Gradient Flows for the $p$-Laplacian Arising from Biological Network Models: A Novel Dynamical Relaxation Approach

TL;DR

The paper develops a scalar PDE model for biological transport networks and derives the continuum energy $\mathcal{E}[c]$ as the $\Gamma$-limit of discrete network energies on equilateral triangulations. It reveals a gradient-flow formulation whose steady states solve the $p$-Laplacian and introduces a finite element discretization together with a novel dynamical relaxation scheme that achieves optimal convergence and mesh-independent performance. The authors establish $\Gamma$-convergence, construct global minimizers from discrete minimizers, and demonstrate numerically that the approach reproduces biologically relevant network patterns while efficiently solving large-exponent $p$-Laplacian equations, even on nonconvex domains. The results indicate robust, scalable performance without adaptive mesh refinement, highlighting the method’s potential for large-scale network formation and nonlinear diffusion problems.

Abstract

We investigate a scalar partial differential equation model for the formation of biological transportation networks. Starting from a discrete graph-based formulation on equilateral triangulations, we rigorously derive the corresponding continuum energy functional as the $Γ$-limit under network refinement and establish the existence of global minimizers. The model possesses a gradient-flow structure whose steady states coincide with solutions of the $p$-Laplacian equation. Building on this connection, we implement finite element discretizations and propose a novel dynamical relaxation scheme that achieves optimal convergence rates in manufactured tests and exhibits mesh-independent performance, with the number of time steps, nonlinear iterations, and linear solves remaining stable under uniform mesh refinement. Numerical experiments confirm both the ability of the scalar model to reproduce biologically relevant network patterns and its effectiveness as a computationally efficient relaxation strategy for solving $p$-Laplacian equations for large exponents $p$.

Figures (7)

• Figure 1: Network formation: Energy (left panel), time step (central), and number of nonlinear iterations as a function of simulation time for 512x512 mesh. The legend in the right panel reports in parentheses the total number of time steps, the total number of nonlinear steps, and the average number of Krylov iterations per Newton step.
• Figure 2: Network formation: Conductivity in logarithmic scale at selected time instances (see Figure \ref{['fig:box_sequence_logs']}).
• Figure 3: MMS errors and convergence rates (in parentheses) for the test cases TC1, TC2, TC3, and TC4 and different error metrics.
• Figure 4: From left to right: time step, $p$-Laplacian energy $1/p \int_\Omega |\nabla u_h|^p - S u_h$ (final value in the legend), $L^2$ norm of $|\nabla u_n|^2 - c_n^{\gamma-1}$, and number of Newton steps as a function of time for different levels of uniform refinement $r$ for TC2 (top row) and TC4 (bottom row). The legends in the right panels report in parentheses the total number of time steps, the total number of nonlinear steps, and the average number of Krylov iterations per Newton step.
• Figure 5: TC5: MMS errors and convergence rates (in parentheses) for different error metrics.

Theorems & Definitions (13)

• Proposition 1
• Proposition 2
• Lemma 3
• Lemma 4
• Proposition 5
• Lemma 6
• Corollary 7
• Lemma 8
• Lemma 9
• Lemma 10