Rigorous continuum limit for the discrete network formation problem

Jan Haskovec, Lisa Maria Kreusser, Peter Markowich

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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 languageEnglish
Pages (from-to)1159-1185
Number of pages27
JournalCommunications in Partial Differential Equations
Issue number11
Early online date17 May 2019
Publication statusPublished - 31 Dec 2019


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

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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