A convergent adaptive finite element method with optimal complexity

Roland Becker (1) , Shipeng Mao (2) , Zhong-Ci Shi (2)

1 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]
2 Institute of Computational Mathematics and Scientific/Engineering Computing

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 suffi- ciently small compared to the error estimator, we mark adequately many edges such that their contributions of local estimator constitute a fixed proportion of the global error estimator. Otherwise we shall reduce the oscillation by marking sufficiently many elements, the local oscillations of which constitute a fixed proportion of the global oscillation. This marking strategy guarantees a strict reduction of the error augmented by the oscillation term. Both, con- vergence rates and optimal complexity of the adaptive finite element method are established, with an explicit expression of the constants in the estimates.

Domaines: Analyse numérique [math.NA]

Roland Becker, Shipeng Mao, Zhong-Ci Shi. A convergent adaptive finite element method with optimal complexity. 2007. ⟨hal-00158888v1⟩