  There are two major theoretical issues for the star formation law (the
relation between the surface densities of molecular gas and star formation rate
on a galaxy scale): (i) At low metallicity, it is not obvious that star-forming
regions are rich in H$_2$ because the H$_2$ formation rate depends on the dust
abundance; and (ii) whether or not CO really traces H$_2$ is uncertain,
especially at low metallicity. To clarify these issues, we use a hydrodynamic
simulation of an isolated disc galaxy with a spatial resolution of a few tens
parsecs. The evolution of dust abundance and grain size distribution is treated
consistently with the metal enrichment and the physical state of the
interstellar medium. We compute the H$_2$ and CO abundances using a subgrid
post-processing model based on the dust abundance and the dissociating
radiation field calculated in the simulation. We find that when the metallicity
is $\lesssim 0.4$ Z$_\odot$ ($t<1$ Gyr), H$_2$ is not a good tracer of star
formation rate because H$_2$-rich regions are limited to dense compact regions.
At $Z\gtrsim 0.8$ Z$_\odot$, a tight star formation law is established for both
H$_2$ and CO. At old ($t \sim 10$ Gyr) ages, we also find that adopting the
so-called MRN grain size distribution with an appropriate dust-to-metal ratio
over the entire disc gives reasonable estimates for the H$_2$ and CO
abundances. For CO, improving the spatial resolution of the simulation is
important while the H$_2$ abundance is not sensitive to sub-resolution
structures at $Z\gtrsim 0.4$ Z$_\odot$.
