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We consider the following repulsive-productive chemotaxis model: find  u  ≥ 0, the cell density, and  v  ≥ 0, the chemical concentration, satisfying   $$ \left\{ \begin{array}{l} \partial_t u - {\Delta} u - \nabla\cdot (u\nabla v)=0 \ \ \text{ in}\ {\Omega},\ t&gt;0,\\ \partial_t v - {\Delta} v + v = u^p \ \ { in}\ {\Omega},\ t&gt;0, \end{array} \right. $$  ∂ t u − Δ u − ∇ ⋅ ( u ∇ v ) = 0 in Ω , t &gt; 0 , ∂ t v − Δ v + v = u p i n Ω , t &gt; 0 , with  p  ∈ (1, 2),    ${\Omega }\subseteq \mathbb {R}^{d}$  Ω ⊆ ℝ d a bounded domain ( d  = 1, 2, 3), endowed with non-flux boundary conditions. By using a regularization technique, we prove the existence of global in time weak solutions of (1) which is regular and unique for  d  = 1, 2. Moreover, we propose two fully discrete Finite Element (FE) nonlinear schemes, the first one defined in the variables ( u , v ) under structured meshes, and the second one by using the auxiliary variable   σ   = ∇ v  and defined in general meshes. We prove some unconditional properties for both schemes, such as mass-conservation, solvability, energy-stability and approximated positivity. Finally, we compare the behavior of these schemes with respect to the classical FE backward Euler scheme throughout several numerical simulations and give some conclusions.

A chemorepulsion model with superlinear production: analysis of the continuous problem and two approximately positive and energy-stable schemes