A finite element discretization is developed for the Cai-Hu model, describing the formation of biological networks. The model consists of a non linear elliptic equation for the pressure $p$ and a non linear reaction-diffusion equation for the conductivity tensor $\mathbb{C}$. The problem requires high resolution due to the presence of multiple scales, the stiffness in all its components and the non linearities. We propose a low order finite element discretization in space coupled with a semi-implicit time advancing scheme. The code is validated with several numerical tests performed with various choices for the parameters involved in the system. In absence of the exact solution, we apply Richardson extrapolation technique to estimate the order of the method.

翻译：针对描述生物网络形成的Cai-Hu模型，本文发展了一种有限元离散化方法。该模型由压力$p$的非线性椭圆方程和电导率张量$\mathbb{C}$的非线性反应扩散方程组成。由于多尺度结构、各分量的刚性以及非线性特性的存在，该问题对数值分辨率要求较高。我们提出一种空间低阶有限元离散格式，结合半隐式时间推进方案。通过若干涉及不同系统参数选择的数值试验验证了代码的有效性。在缺乏精确解的情况下，应用Richardson外推技术估算方法的阶数。