Generalized Prager–Synge identity and robust equilibrated error estimators for discontinuous elements

The well-known Prager–Synge identity is valid in H¹(Ω) and serves as a foundation for developing equilibrated a posteriori error estimators for continuous elements. In this paper, we introduce a new identity, that may be regarded as a generalization of the Prager–Synge identity, to be valid for piecewise H¹(Ω) functions for diffusion problems. For nonconforming finite element approximation of arbitrary odd order, we improve the current methods by proposing a fully explicit approach that recovers an equilibrated flux in H(div; Ø) through a local element-wise scheme. The local efficiency for the recovered flux is robust with respect to the diffusion coefficient jump regardless of its distribution. For discontinuous elements, we note that the typical approach of recovering a H¹ function for the nonconforming error can be proved robust only under some restrictive assumptions. To promote the unconditional robustness of the error estimator with respect to the diffusion coefficient jump, we propose to recover a gradient in H(curl; Ω) space through a simple explicit averaging technique over facets. Our resulting error estimator is proved to be globally reliable and locally efficient regardless of the coefficient distribution. Nevertheless, the reliability constant is no longer to be 1.