As the blood flow characteristics are closely related to various cardiovascular diseases, it is very important to predict them accurate enough in an efficient way. Thus, this paper proposes a one-dimensional spectral element model for human blood vessels with varying cross-sections. The spectral element model is formulated by using the variational approach of finite element formulation. The wave solutions analytically solved in the frequency-domain to satisfy governing equations are used to determine the frequency-dependent interpolation functions. The spectral finite element model is then applied to an example blood vessel to investigate the blood flow rate and blood pressure through the blood vessel.