For a Keller-Segel model for chemotaxis in two spatial dimensions we consider a modification of a positivity preserving fully discrete scheme using a local extremum diminishing flux limiter. We discretize space using piecewise linear finite elements on an quasiuniform triangulation of acute type and time by the backward Euler method. We assume that initial data are sufficiently small in order not to have a blow-up of the solution. Under appropriate assumptions on the regularity of the exact solution and the time step parameter we show existence of the fully discrete approximation and derive error bounds in $L^2$ for the cell density and $H^1$ for the chemical concentration. We also present numerical experiments to illustrate the theoretical results.

翻译：针对二维空间中的Keller-Segel趋化模型，我们研究了一种采用局部极值递减通量限制器的保正全离散格式的改进方案。空间离散采用锐角型拟一致三角剖分上的分片线性有限元，时间离散采用后向欧拉方法。假设初始数据足够小以避免解发生爆破。在精确解正则性和时间步长参数满足适当假设的条件下，我们证明了全离散近似解的存在性，并推导了细胞密度在$L^2$范数下、化学浓度在$H^1$范数下的误差界。同时通过数值实验验证了理论结果。