  We recently proposed a method to calculate the relative magnetic helicity in
a finite volume for a given magnetic field which however required the flux to
be balanced separately on all the sides of the considered volume. In order to
allow finite magnetic fluxes through the boundaries, a Coulomb gauge is
constructed that allows for global magnetic flux balance. We tested and
verified our method in a theoretical fore-free magnetic field model. We apply
the new method to the former calculation data and found a difference of less
than 1.2\%. We also applied our method to the magnetic field above active
region NOAA 11429 obtained by a new photospheric-data-driven MHD model code
GOEMHD3. We analyzed the magnetic helicity evolution in the solar corona using
our new method. It was found that the normalized magnetic helicityis equal to
-0.038 when fast magnetic reconnection is triggered. This value is comparable
to the previous value (-0.029) in the MHD simulations when magnetic
reconnection happened and the observed normalized magnetic helicity (-0.036)
from the eruption of newly emerging active regions. We found that only 8\% of
the accumulated magnetic helicity is dissipated after it is injected through
the bottom boundary. This is in accordance with the Woltjer conjecture. Only
2\% of magnetic helicity injected from the bottom boundary escapes through the
corona. This is consistent with the observation of magnetic clouds, which could
take away magnetic helicity into the interplanetary space, in the case
considered here, several halo CMEs and two X-class solar flares origin from
this active region.
