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 |
|---|---|
| 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 |
Bibliographical note
Funding Information:LMK was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/L016516/1 and the German National Academic Foundation (Studienstiftung des Deutschen Volkes).
Keywords
- 35K55
- 65M60
- 92C42
- Continuum limit
- finite element discretization
- network formation
- Γ-convergence
ASJC Scopus subject areas
- Analysis
- Applied Mathematics
Access to Document
- DiscreteToMacro.PDF
This is an Accepted Manuscript of an article published by Taylor & Francis in Communications in Partial Differential Equations on 17/05/2019, available online: https://www.tandfonline.com/doi/full/10.1080/03605302.2019.1612909.