This paper deals with a fully discrete numerical scheme for the incompressible Chemotaxis(Keller-Segel)-Navier-Stokes system. Based on a discontinuous Galerkin finite element scheme in the spatial directions, a semi-implicit first-order finite difference method in the temporal direction is applied to derive a completely discrete scheme. With the help of a new projection, optimal error estimates in $L^2$ and $H^1$-norms for the cell density, the concentration of chemical substances and the fluid velocity are derived. Further, optimal error bound in $L^2$-norm for the fluid pressure is obtained. Finally, some numerical simulations are performed, whose results confirm the theoretical findings.

翻译：本文研究不可压缩趋化（Keller-Segel）-纳维-斯托克斯系统的全离散数值格式。基于空间方向的间断伽辽金有限元格式，在时间方向应用半隐式一阶有限差分方法，推导出完全离散格式。借助新的投影算子，推导了细胞密度、化学物质浓度和流体速度在$L^2$范数与$H^1$范数下的最优误差估计。进一步获得了流体压力在$L^2$范数下的最优误差界。最后进行了数值模拟，其结果验证了理论结论。