Mixed-dimensional linked models of diffusion: mathematical analysis and stable finite element discretizations

In the context of mathematical modeling, it is sometimes convenient to employ models of different dimensionality simultaneously, even for a single physical phenomenon. However, this type of combination might entail difficulties even when individual models are well-understood, particularly in relation to well-posedness. In this article, we focus on combining two diffusive models, one defined over a continuum and the other one over a curve. The resulting problem is of mixed-dimensionality, where here the low-dimensional problem is embedded within the high-dimensional one. We show unconditional stability and convergence of the continuous and discrete linked problems discretized by mixed finite elements. The theoretical results are highlighted with numerical examples illustrating the effects of linking diffusive models. As a side result, we show that the methods introduced in this article can be used to infer the solution of diffusive problems with incomplete data. The findings of this article are of particular interest to engineers and applied mathematicians and open an avenue for further research connected to the field of data science.