In this paper, we present the design of a posteriori error estimator for the plasmon phenomenon on the graphene surface and propose a method to achieve local high-precision numerical calculations when plasmon phenomena occur on the graphene surface. We provide a lower bound estimate for the posteriori error estimator, along with a proof of convergence. Specifically, the constructed posterior error estimator enables local refinement in regions where the error is significant at the graphene interface. Firstly, we outline the construction of the posterior error estimator and provide the proof of its lower bound. Secondly, we establish the convergence of the Adaptive Edge Finite Element Method  (AEFEM). Finally, we present numerical results that validate the effectiveness of the error estimator.