一个抛物型变分不等式系统模拟生物膜生长的数值分析
Numerical Analysis of a Parabolic Variational Inequality System Modeling Biofilm Growth at the Porescale
摘要：
在本文中，我们考虑了一个系统的两个耦合的非线性扩散-反应偏微分方程（PDE）模型的生物膜的生长和消耗的营养。在感兴趣的尺度上，生物膜密度受到逐点约束，因此生物膜PDE被构造为抛物型变分不等式。我们推导出严格的误差估计的有限元（FE）近似耦合的非线性系统，并确认实验的数值逼近收敛于预测的速度。我们还展示了模拟，其中我们跟踪的自由边界在类似的孔隙尺度几何形状的域，并在其中我们测试不同的建模假设。
英文摘要：
In this paper we consider a system of two coupled nonlinear diffusion--reaction partial differential equations (PDEs) which model the growth of biofilm and consumption of the nutrient. At the scale of interest the biofilm density is subject to a pointwise constraint, thus the biofilm PDE is framed as a parabolic variational inequality. We derive rigorous error estimates for a finite element (FE) approximation to the coupled nonlinear system and confirm experimentally that the numerical approximation converges at the predicted rate. We also show simulations in which we track the free boundary in the domains which resemble the pore scale geometry and in which we test the different modeling assumptions.
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