Analysis of fully discrete Crank-Nicolson finite element methods for a stochastic Keller-Segel chemotaxis system with gradient-type multiplicative noise

2025/07/20 by Liet Vo, Vo, Liet
Biochemistry, Genetics and Molecular Biology · Mathematics · #FOS: Mathematics #Gene Regulatory Network Analysis #Mathematical Biology Tumor Growth #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.2507.15103

openalex publication_date 2025/07/20 · openalex created_date 2025/10/16 · openalex updated_date 2026/07/28

Abstract

We develop and analyze numerical methods for a stochastic Keller-Segel system perturbed by Stratonovich noise, which models chemotactic behavior under randomly fluctuating environmental conditions. The proposed fully discrete scheme couples a Crank-Nicolson time discretization with a splitting mixed finite element method in space. We rigorously prove the stability of the numerical scheme and establish strong convergence rates of order O(k^1/2 + k^-1/2h²), where k and h denote the time and spatial step sizes, respectively. Notably, the presence of stochastic forcing leads to an inverse dependence on k in the error estimates, distinguishing the convergence behavior from that of the deterministic case. Numerical experiments are presented to validate the theoretical results and demonstrate the effectiveness and accuracy of the proposed methods.

Citations

• Higher order time discretization method for the stochastic Stokes equations with multiplicative noise
• High moment and pathwise error estimates for fully discrete mixed finite element approximations of the Stochastic Stokes Equations with Multiplicative Noises
• Positivity-preserving and energy-dissipative finite difference schemes for the Fokker-Planck and Keller-Segel equations
• The one-dimensional stochastic Keller--Segel model with time-homogeneous spatial Wiener processes
• Numerical analysis for a chemotaxis-Navier-Stokes system

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