## Abstract

Motivated by recent papers describing the formation of biological transport networks we study a discrete model proposed by Hu and Cai consisting of an energy consumption function constrained by a linear system on a graph. For the spatially two-dimensional rectangular setting we prove the rigorous continuum limit of the constrained energy functional as the number of nodes of the underlying graph tends to infinity and the edge lengths shrink to zero uniformly. The proof is based on reformulating the discrete energy functional as a sequence of integral functionals and proving their Γ-convergence towards a continuum energy functional.

Original language | English |
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Pages (from-to) | 1159-1185 |

Number of pages | 27 |

Journal | Communications in Partial Differential Equations |

Volume | 44 |

Issue number | 11 |

Early online date | 17 May 2019 |

DOIs | |

Publication status | Published - 31 Dec 2019 |

## Keywords

- 35K55
- 65M60
- 92C42
- Continuum limit
- finite element discretization
- network formation
- Γ-convergence

## ASJC Scopus subject areas

- Analysis
- Applied Mathematics