Convergence and quasi-optimality of adaptive nonconforming finite element methods for some nonsymmetric and indefinite problems |
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Authors: | Huangxin Chen Xuejun Xu Ronald H W Hoppe |
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Institution: | 1. LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China 2. Institut für Mathematik, Universit?t Augsburg, 86159, Augsburg, Germany 3. Department of Mathematics, University of Houston, Houston, TX, 77204-3008, USA
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Abstract: | Recently an adaptive nonconforming finite element method (ANFEM) has been developed by Carstensen and Hoppe (in Numer Math
103:251–266, 2006). In this paper, we extend the result to some nonsymmetric and indefinite problems. The main tools in our
analysis are a posteriori error estimators and a quasi-orthogonality property. In this case, we need to overcome two main
difficulties: one stems from the nonconformity of the finite element space, the other is how to handle the effect of a nonsymmetric
and indefinite bilinear form. An appropriate adaptive nonconforming finite element method featuring a marking strategy based
on the comparison of the a posteriori error estimator and a volume term is proposed for the lowest order Crouzeix–Raviart
element. It is shown that the ANFEM is a contraction for the sum of the energy error and a scaled volume term between two
consecutive adaptive loops. Moreover, quasi-optimality in the sense of quasi-optimal algorithmic complexity can be shown for
the ANFEM. The results of numerical experiments confirm the theoretical findings. |
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