  We consider an iteration method for solving an elliptic type boundary value
problem $\mathcal{A} u=f$, where a positive definite operator $\mathcal{A}$ is
generated by a quasi--periodic structure with rapidly changing coefficients
(typical period is characterized by a small parameter $\epsilon$) . The method
is based on using a simpler operator $\mathcal{A}_0$ (inversion of
$\mathcal{A}_0$ is much simpler than inversion of $\mathcal{A}$), which can be
viewed as a preconditioner for $\mathcal{A}$. We prove contraction of the
iteration method and establish explicit estimates of the contraction factor
$q$. Certainly the value of $q$ depends on the difference between $\mathcal{A}$
and $\mathcal{A}_0$. For typical quasi--periodic structures, we establish
simple relations that suggest an optimal $\mathcal{A}_0$ (in a selected set of
"simple" structures) and compute the corresponding contraction factor. Further,
this allows us to deduce fully computable two--sided a posteriori estimates
able to control numerical solutions on any iteration. The method is especially
efficient if the coefficients of $\mathcal{A}$ admit low rank representations
and algebraic operations are performed in tensor structured formats. Under
moderate assumptions the storage and solution complexity of our approach
depends only weakly (merely linear-logarithmically) on the frequency parameter
$1/\epsilon$, providing the FEM approximation of the order of
$O(\epsilon^{1+p})$, $p>0$.
