In this paper, we propose a numerical method for the elastic equation governing the blood driven motion of human carotid atherosclerotic plaques. The proposed schema combines a spectral element method for the spatial discretization and a Newmark-like method for the time discretization. We obtain the optimal error estimate for the numerical solution to the full-discrete problem, showing that the convergence is exponential in space and of second order in time. It is also proven that the overall schema is unconditionally stable in certain cases. Finally, we present some numerical examples to verify the error estimates, and carry out a simulation in an application domain extracted from practical experiments.