Error analysis of Petrov-Galerkin immersed finite element methods
Error analysis of Petrov-Galerkin immersed finite element methods
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DOI：
10.1016/j.cma.2022.115744
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发表时间：
2023-02
影响因子：
7.2
通讯作者：
Cuiyu He;Shun Zhang;Xu Zhang
中科院分区：
文献类型：
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作者：
Cuiyu He;Shun Zhang;Xu Zhang
在经典的Petrov-Galerkin(PG)浸没有限元方法缺乏局部正性的基础上，通过引入稳定项，设计并分析了一种新的稳定的二阶椭圆界面问题的PG浸没有限元方法。Petrov-Galerkin浸没有限元方法对试验空间采用浸没有限元函数，对试验空间采用标准有限元函数。本文分析了该方法的优先误差估计和后验误差估计。我们证明了能量范数的连续性、非连续性条件和优先误差估计。所提出的后验误差估计器是可靠和有效的，其可靠性和效率常数与界面位置无关。大量的数值结果证实了数值格式的最优收敛，并表明了数值格式对于界面-网格交点和系数对比度的稳健性，尽管关于界面-网格交点的inf-sup常数的稳健性在理论上还没有得到证明。
This paper designs and analyzes a new and stable Petrov–Galerkin (PG) immersed finite element method (IFEM) for the second-order elliptic interface problems by introducing stabilization terms based on the classical PG-IFEM, which lacks the local positivity. The Petrov–Galerkin immersed finite element method uses the immersed finite element functions for the trial space and the standard finite element functions for the test space. Both thea priorianda posteriorierror estimates of the method are analyzed in this paper. We prove the continuity and inf-sup condition and thea priorierror estimate of the energy norm. The proposeda posteriorierror estimator is proved to be both reliable and efficient, with both reliability and efficiency constants independent of the location of the interface. Extensive numerical results confirm the numerical scheme’s optimal convergence and indicate the robustness with respect to the interface-mesh intersection and the coefficient contrast, despite the robustness of the inf-sup constant with respect to the interface-mesh intersection has yet been theoretically proved.