Rigorous continuum limit for the discrete network formation problem
Accepted version
Peer-reviewed
Repository URI
Repository DOI
Change log
Authors
Haskovec, J
Kreusser, LM
Markowich, P
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.
Description
Keywords
Continuum limit, Gamma-convergence, finite element discretization, network formation
Journal Title
Communications in Partial Differential Equations
Conference Name
Journal ISSN
0360-5302
1532-4133
1532-4133
Volume Title
44
Publisher
Taylor & Francis
Publisher DOI
Rights
All rights reserved
Sponsorship
Engineering and Physical Sciences Research Council (EP/L016516/1)
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).