Unconditional structure preserving fully discrete finite element method for the Keller–Segel equations on closed surfaces

In this paper, we study unconditional structure preserving fully discrete finite element method (FEM) for the Keller–Segel equations on closed surfaces. Based on the equivalent form got from the Slotboom transformation, applying the surface finite element method (SFEM) and the Euler scheme in space and time, we present a fully discrete scheme for the Keller–Segel equations. Then, by modifying the approximation of the exponential function with piecewise local geometric means and introducing two auxiliary variables with respect to the Slotboom transformation to split the coupling, and applying the lump process to the mass matrix, we get a new structure preserving fully discrete finite element scheme. The proposed scheme is rigorously proved to be stable, unconditionally mass conservative, positivity preserving and energy dissipative. Moreover, at each time step, only the linear elliptic equation with constant coefficient needs to be solved, which can be implemented efficiently. Finally, a number of numerical experiments are shown to demonstrate the theoretical analysis.