AI 中文总结
本文针对带乘性噪声与逻辑增长项的随机Keller-Segel系统，提出一种分裂混合有限元方法，解耦原耦合系统以降低计算成本，经理论分析与数值实验验证了其收敛性及对相关特性的捕捉能力。
AI 中文摘要
本文针对一类带乘性噪声驱动的逻辑增长项的随机Keller-Segel系统，提出并分析了一种分裂混合有限元方法。通过引入代表化学梯度的辅助变量并结合时间滞后分裂策略，该方法将原耦合系统解耦为一系列更简单的子问题，从而消除了Ladyzhenskaya-Babuška-Brezzi稳定性约束，允许对所有未知量使用连续分段线性有限元空间，且避免在每个时间步求解完全耦合的非线性系统，大幅降低了计算成本。结合隐式欧拉时间离散化，该方法为随机Keller-Segel系统生成了全离散数值格式。结合局部化技术与合适的随机稳定性论证，本文建立了全离散近似的最优强误差估计，并证明了具有显式收敛速率的依概率收敛。数值实验验证了理论收敛速率，且表明该方法成功捕捉到逻辑增长项诱导的全局有界性，以及乘性噪声对趋化聚集的影响。
英文摘要
In this paper, we propose and analyze a splitting mixed finite element method for a stochastic Keller--Segel system with logistic growth driven by multiplicative noise. By introducing an auxiliary variable representing the chemical gradient together with a time-lagged splitting strategy, the proposed method decouples the original coupled system into a sequence of simpler subproblems. Consequently, it eliminates the Ladyzhenskaya--Babuška--Brezzi stability constraint, permits the use of continuous piecewise linear finite element spaces for all unknowns, and avoids solving a fully coupled nonlinear system at each time step, thereby significantly reducing the computational cost. Combined with an implicit Euler time discretization, the proposed approach yields a fully discrete numerical scheme for the stochastic Keller--Segel system. Using a localization technique together with suitable stochastic stability arguments, we establish optimal strong error estimates for the fully discrete approximations and prove convergence in probability with explicit convergence rates. Numerical experiments verify the theoretical convergence rates and demonstrate that the proposed method successfully captures the global boundedness induced by the logistic growth term as well as the influence of multiplicative noise on chemotactic aggregation.
Comments31 pages
