Partial asymptotic decomposition of the domain for the diffusion–discrete absorption

We consider the diffusion–discrete absorption equation, which is an approximate model of the diffusion of a substance in a solution containing a chain of cells absorbing the substance; the size of the cells is much smaller than the distance $h$ between them, and this distance is small compared to the length of the chain. The diffusion–discrete absorption equation contains the standard diffusion term and a discrete point absorption, which is described by the sum of a large number of Dirac delta functions with supports on nonuniform grid multiplied by an unknown function (concentration). We study the possibility of a partial asymptotic decomposition of the domain for the diffusion–discrete absorption equation: it is required to preserve the discrete description of the absorption on a part of the domain and pass to a continuous description on the greater part of the domain. This combination of the macroscopic and microscopic descriptions in one model is characteristic of multiscale modeling. We obtain an error estimate for the partially continuous model with respect to the original model with completely discrete absorption.