  A numerical procedure providing guaranteed two-sided bounds on the effective
coefficients of elliptic partial differential operators is presented. The upper
bounds are obtained in a standard manner through the variational formulation of
the problem and by applying the finite element method. To obtain the lower
bounds we formulate the dual variational problem and introduce appropriate
approximation spaces employing the finite element method as well. We deal with
the 3D setting, which has been rarely considered in the literature so far. The
theoretical justification of the procedure is presented and supported with
illustrative examples.
