Conservative upwind finite-element method for a simplified Keller–Segel system modelling chemotaxis

Finite-element approximation for a non-linear parabolic–elliptic system is considered. The system describes the aggregation of slime moulds resulting from their chemotactic features and is called a simplified Keller–Segel system. Applying an upwind technique, first we present a finite-element scheme that satisfies both positivity and mass conservation properties. Consequently, if the triangulation is of acute type, our finite-element approximation preserves the L1 norm, which is an important property of the original system. Then, under some assumptions on the regularity of a solution and on the triangulation, we establish error estimates in Lp × W1, ∞ with a suitable p > d, where d is the dimension of a spatial domain. Our scheme is well suited for practical computations. Some numerical examples that validate our theoretical results are also presented.