Convergence of a standard adaptive nonconforming finite element method with optimal complexity

Authors:
Shipeng Mao;Xuying Zhao;Zhongci Shi
Affiliations:
LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China and Graduate Unive ...;LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
Venue:
Applied Numerical Mathematics
Year:
2010

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Abstract

In this paper, we analyze the convergence and optimal complexity of the usual simple adaptive nonconforming finite element method by using Dorfler collective marking strategy. Based on several basic ingredients, such as the estimator reduction, quasi-orthogonality, local upper bound and so on, we eventually show the convergence of the adaptive algorithm by establishing the reduction of some total error and the quasi-optimal convergence rate. Our analysis does not need the relation between the nonconforming P"1 element and the mixed Raviart-Thomas element. The results of numerical experiments confirm that our adaptive algorithm is optimal.