<p>Recently, Benzi et al. proposed several iterative methods for solving the double saddle point system arising from the modeling of liquid crystal directors using finite elements. To evaluate the structured backward stability of these numerical algorithms, in this paper we consider the structured backward error analysis for the double saddle point system. Based on the structured backward error, we also present a new termination criterion used in the iterative method to verify whether the current iterative solution is an exact solution of the acceptable perturbed structure-preserving double saddle point system. Numerical examples show that the structured backward error can be easily used to test the stability of practical algorithms and the new stopping criterion is more suitable for the iteration methods for solving the double saddle point problems than the typical ones.</p>