趋化性-Navier-Stokes系统的数值分析
Numerical analysis for a chemotaxis-Navier-Stokes system
摘要：
在本文中，我们开发了一个数值方法来近似$d$维趋化性Navier-Stokes系统，$d=2,3$，模拟细胞在不可压缩流体中的游动。该模型描述了在化学信号以与生物体的量成比例的速率消耗的情况下的趋化性-流体相互作用。我们构造了基于有限元法的数值逼近，并分析了一些误差估计和收敛到弱解。为了构造数值方案，我们使用分裂技术来处理细胞密度方程中的化学吸引项，从而引入一个新的变量，由化学浓度的梯度给出。有了等价的模型，我们考虑一个全离散的有限元逼近，它是适定性的，它是质量守恒的。我们得到了一致估计并分析了格式的收敛性。最后，我们提出了一些数值模拟，以验证我们的计划的良好行为，以及检查数值的误差估计证明在我们的理论分析。
英文摘要：
In this paper we develop a numerical scheme for approximating a $d$-dimensional chemotaxis-Navier-Stokes system, $d=2,3$, modeling cellular swimming in incompressible fluids. This model describes the chemotaxis-fluid interaction in cases where the chemical signal is consumed with a rate proportional to the amount of organisms. We construct numerical approximations based on the Finite Element method and analyze some error estimates and convergence towards weak solutions. In order to construct the numerical scheme, we use a splitting technique to deal with the chemo-attraction term in the cell-density equation, leading to introduce a new variable given by the gradient of the chemical concentration. Having the equivalent model, we consider a fully discrete Finite Element approximation which is well-posed and it is mass-conservative. We obtain uniform estimates and analyze the convergence of the scheme. Finally, we present some numerical simulations to verify the good behavior of our scheme, as well as to check numerically the error estimates proved in our theoretical analysis.
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