Convergence analysis for the homogenization problem of elliptic stokes equations and computation

Abstract

We study the convergence rate of an asymptotic expansion for the elliptic, parabolic and Stokes operators with rapidly oscillating coefficients. First we propose homogenized expansions which are convolution forms of Green function and given force term of elliptic equation. Then, using local $L^p$-theory, the growth rate of the perturbation of Green function is found. From the representation of elliptic solution by Green function, we estimate the convergence rate in $L^p$ space of the homogenized expansions to the exact solution. Next, we consider $L^{2}(0,T:H^1(Ω))$ or $L^{∞}( Ω × (0,T))$ convergence rate of the first order approximation for parabolic homogenization problems. Furthermore we deal with the Stokes equations with periodic viscosity and study its regularity. Finally, we present the numerical example.