  We study rates of convergence of solutions in L^2 and H^{1/2} for a family of
elliptic systems {L_\epsilon} with rapidly oscillating oscillating coefficients
in Lipschitz domains with Dirichlet or Neumann boundary conditions. As a
consequence, we obtain convergence rates for Dirichlet, Neumann, and Steklov
eigenvalues of {L_\epsilon}. Most of our results, which rely on the recently
established uniform estimates for the L^2 Dirichlet and Neumann problems in
\cite{12,13}, are new even for smooth domains.
