  Traveling across several order of magnitude in distance, relativistic jets
from strong gravity region to asymptotic flat spacetime region are believed to
consist of several general relativistic magnetohydrodynamic (GRMHD) processes.
We present a semi-analytical approach for modeling the global structures of a
trans-fast magnetosonic relativistic jet, which should be ejected from a plasma
source nearby a black hole in a funnel region enclosed by dense accreting flow
and also disk corona around the black hole. Our model consistently includes the
inflow and outflow part of the GRMHD solution along the magnetic field lines
penetrating the black hole horizon. After the rotational energy of the black
hole is extracted electromagnetically by the negative energy GRMHD inflow, the
huge electromagnetic energy flux then propagates from the inflow to the outflow
region across the plasma source, and in the outflow region the electromagnetic
energy converts to the fluid kinetic energy. Eventually, the accelerated
outflow must exceed the fast-magnetosonic wave speed. We apply the
semi-analytical trans-fast magnetosonic flow model to the black hole
magnetosphere for both parabolic and split-monopole magnetic field
configurations, and discuss the general flow properties; that is, jet
acceleration, jet magnetization, and the locations of some characteristic
surfaces of the black hole magnetosphere. We have confirmed that, at large
distance, the GRMHD jet solutions are in good agreement with the previously
known trans-fast special relativistic magnetohydrodynamic (SRMHD) jet
properties, as expected. The flexibility of the model provides a prompt and
heuristic way to approximate the global GRMHD trans-fast magnetosonic jet
properties.
