Improved ZZ a posteriori error estimators for diffusion problems: Discontinuous elements

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.