Abstract
We introduce a general, analytical framework to express and to approximate partial differential equations (PDEs) numerically on graphs and networks of surfaces – generalized by the term hypergraphs. To this end, we consider PDEs on hypergraphs as singular limits of PDEs in networks of thin domains (such as fault planes, pipes, etc.), and we observe that (mixed) hybrid formulations offer useful tools to formulate such PDEs. Thus, our numerical framework is based on hybrid finite element methods (in particular, the class of hybrid discontinuous Galerkin methods).
Highlights
This manuscript establishes a general approach to formulate partial differential equations (PDEs) on networks ofsurfaces, referred to as hypergraphs
Singular limits leading to hypergraphs for fluid equations can be found in [29], where a Kirchhoff law in a junction of thin pipes is derived, and [30] where junctions of thin pipes and plates are treated using the method of two-scale convergence
We have motivated the formulation of PDEs on geometric hypergraphs, which generalize the notions of “domains”, “graphs”, and “network of surfaces”
Summary
This manuscript establishes a general approach to formulate partial differential equations (PDEs) on networks of (hyper)surfaces, referred to as hypergraphs. Singular limits leading to hypergraphs for fluid equations can be found in [29], where a Kirchhoff law in a junction of thin pipes is derived, and [30] where junctions of thin pipes and plates are treated using the method of two-scale convergence. The remainder of this manuscript is structured as follows: First, we discuss conservation equations on hypergraphs. The publication is wrapped up, by a section on possible conclusions
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