The Flux-corrected Finite Element Method For Biology Chemotaxis Models

Chemotaxis is of great significance to the growth and development of organisms,along with the production and life of human beings.In order to study it quantitatively,many biology chemotaxis models have been proposed.Positivity-preserving,strong cou-pling and nonlinearity are main difficulties for solving these models.To deal with these difficulties,solve these models and simulate biology chemotaxis phenomena,we do the following study based on the flux-corrected finite element method(FC-FEM):At first,we solve and simulate the two-and three-dimensional chemotaxis models based on the FC-FEM,including the Keller-Segel model,the pattern model and the ag-gregation model.These models describe the chemotaxis of cells or microorganisms them-selves.In order to solve the difficulty of strong coupling and nonlinearity,a semi-discrete scheme is obtained by combining the backward Euler method,linearization method and decoupling strategy.Besides,the two-or three-dimensional semi-discrete problem is transformed into two or three one-dimensional subproblems by an operator splitting method,which can improve the computational efficiency.Then,the FC-FEM is used to obtain a positive and full discrete scheme for each one-dimensional equation.Numerical experiments show that this method is not only accurate,but also evidently reduces the computational cost of high-dimensional chemotaxis models.By this method,we capture the blow-up phenomena of the solution of two-and three-dimensional Keller-Segel mod-els.Moreover,the pattern formation and aggregation of bacteria under chemotaxis are simulated,the effect of chemosensitivity coefficient on pattern formation is also explored.Next,we develop the FC-FEM to solve and simulate the two-dimensional coupled chemotaxis-fluid model.Such model couples chemotaxis equations with fluid equations to describe chemotaxis of cells or microorganisms affected by fluid motion.To decouple the velocity and pressure in the fluid equation,the semi-discrete scheme is obtained by the backward Euler method and the pressure-corrected method.After that,the FC-FEM is used to obtain a positive and full discrete scheme based on the semi-discrete scheme.The accuracy and positivity of this method were tested through a series of numerical experiments.Additionally,we simulate the biological convection phenomenon of bacillus subtilis,and find that the oxygen truncation function with regularity is helpful for us to simulate that phenomenon well.