  In this work, we theoretically and numerically discuss the time fractional
subdiffusion-normal transport equation, which depicts a crossover from
sub-diffusion (as $t\rightarrow 0$) to normal diffusion (as $t\rightarrow
\infty$). Firstly, the well-posedness and regularities of the model are studied
by using the bivariate Mittag-Leffler function. Theoretical results show that
after introducing the first-order derivative operator, the regularity of the
solution can be improved in substance. Then, a numerical scheme with
high-precision is developed no matter the initial value is smooth or
non-smooth. More specifically, we use the contour integral method (CIM) with
parameterized hyperbolic contour to approximate the temporal local and
non-local operators, and employ the standard Galerkin finite element method for
spacial discretization. Rigorous error estimates show that the proposed
numerical scheme has spectral accuracy in time and optimal convergence order in
space. Besides, we further improve the algorithm and reduce the computational
cost by using the barycentric Lagrange interpolation. Finally, the obtained
theoretical results as well as the acceleration algorithm are verified by
several 1-D and 2-D numerical experiments, which also show that the numerical
scheme developed in this paper is effective and robust.
