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.

中文翻译：

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在孔隙尺度模拟生物膜生长的抛物线变分不等式系统的数值分析

在本文中，我们考虑了一个由两个耦合的非线性扩散-反应偏微分方程 (PDE) 组成的系统，该系统模拟了生物膜的生长和营养物的消耗。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 被框定为抛物线变分不等式。我们为耦合非线性系统的有限元 (FE) 近似推导出严格的误差估计，并通过实验确认数值近似以预测速率收敛。我们还展示了模拟，其中我们跟踪类似于孔隙尺度几何结构的域中的自由边界，并在其中测试不同的建模假设。