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.

翻译：本文提出了一种能够为椭圆型偏微分算子的有效系数提供保证性双侧界的数值过程。上界通过问题的变分公式及有限元法的标准方式获得。为得到下界，我们构造了对偶变分问题，并同样采用有限元法引入适当的近似空间。本文处理了此前文献中较少涉及的三维情形。文中给出了该过程的理论依据，并通过算例验证。