Characteristic splitting mixed finite element analysis of Keller–Segel chemotaxis models

Chemotaxis models are described by a system of coupled nonlinear partial differential equations: a convection–diffusion type cell density equation and a parabolic or reaction–diffusion type chemical concentration equation. This paper is focused on development of a new numerical method for these models. In this method, a splitting mixed element technique is used to solve the chemical concentration equation, where the LBB condition is not necessary and the flux equation is separated from the chemical concentration equation. Moreover a mass-conservative characteristic finite element method is used to solve the cell density equation, avoiding nonphysically oscillations and keeping mass balance globally. The coefficient matrices of the discrete system are symmetric and positive definite. The convergence of this method is analyzed and the corresponding error estimates are derived for pre-blow-up time since we assume boundedness of the exact solution. Numerical simulations are performed to verify that our method can obtain accurate, nonnegative, oscillation-free solutions and has the ability to recover blowing-up solutions.