Convergence and quasi-optimal complexity of a simple adaptive finite element method
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 43 (2009) no. 6, p. 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 : https://doi.org/10.1051/m2an/2009036
Classification:  65N12,  65N15,  65N30,  65N50
Keywords: adaptive finite elements, a posteriori error analysis, convergence of adaptive algorithms, complexity estimates
@article{M2AN_2009__43_6_1203_0,
     author = {Becker, Roland and Mao, Shipeng},
     title = {Convergence and quasi-optimal complexity of a simple adaptive finite element method},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {43},
     number = {6},
     year = {2009},
     pages = {1203-1219},
     doi = {10.1051/m2an/2009036},
     zbl = {pre05636852},
     mrnumber = {2588438},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2009__43_6_1203_0}
}
Becker, Roland; Mao, Shipeng. Convergence and quasi-optimal complexity of a simple adaptive finite element method. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 43 (2009) no. 6, pp. 1203-1219. doi : 10.1051/m2an/2009036. http://www.numdam.org/item/M2AN_2009__43_6_1203_0/

[1] I. Babuška and W.C. Rheinboldt, Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15 (1978) 736-754. | MR 483395 | Zbl 0398.65069

[2] R. Becker and S. Mao, An optimally convergent adaptive mixed finite element method. Numer. Math. 111 (2008) 35-54. | MR 2448202 | Zbl 1159.65088

[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. Computing 55 (1995) 271-288. | MR 1370103 | Zbl 0848.65074

[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. | MR 2480083 | Zbl 1171.65073

[6] P. Binev, W. Dahmen and R. Devore, Adaptive finite element methods with convergence rates. Numer. Math. 97 (2004) 219-268. | MR 2050077 | Zbl 1063.65120

[7] J.H. Bramble and J.E. Pasciak, New estimates for multilevel algorithms including the v-cycle. Math. Comp. 60 (1995) 447-471. | MR 1176705 | Zbl 0783.65081

[8] C. Carstensen, Quasi-interpolation and a posteriori error analysis in finite element methods. ESAIM: M2AN 33 (1999) 1187-1202. | Numdam | MR 1736895 | Zbl 0948.65113

[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 1706735 | Zbl 0938.65124

[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. | MR 2421046 | Zbl 1176.65122

[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 520174 | Zbl 0383.65058

[12] A. Cohen, W. Dahmen and R. Devore, Adaptive wavelet methods for elliptic operator equations: Convergence rates. Math. Comput. 70 (2001) 27-75. | MR 1803124 | Zbl 0980.65130

[13] R. Devore, Nonlinear approximation. Acta Numer. 7 (1998) 51-150. | MR 1689432 | Zbl 0931.65007

[14] W. Dörfler, A convergent adaptive algorithm for Poisson's equation. SIAM J. Numer. Anal. 33 (1996) 1106-1124. | MR 1393904 | Zbl 0854.65090

[15] W. Dörfler and R.H. Nochetto, Small data oscillation implies the saturation assumption. Numer. Math. 91 (2002) 1-12. | MR 1896084 | Zbl 0995.65109

[16] K. Eriksson, D. Estep, P. Hansbo and C. Johnson, Introduction to adaptive methods for differential equations. Acta Numer. 4 (1995) 105-158. | MR 1352472 | Zbl 0829.65122

[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. | MR 1770058 | Zbl 0970.65113

[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. | MR 2413035 | Zbl 1153.65111

[19] R. Stevenson, Optimality of a standard adaptive finite element method. Found. Comput. Math. 7 (2007) 245-269. | MR 2324418 | Zbl 1136.65109

[20] R. Verfürth, A review of a posteriori error estimation and adaptive mesh-refinement techniques. John Wiley/Teubner, New York-Stuttgart (1996). | Zbl 0853.65108

[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. A 49 (2006) 1405-1429. | MR 2287269 | Zbl 1112.65104