Fully discrete stabilized mixed finite element method for chemotaxis equations on surfaces

In this paper, we study fully discrete stabilized mixed finite element methods for solving chemotaxis equations on surfaces. By defining two global fluxes as new vector-valued variables, the original equations are firstly transformed into an equivalent first order mixed form. Then, employing the stabilized mixed finite element method for the spatial discretization, and the backward Euler and backward differential formulation of second order (BDF2) schemes for the temporal discretization, respectively, we propose the first and second order fully discrete stabilized finite element methods for the models. Furthermore, by utilizing a modified scalar auxiliary variable (SAV) method to balance the errors in the stabilizing and decoupling processes, we prove the stabilities and the inf-sup conditions of the stabilized methods. Finally, several numerical examples are presented to verify the efficiencies of the proposed methods.