A Unified Analysis of Quasi-Optimal Convergence for Adaptive Mixed Finite Element Methods
A Unified Analysis of Quasi-Optimal Convergence for Adaptive Mixed Finite Element Methods
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DOI：
10.1137/16m105513x
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发表时间：
2016-01
期刊：
影响因子：
--
通讯作者：
Jun Hu;Guozhu Yu
中科院分区：
文献类型：
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作者：
Jun Hu;Guozhu Yu
本文给出了当有限元空间和相应的后验误差估计满足五个假设时，一类问题的自适应混合有限元方法的收敛和最优性的统一分析。我们证明了这五个条件对于所考虑的自适应算法的收敛和最优性是充分的。分析的主要内容是一种分析离散可靠性和准正交性的新方法。这一新方法源于对离散位移和应力空间范数的适当而自然的选择，即前者的离散H^1范数和后者的L^2范数，以及新定义的从较粗网格上的离散应力空间到较细网格上的离散散度自由空间的投影算子。作为应用，我们在二维和三维证明了Poisson和Stokes问题的Raviart-Thomas和Brezzi-Douglas-Marini元素的这五个假设。
In this paper, we present a unified analysis of both convergence and optimality of adaptive mixed finite element methods for a class of problems when the finite element spaces and corresponding a posteriori error estimates under consideration satisfy five hypotheses. We prove that these five conditions are sufficient for convergence and optimality of the adaptive algorithms under consideration. The main ingredient for the analysis is a new method to analyze both discrete reliability and quasi-orthogonality. This new method arises from an appropriate and natural choice of the norms for both the discrete displacement and stress spaces, namely, a mesh-dependent discrete $H^1$ norm for the former and a $L^2$ norm for the latter, and a newly defined projection operator from the discrete stress space on the coarser mesh onto the discrete divergence free space on the finer mesh. As applications, we prove these five hypotheses for the Raviart--Thomas and Brezzi--Douglas--Marini elements of the Poisson and Stokes problems in both 2D and 3D.