In this talk, we present superconvergence analysis of a backward differentiation formula (BDF) finite element method for the nonlinear Klein–Gordon–Schrödinger equation. A linearized fully discrete scheme is introduced to solve the nonlinear system. To overcome the constraint on the ratio between the spatial mesh size  h  and time step  \tau , we employ a time-discrete system to separate the error into temporal and spatial parts. By carefully recombining key terms, we bound the temporal error in the  L^2 -norm and the spatial error in the  H^1 -norm. Using Ritz projection and interpolation techniques, we derive superconvergence of order \( O(h^2) \) in the  H^1 -norm for the original variable. A numerical example is provided to validate the theoretical results.