Mathematical modelling, analysis and numerical approximation of second order elliptic problems with inclusions

Many biological and geological systems can be modelled as porous media with small inclusions. Vascularized tissue, roots embedded in soil or fractured rocks are examples of such systems. In these applications, tissue, soil or rocks are considered to be porous media, while blood vessels, roots or fractures form small inclusions. To model flow pro- cesses in thin inclusions, one-dimensional (1D) models of Darcy- or Poiseuille type have been used, whereas Darcy-equations of higher dimension have been considered for the flow processes within the porous matrix. A coupling between flow in the porous matrix and the inclusions can be achieved by setting suitable source terms for the corresponding models, where the source term of the higher-dimensional model is concentrated on the centre lines of the inclusions. In this paper, we investigate an alternative coupling scheme. Here, the source term lives on the boundary of the inclusions. By doing so, we lift the dimension by one and thus increase the regularity of the solution. We show that this model can be derived from a full-dimensional model and the occurring modelling errors are estimated. Furthermore, we prove the well-posedness of the variational formulation and discuss the convergence behaviour of standard finite element methods with respect to this model. Our theoretical results are confirmed by numerical tests. Finally, we demonstrate how the new coupling concept can be used to simulate stationary flow through a capillary network embedded in a biological tissue.