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

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 $\Gamma$-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 $p$.