Convergence and quasi-optimal complexity of a simple adaptive finite element method
ESAIM: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 6, pp. 1203-1219.

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.

DOI : 10.1051/m2an/2009036
Classification : 65N12, 65N15, 65N30, 65N50
Mots-clés : adaptive finite elements, a posteriori error analysis, convergence of adaptive algorithms, complexity estimates 
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     author = {Becker, Roland and Mao, Shipeng},
     title = {Convergence and quasi-optimal complexity of a simple adaptive finite element method},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
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     volume = {43},
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     url = {http://archive.numdam.org/articles/10.1051/m2an/2009036/}
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Becker, Roland; Mao, Shipeng. Convergence and quasi-optimal complexity of a simple adaptive finite element method. ESAIM: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 6, pp. 1203-1219. doi : 10.1051/m2an/2009036. http://archive.numdam.org/articles/10.1051/m2an/2009036/

[1] I. Babuška and W.C. Rheinboldt, Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15 (1978) 736-754. | MR | Zbl

[2] R. Becker and S. Mao, An optimally convergent adaptive mixed finite element method. Numer. Math. 111 (2008) 35-54. | MR | Zbl

[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. Computing 55 (1995) 271-288. | MR | Zbl

[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. | MR | Zbl

[6] P. Binev, W. Dahmen and R. Devore, Adaptive finite element methods with convergence rates. Numer. Math. 97 (2004) 219-268. | MR | Zbl

[7] J.H. Bramble and J.E. Pasciak, New estimates for multilevel algorithms including the v-cycle. Math. Comp. 60 (1995) 447-471. | MR | Zbl

[8] C. Carstensen, Quasi-interpolation and a posteriori error analysis in finite element methods. ESAIM: M2AN 33 (1999) 1187-1202. | Numdam | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[11] 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). | MR | Zbl

[12] A. Cohen, W. Dahmen and R. Devore, Adaptive wavelet methods for elliptic operator equations: Convergence rates. Math. Comput. 70 (2001) 27-75. | MR | Zbl

[13] R. Devore, Nonlinear approximation. Acta Numer. 7 (1998) 51-150. | MR | Zbl

[14] W. Dörfler, A convergent adaptive algorithm for Poisson's equation. SIAM J. Numer. Anal. 33 (1996) 1106-1124. | MR | Zbl

[15] W. Dörfler and R.H. Nochetto, Small data oscillation implies the saturation assumption. Numer. Math. 91 (2002) 1-12. | MR | Zbl

[16] K. Eriksson, D. Estep, P. Hansbo and C. Johnson, Introduction to adaptive methods for differential equations. Acta Numer. 4 (1995) 105-158. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[19] R. Stevenson, Optimality of a standard adaptive finite element method. Found. Comput. Math. 7 (2007) 245-269. | MR | Zbl

[20] R. Verfürth, A review of a posteriori error estimation and adaptive mesh-refinement techniques. John Wiley/Teubner, New York-Stuttgart (1996). | Zbl

[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. A 49 (2006) 1405-1429. | MR | Zbl

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