Sharp Convergence Rates for Darcy's Law

Abstract

This paper is concerned with Darcy's law for an incompressible viscous fluid flowing in a porous medium. We establish the sharp $O(\sqrt{\e})$ convergence rate in a periodically perforated and bounded domain in $R^d$ for $d\ge 2$, where $\e$ represents the size of solid obstacles. This is achieved by constructing two boundary correctors to control the boundary layers created by the incompressibility condition and the discrepancy of boundary values between the solution and the leading term in its asymptotic expansion. One of the correctors deals with the tangential boundary data, while the other handles the normal boundary data.