  We investigate the gravitational fragmentation of expanding shells driven by
HII regions using the three-dimensional Lagrangian simulation codes based on
the Riemann solver, called Godunov smoothed particle hydrodynamics. The ambient
gas is assumed to be uniform. In order to attain high resolution to resolve the
geometrically thin dense shell, we calculate not the whole but a part of the
shell. We find that perturbations begin to grow earlier than the prediction of
the linear analysis under the thin-shell approximation. The wavenumber of the
most unstable mode is larger than that in the thin-shell linear analysis. The
development of the gravitational instability is accompanied by the significant
deformation of the contact discontinuity. These results are consistent with a
linear analysis presented by Iwasaki et al. (2011) that have taken into account
the density profile across the thickness and approximate shock and contact
discontinuity boundary conditions. We derive useful analytic formulae for the
fragment scale and the epoch when the gravitational instability begins to grow.
