Comparison of two finite element schemes for a chemo-repulsion system with quadratic production

Abstract

In this paper we propose two fully discrete Finite Elements (FE) schemes for a repulsive chemotaxis model with a quadratic production term. The first one, which corresponds to the backward Euler in time with FE in space, is energy-stable in the primitive variables of the model, under a “compatibility” condition on the FE spaces. The second one, which is obtained modifying the scheme proposed in [13] by applying a regularization procedure, has an “approximated positivity” property which is obtained from discrete energy estimates and an additional estimate for a singular functional. These properties are not available in previous approaches. Additionally, we study the well-posedness and the long time behaviour of the schemes, obtaining exponential convergence to constant states as in the continuous problem. Finally, we compare the numerical schemes throughout several numerical simulations, which are in agreement with the theoretical results.