Simulation of blood flows in the human artery is a promising tool for understanding the hemodynamics. The blood flow is often smooth in a healthy artery, but may become locally chaotic in a diseased artery with stenosis, and as a result, a traditional solver may take many iterations to converge or does not converge at all. To overcome the problem, we develop a nonlinearly preconditioned Newton method in which the variables associated with the stenosis are iteratively eliminated and then a global Newton method is applied to the smooth part of the system. More specifically, we model the blood flow in a patient-specific artery based on the unsteady incompressible Navier-Stokes equations with resistive boundary conditions discretized by a fully implicit finite element method. The resulting nonlinear system at each time step is solved by using an inexact Newton method with a domain decomposition based Jacobian solver. To improve the convergence and robustness of the Newton method for arteries with stenosis, we develop an adaptive restricted region-based nonlinear elimination preconditioner which performs subspace correction to remove the local high nonlinearities. Numerical experiments for several cerebral arteries are presented to demonstrate the superiority of the proposed algorithm over the classical method with respect to some physical and numerical parameters. We also report the parallel scalability of the proposed algorithm on a supercomputer with thousands of processor cores.