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 $p$.

翻译：我们研究了一种用于生物输运网络形成的标量偏微分方程模型。从等边三角剖分上的离散图模型出发，我们严格推导了在网络细化下作为$\Gamma$-极限的连续能量泛函，并证明了全局极小解的存在性。该模型具有梯度流结构，其稳态解与$p$-Laplacian方程的解一致。基于这一联系，我们实现了有限元离散化，并提出了一种新颖的动态松弛格式。该格式在构造测试中达到了最优收敛速率，并展现出网格无关的性能——在均匀网格细化下，时间步数、非线性迭代次数和线性求解次数保持稳定。数值实验证实了该标量模型既能重现生物学相关的网络模式，又能作为求解大指数$p$的$p$-Laplacian方程的高效计算松弛策略。