  We compute the electron $g$ factor to the $\mathcal{O}(\alpha^5)$ order on
the lattice in quenched QED. We first study finite volume corrections in
various IR regularization methods to discuss which regularization is optimal
for our purpose. We find that in QED$_L$ the finite volume correction to the
effective mass can have different parametric dependences depending on the size
of Euclidean time $t$ and match the `naive on-shell result' only at very large
$t$ region, $t \gg L$. We adopt finite photon mass regularization to suppress
finite volume effects exponentially and also discuss our strategy for selecting
simulation parameters and the order of extrapolations to efficiently obtain the
$g$ factor. We perform lattice simulation using small lattices to test
feasibility of our calculation strategy. This study can be regarded as an
intermediate step toward giving the five-loop coefficient independently of the
preceding studies.
