A fully decoupled and structure-preserving relaxation Crank--Nicolson finite element method for Gross--Pitaevskii--Poisson model

We propose a fully decoupled, structure-preserving relaxation Crank--Nicolson finite element method (FEM) for the coupled Gross--Pitaevskii--Poisson (GPP) system modeling ultracold plasmas. By introducing suitable auxiliary variables to reformulate the nonlinear interaction and charge density terms, the original system is recast into an equivalent form that enables a linear, fully decoupled numerical scheme. The proposed method preserves key physical invariants, including the mass of each component and a modified discrete energy, at the fully discrete level. We establish the well-posedness and uniqueness of the scheme and rigorously derive optimal error estimates, achieving second-order accuracy in time and optimal $(k+1)$-th order convergence in space for $P^k$ finite element approximations. Numerical experiments confirm the theoretical results and demonstrate the effectiveness of the method in preserving conservation properties and accurately capturing complex dynamical behaviors of the coupled GPP system.