Local a Priori Estimate on the General Scale Subdomains

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.