  We propose a hybrid approach to solve the high-frequency Helmholtz equation
with point source terms in smooth heterogeneous media. The method is based on
the ray-based finite element method (ray-FEM), whose original version can not
handle the singularity close to point sources accurately. This pitfall is
addressed by combining the ray-FEM, which is used to compute the smooth
far-field of the solution accurately, with a high-order asymptotic expansion
close to the point source, which is used to properly capture the singularity of
the solution in the near-field. The method requires a fixed number of grid
points per wavelength to accurately represent the wave field with an asymptotic
convergence rate of $\mathcal{O}(\omega^{-1/2})$, where $\omega$ is the
frequency parameter in the Helmholtz equation. In addition, a fast
sweeping-type preconditioner is used to solve the resulting linear system.
  We present numerical examples in 2D to show both accuracy and efficiency of
our method as the frequency increases. In particular, we provide numerical
evidence of the convergence rate, and we show empirically that the overall
complexity is $\mathcal{O}(\omega^2)$ up to a poly-logarithmic factor.
