Modeling, Structure and Discretization of Hierarchical Mixed-Dimensional Partial Differential Equations
Mixed-dimensional partial differential equations arise in several physical applications, wherein parts of the domain have extreme aspect ratios. In this case, it is often appealing to model these features as lower-dimensional manifolds embedded into the full domain. Examples are fractured and composite materials, but also wells (in geological applications), plant roots, or arteries and veins.
In this manuscript, we survey the structure of mixed-dimensional PDEs in the context where the sub-manifolds are a single dimension lower than the full domain, including the important aspect of intersecting sub-manifolds, leading to a hierarchy of successively lower-dimensional sub-manifolds. We are particularly interested in partial differential equations arising from conservation laws. Our aim is to provide an introduction to such problems, including the mathematical modeling, differential geometry, and discretization.
The authors wish to thank Gunnar Fløystad, Eirik Keilegavlen, Jon Eivind Vatne and Ivan Yotov for valuable comments and discussions on this topic. The authors also with to thank the two anonymous reviewers who provided helpful comments on the initial version of this manuscript. This research is funded in part by the Norwegian Research Council grants: 233736 and 250223.
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