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Convergence and quasi-optimal complexity of a simple adaptive finite element method

Published online by Cambridge University Press:  21 August 2009

Roland Becker  and
Shipeng Mao
Roland Becker
Affiliation:
Laboratoire de Mathématiques Appliquées and INRIA Bordeaux Sud-Ouest Concha, Université de Pau, 64013 Pau Cedex, France. roland.becker@univ-pau.fr; beckerroland@me.com
Shipeng Mao
Affiliation:
Institute of Computational Mathematics and INRIA Bordeaux Sud-Ouest Concha, Chinese Academy of Sciences (CAS), Beijing, 100190, P. R. China. maosp@lsec.cc.ac.cn
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Abstract

We prove convergence and quasi-optimal complexity of an adaptive finite element algorithm on triangular meshes with standard mesh refinement. Our algorithm is based on an adaptive marking strategy. In each iteration, a simple edge estimator is compared to an oscillation term and the marking of cells for refinement is done according to the dominant contribution only. In addition, we introduce an adaptive stopping criterion for iterative solution which compares an estimator for the iteration error with the estimator for the discretization error.

Keywords

Adaptive finite elementsa posteriori error analysisconvergence of adaptive algorithmscomplexity estimates.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 43 , Issue 6 , November 2009 , pp. 1203 - 1219
Copyright
© EDP Sciences, SMAI, 2009

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References

Babuška, I. and Rheinboldt, W.C., Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15 (1978) 736–754. CrossRef
Becker, R. and Mao, S., An optimally convergent adaptive mixed finite element method. Numer. Math. 111 (2008) 35–54. CrossRef
R. Becker and D. Trujillo, Convergence of an adaptive finite element method on quadrilateral meshes. Research Report RR-6740, INRIA, France (2008).
Becker, R., Johnson, C. and Rannacher, R., Adaptive error control for multigrid finite element methods. Computing 55 (1995) 271–288. CrossRef
Becker, R., Mao, S. and Shi, Z.-C., A convergent adaptive finite element method with optimal complexity. Electron. Trans. Numer. Anal. 30 (2008) 291–304.
Binev, P., Dahmen, W. and DeVore, R., Adaptive finite element methods with convergence rates. Numer. Math. 97 (2004) 219–268. CrossRef
Bramble, J.H. and Pasciak, J.E., New estimates for multilevel algorithms including the v-cycle. Math. Comp. 60 (1995) 447–471.
Carstensen, C., Quasi-interpolation and a posteriori error analysis in finite element methods. ESAIM: M2AN 33 (1999) 1187–1202. CrossRef
Carstensen, C. and Verfürth, R., Edge residuals dominate a posteriori error estimates for low order finite element methods. SIAM J. Numer. Anal. 36 (1999) 1571–1587. CrossRef
Cascon, J.M., Kreuzer, Ch., Nochetto, R.N. and Siebert, K.G., Quasi-optimal convergence rate for an adaptive finite element method. SIAM J Numer. Anal. 46 (2008) 2524–2550. CrossRef
P.G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications 4. Amsterdam, New York, Oxford: North-Holland Publishing Company (1978).
Cohen, A., Dahmen, W. and DeVore, R., Adaptive wavelet methods for elliptic operator equations: Convergence rates. Math. Comput. 70 (2001) 27–75. CrossRef
DeVore, R., Nonlinear approximation. Acta Numer. 7 (1998) 51–150. CrossRef
Dörfler, W., A convergent adaptive algorithm for Poisson's equation. SIAM J. Numer. Anal. 33 (1996) 1106–1124. CrossRef
Dörfler, W. and Nochetto, R.H., Small data oscillation implies the saturation assumption. Numer. Math. 91 (2002) 1–12.
Eriksson, K., Estep, D., Hansbo, P. and Johnson, C., Introduction to adaptive methods for differential equations. Acta Numer. 4 (1995) 105–158. CrossRef
Morin, P., Nochetto, R.H. and Siebert, K.G., Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal. 38 (2000) 466–488. CrossRef
Morin, P., Siebert, K.G. and Veeser, A., A basic convergence result for conforming adaptive finite elements. Math. Models Methods Appl. Sci. 18 (2008) 707–737. CrossRef
Stevenson, R., Optimality of a standard adaptive finite element method. Found. Comput. Math. 7 (2007) 245–269. CrossRef
R. Verfürth, A review of a posteriori error estimation and adaptive mesh-refinement techniques. John Wiley/Teubner, New York-Stuttgart (1996).
Wu, H. and Chen, Z., Uniform convergence of multigrid v-cycle on adaptively refined finite element meshes for second order elliptic problems. Sci. China Ser. A 49 (2006) 1405–1429. CrossRef