A fully finite element approximation of the Keller–Segel model, including self- and cross-diffusion terms and a logistic source, has been considered. The existence of a fully finite element approximation of the Keller–Segel model, some stability bounds of the solution and the convergence of the approximated solution have been shown. As a result, the existence of a solution to the nonlinear Keller–Segel model in ${\mathbb{R}}^d$, $d\le 3$ has been demonstrated. Moreover, an iterative approach to solving the resulted nonlinear discrete system has been introduced. Finally, some numerical results have been shown.

中文翻译：

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带有附加自扩散和交叉扩散项以及逻辑源的 Keller-Segel 模型的有限元逼近

已经考虑了 Keller-Segel 模型的完全有限元近似，包括自扩散项和交叉扩散项以及逻辑源。已经显示了 Keller-Segel 模型的完全有限元近似的存在、解的一些稳定性边界和近似解的收敛性。因此，非线性 Keller-Segel 模型的解存在于${\mathbb{电阻}}^d$, $d\le 3$已经证明。此外，还引入了求解所得非线性离散系统的迭代方法。最后，给出了一些数值结果。