We study how and to what extent the existence of non-local diffusion affects the transport of chemical species in a composite medium. For our purposes, we prescribe the mass flux to obey a two-scale, non-local constitutive law featuring derivatives of fractional order, and we employ the asymptotic homogenisation technique to obtain an overall description of the species’ evolution. As a result, the non-local effects at the micro-scale are ciphered in the effective diffusivity, while at the macro-scale the homogenised problem features an integro-differential equation of fractional type. In particular, we prove that in the limit case in which the non-local interactions are neglected, classical results of asymptotic homogenisation theory are re-obtained. Finally, we perform numerical simulations to show the impact of the fractional approach on the overall diffusion of species in a composite medium. To this end, we consider two simplified benchmark problems, and report some details of the numerical schemes based on finite element methods.

我们研究了非局部扩散的存在如何以及在何种程度上影响复合介质中化学物质的运输。出于我们的目的，我们规定质量通量服从具有分数阶导数的两尺度，非局部本构定律，并且我们采用渐近均质化技术来获得该物种进化的总体描述。结果，对微观尺度的非局部效应进行了加密在有效扩散率方面，而在宏观尺度上，均质化问题具有分数阶积分微分方程。特别是，我们证明了在忽略非局部相互作用的极限情况下，重新获得了渐近同质化理论的经典结果。最后，我们进行数值模拟以显示分数方法对复合介质中物种总体扩散的影响。为此，我们考虑了两个简化的基准问题，并报告了基于有限元方法的数值方案的一些细节。