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 {verified} 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，本文提出了一种有限元离散化方法。该模型包含压力$p$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。由于多尺度现象、各分量的刚性与非线性特性，该问题需要高分辨率求解。我们采用空间上的低阶有限元离散化，结合半隐式时间推进格式。通过选取系统中不同参数组合进行多个数值测试，对代码进行了验证。在无精确解的情况下，应用Richardson外推技术估计了方法的收敛阶。