  Measurements and theoretical calculations of meson form factors are essential
for our understanding of internal hadron structure and QCD, the dynamics that
bind the quarks in hadrons. The pion electromagnetic form factor has been
measured at small space-like momentum transfer $|q^2| < 0.3$~GeV$^2$ by pion
scattering from atomic electrons and at values up to $2.5$~GeV$^2$ by
scattering electrons from the pion cloud around a proton. On the other hand, in
the limit of very large (or infinite) $Q^2=-q^2$, perturbation theory is
applicable. This leaves a gap in the intermediate $Q^2$ where the form factors
are not known.
  As a part of their 12 GeV upgrade Jefferson Lab will measure pion and kaon
form factors in this intermediate region, up to $Q^2$ of $6$~GeV$^2$. This is
then an ideal opportunity for lattice QCD to make an accurate prediction ahead
of the experimental results. Lattice QCD provides a from-first-principles
approach to calculate form factors, and the challenge here is to control the
statistical and systematic uncertainties as errors grow when going to higher
$Q^2$ values.
  Here we report on a calculation that tests the method using an $\eta_s$
meson, a 'heavy pion' made of strange quarks, and also present preliminary
results for kaon and pion form factors. We use the $n_f=2+1+1$ ensembles made
by the MILC collaboration and Highly Improved Staggered Quarks, which allows us
to obtain high statistics. The HISQ action is also designed to have small
discretisation errors. Using several light quark masses and lattice spacings
allows us to control the chiral and continuum extrapolation and keep systematic
errors in check.
