  The local a priori estimate for the finite element approximation is essential
for underlying the local and parallel technique. It is well known that the
constant coefficients in the inequality is independent of the mesh size. But it
is not so clear whether the constant depends on the scale of the local
subdomains. The aim of this note is to derive a new local a priori estimate on
the general scale domains. We also show that the dependence of the constant
appearing in the local a priori estimate on the scale of the subdomains.
