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

Roland Becker; Shipeng Mao

ESAIM: Mathematical Modelling and Numerical Analysis (2009)

  • Volume: 43, Issue: 6, page 1203-1219
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Becker, Roland, and Mao, Shipeng. "Convergence and quasi-optimal complexity of a simple adaptive finite element method." ESAIM: Mathematical Modelling and Numerical Analysis 43.6 (2009): 1203-1219. <http://eudml.org/doc/250644>.

@article{Becker2009,
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. },
author = {Becker, Roland, Mao, Shipeng},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Adaptive finite elements; a posteriori error analysis; convergence of adaptive algorithms; complexity estimates.; adaptive finite elements; a posteriori error analysis; complexity estimates; local mesh refinements},
language = {eng},
month = {8},
number = {6},
pages = {1203-1219},
publisher = {EDP Sciences},
title = {Convergence and quasi-optimal complexity of a simple adaptive finite element method},
url = {http://eudml.org/doc/250644},
volume = {43},
year = {2009},
}

TY - JOUR
AU - Becker, Roland
AU - Mao, Shipeng
TI - Convergence and quasi-optimal complexity of a simple adaptive finite element method
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2009/8//
PB - EDP Sciences
VL - 43
IS - 6
SP - 1203
EP - 1219
AB - 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.
LA - eng
KW - Adaptive finite elements; a posteriori error analysis; convergence of adaptive algorithms; complexity estimates.; adaptive finite elements; a posteriori error analysis; complexity estimates; local mesh refinements
UR - http://eudml.org/doc/250644
ER -

References

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  1. I. Babuška and W.C. Rheinboldt, Error estimates for adaptive finite element computations. SIAM J. Numer. Anal.15 (1978) 736–754.  Zbl0398.65069
  2. R. Becker and S. Mao, An optimally convergent adaptive mixed finite element method. Numer. Math.111 (2008) 35–54.  Zbl1159.65088
  3. R. Becker and D. Trujillo, Convergence of an adaptive finite element method on quadrilateral meshes. Research Report RR-6740, INRIA, France (2008).  
  4. R. Becker, C. Johnson and R. Rannacher, Adaptive error control for multigrid finite element methods. Computing55 (1995) 271–288.  Zbl0848.65074
  5. R. Becker, S. Mao and Z.-C. Shi, A convergent adaptive finite element method with optimal complexity. Electron. Trans. Numer. Anal.30 (2008) 291–304.  Zbl1171.65073
  6. P. Binev, W. Dahmen and R. DeVore, Adaptive finite element methods with convergence rates. Numer. Math.97 (2004) 219–268.  Zbl1063.65120
  7. J.H. Bramble and J.E. Pasciak, New estimates for multilevel algorithms including the v-cycle. Math. Comp.60 (1995) 447–471.  Zbl0783.65081
  8. C. Carstensen, Quasi-interpolation and a posteriori error analysis in finite element methods. ESAIM: M2AN33 (1999) 1187–1202.  
  9. C. Carstensen and R. Verfürth, Edge residuals dominate a posteriori error estimates for low order finite element methods. SIAM J. Numer. Anal.36 (1999) 1571–1587.  Zbl0938.65124
  10. J.M. Cascon, Ch. Kreuzer, R.N. Nochetto and K.G. Siebert, Quasi-optimal convergence rate for an adaptive finite element method. SIAM J Numer. Anal.46 (2008) 2524–2550.  Zbl1176.65122
  11. P.G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications4. Amsterdam, New York, Oxford: North-Holland Publishing Company (1978).  
  12. A. Cohen, W. Dahmen and R. DeVore, Adaptive wavelet methods for elliptic operator equations: Convergence rates. Math. Comput.70 (2001) 27–75.  Zbl0980.65130
  13. R. DeVore, Nonlinear approximation. Acta Numer.7 (1998) 51–150.  
  14. W. Dörfler, A convergent adaptive algorithm for Poisson's equation. SIAM J. Numer. Anal.33 (1996) 1106–1124.  Zbl0854.65090
  15. W. Dörfler and R.H. Nochetto, Small data oscillation implies the saturation assumption. Numer. Math.91 (2002) 1–12.  Zbl0995.65109
  16. K. Eriksson, D. Estep, P. Hansbo and C. Johnson, Introduction to adaptive methods for differential equations. Acta Numer.4 (1995) 105–158.  Zbl0829.65122
  17. P. Morin, R.H. Nochetto and K.G. Siebert, Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal.38 (2000) 466–488.  Zbl0970.65113
  18. P. Morin, K.G. Siebert and A. Veeser, A basic convergence result for conforming adaptive finite elements. Math. Models Methods Appl. Sci.18 (2008) 707–737.  Zbl1153.65111
  19. R. Stevenson, Optimality of a standard adaptive finite element method. Found. Comput. Math.7 (2007) 245–269.  Zbl1136.65109
  20. R. Verfürth, A review of a posteriori error estimation and adaptive mesh-refinement techniques. John Wiley/Teubner, New York-Stuttgart (1996).  Zbl0853.65108
  21. H. Wu and Z. Chen, Uniform convergence of multigrid v-cycle on adaptively refined finite element meshes for second order elliptic problems. Sci. China Ser. A49 (2006) 1405–1429.  Zbl1112.65104

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