  We introduce a new constructive method for establishing lower bounds on
convergence rates of periodic homogenization problems associated with
divergence type elliptic operators. The construction is applied in two
settings. First, we show that solutions to boundary layer problems for
divergence type elliptic equations set in halfspaces and with infinitely smooth
data, may converge to their corresponding boundary layer tails as slow as one
wish depending on the position of the hyperplane. Second, we construct a
Dirichlet problem for divergence type elliptic operators set in a bounded
domain, and with all data being $C^\infty$-smooth, for which the boundary value
homogenization holds with arbitrarily slow speed.
