Fully discrete finite element approximation for the projection method to solve the Chemotaxis-Fluid System
Chenyang Li, Ping Lin, Haibiao Zheng
TL;DR
The paper addresses the numerical solution of a chemotaxis–Navier–Stokes system by introducing a first-order pressure-correction finite element method. It uses a Mini element for the velocity–pressure pair and linear elements for cell density and chemical concentration, with backward Euler time stepping and semi-implicit treatment of nonlinearities. A rigorous a priori error analysis yields an $L^2$-error bound of order $O(\tau^2 + h^4)$ under a mild time-step restriction, and numerical experiments verify stability, convergence, and physiologically meaningful chemotaxis–fluid dynamics. This work provides the first complete error analysis for a projection-based discretization of the fully coupled chemotaxis–fluid model and offers a practical, efficient tool for simulating such multi-physics systems.
Abstract
In this paper, we investigate a chemotaxis-fluid interaction model governed by the incompressible Navier-Stokes equations coupled with the classical Keller-Segel chemotaxis system. To numerically solve this coupled system, we develop a pressure-correction projection finite element method based on a projection framework. The proposed scheme employs a backward Euler method for temporal discretization and a mixed finite element method for spatial discretization. Nonlinear terms are treated semi-implicitly to enhance computational stability and efficiency. We further establish rigorous error estimates for the fully discrete scheme, demonstrating the convergence of the numerical method. A series of numerical experiments are conducted to validate the stability, accuracy, and effectiveness of the proposed method. The results confirm the scheme's capability to capture the essential dynamical behaviors and characteristic features of the chemotaxis-fluid system.