Geodesic
A convergent adaptive finite element method with optimal complexity
Electronic transactions on numerical analysis, Tome 30 (2008), pp. 291-304
In this paper, we introduce and analyze a simple adaptive finite element method for second order elliptic partial differential equations. The marking strategy depends on whether the data oscillation is sufficiently small compared to the error estimator in the current mesh. If the oscillation is small compared to the error estimator, we mark as many edges such that their contributions to the local estimator are at least a fixed proportion of the global error estimator (bulk criterion for the estimator). Otherwise, we reduce the oscillation by marking sufficiently many elements, such that the oscillations of the marked cells are at least a fixed proportion of the global oscillation (bulk criterion for the oscillation). This marking strategy guarantees a strict reduction of the error augmented by the oscillation term. Both convergence rates and optimal complexity of the adaptive finite element method are established, with an explicit expression of the constants in the estimates.
Classification : 65N12, 65N15, 65N30, 65N50
Keywords: adaptive finite element method, a posteriori error estimator, convergence rate, optimal computational complexity 
@article{ETNA_2008__30__a7,
     author = {Becker,  Roland and Mao,  Shipeng and Shi,  Zhong-Ci},
     title = {A convergent adaptive finite element method with optimal complexity},
     journal = {Electronic transactions on numerical analysis},
     pages = {291--304},
     year = {2008},
     volume = {30},
     zbl = {1171.65073},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2008__30__a7/}
}
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Becker,  Roland; Mao,  Shipeng; Shi,  Zhong-Ci. A convergent adaptive finite element method with optimal complexity. Electronic transactions on numerical analysis, Tome 30 (2008), pp. 291-304. http://geodesic.mathdoc.fr/item/ETNA_2008__30__a7/