  In this paper we study the homogenization of the Dirichlet problem for the
Stokes equations in a perforated domain with multiple microstructures. First,
under the assumption that the interface between subdomains is a union of
Lipschitz surfaces, we show that the effective velocity and pressure are
governed by a Darcy law, where the permeability matrix is piecewise constant.
The key step is to prove that the effective pressure is continuous across the
interface, using Tartar's method of test functions. Secondly, we establish the
sharp error estimates for the convergence of the velocity and pressure,
assuming the interface satisfies certain smoothness and geometric conditions.
This is achieved by constructing two correctors. One of them is used to correct
the discontinuity of the two-scale approximation on the interface, while the
other is used to correct the discrepancy between boundary values of the
solution and its approximation.
