  The goal of this paper is to show that the local Gr\"obner fan is a
polyhedral fan for ideals in the ring of power series and the homogenized ring
of analytic differential operators. We will also discuss about relations
between the local Gr\"obner fan and the (global) Gr\"obner fan for a given
ideal and algorithms of computing local Gr\"obner fans. In rings of
differential operators, finiteness and convexity of local Gr\"obner fans were
firstly proved by Assi, Castro-Jim\'enez and Granger. But they did not prove
that they are polyhedral fans.
