论文标题
分析经典凯勒的完全离散近似 - 识别模型：较低和先验边界
Analysis of a fully discrete approximation for the classical Keller--Segel model: lower and a priori bounds
论文作者
论文摘要
本文致力于为经典的凯勒（Keller）构建近似解决方案 - 鉴定\ emph {emotaxis}。它由一个非线性抛物线方程系统组成，其中未知数是细胞的平均密度（或生物体），该密度是保守变量和趋化剂的平均密度。 数值建议由粗糙的有限元方法以及质量大块技术和半副欧拉的时间整合组成。结果方案原来是线性的，并将变量的计算分解。近似解决方案保持下限 - 细胞密度的阳性和趋化剂密度的非神经性 - 在$ l^1（ω）$ - norm中有限，满足离散的能量定律，并具有\ emph {a a a a a a a a a emph {a a a a a。后者是通过离散的Moser-trudinger不等式来实现的。据我们所知，我们的数值方法是在同一时间涉及所有前所述属性的文献中可以遇到的第一种方法。此外，进行了一些数值示例以支持和补充理论结果。
This paper is devoted to constructing approximate solutions for the classical Keller--Segel model governing \emph{chemotaxis}. It consists of a system of nonlinear parabolic equations, where the unknowns are the average density of cells (or organisms), which is a conserved variable, and the average density of chemoattractant. The numerical proposal is made up of a crude finite element method together with a mass lumping technique and a semi-implicit Euler time integration. The resulting scheme turns out to be linear and decouples the computation of variables. The approximate solutions keep lower bounds -- positivity for the cell density and nonnegativity for the chemoattractant density --, are bounded in the $L^1(Ω)$-norm, satisfy a discrete energy law, and have \emph{ a priori} energy estimates. The latter is achieved by means of a discrete Moser--Trudinger inequality. As far as we know, our numerical method is the first one that can be encountered in the literature dealing with all of the previously mentioned properties at the same time. Furthermore, some numerical examples are carried out to support and complement the theoretical results.
