In this article, a numerical analysis of the asymptotic behavior of the discrete energy associated with a dissipative coupled wave system is conducted. The numerical approximation of the system is constructed using the P1 finite element method for spatial discretization, combined with the implicit Euler scheme for time integration. An a priori error analysis is established, showing that, under extra regularity assumptions on the continuous solution, the numerical scheme exhibits linear convergence. Then, for the first time in the literature, the exponential decay of the fully discrete energy is shown using the energy method. Finally, several numerical simulations are provided to illustrate the convergence behavior and to analyze the evolution of the discrete energy.
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