Partial differential equations on hypergraphs and networks of surfaces: derivation and hybrid discretizations

TL;DR

This paper develops a unified analytical and numerical framework for solving partial differential equations on hypergraphs and surface networks, using hybrid finite element methods to handle complex geometries and limits.

Contribution

It introduces a novel approach to formulate and approximate PDEs on hypergraphs as limits of PDEs on thin domain networks, employing hybrid discretizations.

Findings

Framework effectively models PDEs on hypergraphs and surface networks.

Hybrid finite element methods provide accurate numerical solutions.

The approach bridges PDE theory and complex network geometries.

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).