090/S0025-5718-06-01829-1
- MathSciNet review: 2219017