论文标题
不平等元素的普遍的布拉格同步不平等和平衡的误差估计器
Generalized Prager-Synge Inequality and Equilibrated Error Estimators for Discontinuous Elements
论文作者
论文摘要
众所周知的Prager-synge身份在$ H^1（ω）$中有效，并为连续元素的后验误差估计器提供了平衡的基础。在本文中，我们引入了一种新的不平等，可以被视为布拉格 - 同步身份的概括，对于分段$ h^1（ω）$函数对于扩散问题有效。事实证明，不平等在两个维度上是身份。 对于不合格的有限元元素的近似值，我们提出了一种完全显式的方法，该方法通过局部元素方面的方案在$ h（div;ω）$中恢复了平衡的通量，并通过$ h（curl;ω）$恢复梯度通过简单的平均技术换取。然后证明所得的误差估计器在全球上是可靠的，并且在局部效率高。此外，可靠性和效率常数与扩散系数的跳跃无关，无论其分布如何。
The well-known Prager-Synge identity is valid in $H^1(Ω)$ and serves as a foundation for developing equilibrated a posteriori error estimators for continuous elements. In this paper, we introduce a new inequality, that may be regarded as a generalization of the Prager-Synge identity, to be valid for piecewise $H^1(Ω)$ functions for diffusion problems. The inequality is proved to be identity in two dimensions. For nonconforming finite element approximation of arbitrary odd order, we propose a fully explicit approach that recovers an equilibrated flux in $H(div; Ω)$ through a local element-wise scheme and that recovers a gradient in $H(curl;Ω)$ through a simple averaging technique over edges. The resulting error estimator is then proved to be globally reliable and locally efficient. Moreover, the reliability and efficiency constants are independent of the jump of the diffusion coefficient regardless of its distribution.
