This paper introduces and analyzes an equilibrated a posteriori error estimator for mixed finite element approximations to the diffusion problem in two dimensions. The estimator, which is a generalization of those in Braess and Schöberl (Math Comput 77:651–672, 2008) and Cai and Zhang (SIAM J Numer Anal 50(1):151–170, 2012), is based on the Prager–Synge identity and on a local recovery of a gradient in the curl free subspace of the \(H(\text {curl})\)-confirming finite element spaces. The resulting estimator admits guaranteed reliability, and its robust local efficiency is proved under the quasi-monotonicity condition of the diffusion coefficient. Numerical experiments are given to confirm the theoretical results.

本文介绍并分析了二维扩散问题的混合有限元逼近的平衡后验误差估计。估算器是Praess和Schöberl（Math Comput 77：651–672，2008）和Cai and Zhang（SIAM J Numer Anal 50（1）：151–170，2012）中的推广，基于Prager –Synge身份和\（H（\ text {curl}）\）的无卷曲子空间中的梯度的局部恢复，从而确定有限元空间。所得的估计器具有保证的可靠性，并且在扩散系数的准单调性条件下证明了其鲁棒的局部效率。数值实验证实了理论结果。