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).
Keywords: Conservation equations, hypergraphs, hybrid discontinuous Galerkin
@article{M2AN_2022__56_2_505_0,
author = {Rupp, Andreas and Gahn, Markus and Kanschat, Guido},
title = {Partial differential equations on hypergraphs and networks of surfaces: {Derivation} and hybrid discretizations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {505--528},
year = {2022},
publisher = {EDP-Sciences},
volume = {56},
number = {2},
doi = {10.1051/m2an/2022011},
mrnumber = {4385103},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2022011/}
}
TY - JOUR AU - Rupp, Andreas AU - Gahn, Markus AU - Kanschat, Guido TI - Partial differential equations on hypergraphs and networks of surfaces: Derivation and hybrid discretizations JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2022 SP - 505 EP - 528 VL - 56 IS - 2 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2022011/ DO - 10.1051/m2an/2022011 LA - en ID - M2AN_2022__56_2_505_0 ER -
%0 Journal Article %A Rupp, Andreas %A Gahn, Markus %A Kanschat, Guido %T Partial differential equations on hypergraphs and networks of surfaces: Derivation and hybrid discretizations %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2022 %P 505-528 %V 56 %N 2 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2022011/ %R 10.1051/m2an/2022011 %G en %F M2AN_2022__56_2_505_0
Rupp, Andreas; Gahn, Markus; Kanschat, Guido. Partial differential equations on hypergraphs and networks of surfaces: Derivation and hybrid discretizations. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 56 (2022) no. 2, pp. 505-528. doi: 10.1051/m2an/2022011
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