Abstract In Cai, He, and Zhang (2017), we studied an improved Zienkiewicz-Zhu (ZZ) a posteriori error estimator for conforming linear finite element approximation to diffusion problems. The estimator is more efficient than the original ZZ estimator for non-smooth problems, but with comparable computational costs. This paper extends the improved ZZ estimator for discontinuous linear finite element approximations including both nonconforming and discontinuous elements. In addition to post-processing a flux, we further explicitly recover a gradient in the H ( curl ) conforming finite element space. The resulting error estimator is proved, theoretically and numerically, to be efficient and reliable with constants independent of the jump of the coefficient regardless of its distribution.

中文翻译：

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改进的 ZZ 后验误差估计器用于扩散问题：不连续元素

摘要 在 Cai、He 和 Zhang (2017) 中，我们研究了改进的 Zienkiewicz-Zhu (ZZ) 后验误差估计器，用于使线性有限元近似符合扩散问题。对于非平滑问题，估计器比原始 ZZ 估计器更有效，但计算成本相当。本文扩展了改进的 ZZ 估计器，用于不连续线性有限元近似，包括非一致元素和不连续元素。除了对通量进行后处理之外，我们还进一步明确地恢复了符合 H ( curl ) 的有限元空间中的梯度。由此产生的误差估计器在理论上和数值上都被证明是有效和可靠的，其常数与系数的跳跃无关，而不管其分布如何。