Unconditionally energy stable fully discrete schemes for a chemo-repulsion model

Analysis of a chemo-repulsion model with nonlinear production: The continuous problem and unconditionally energy stable fully discrete schemes

“…Proof. The proof follows as in Theorem 4.5 of [15], by using the Leray-Schauder fixed point theorem.…”

“…The hypotheses of the Leray-Schauder fixed point theorem are satisfied as in Theorem 3.11 of [15], but applying in this case Lemma 4.6 in order to prove the continuity of the operator R.…”

“…This kind of formulation considering σ = ∇v as auxiliary variable has been used in the construction of numerical schemes for other chemotaxis models (see for instance [14,15,23]). Once problem ( 78) is solved, we can recover v ε from u ε by solving…”

“…In [13,14], unconditionally energy stable time-discrete numerical schemes and fully discrete FE schemes for a chemo-repulsion model with quadratic production have been analyzed. In [15], the authors studied unconditionally energy stable fully discrete FE schemes for a chemo-repulsion model with linear production. However, as far as we know, for the chemo-repulsion model with production term u p (2) there are not works studying energy-stable numerical schemes.…”

“…(Unconditional existence)There exists at least one solution (u n ε , σ n ε ) of scheme USε.Proof. The proof follows as in Theorem 4.5 of[15], by using the Leray-Schauder fixed point theorem.Lemma 4.21. (Conditional uniqueness) If k f (h, ε) < 1 (where f (h, ε) ↑ +∞ when h ↓ 0 or ε ↓ 0), then the solution (u n ε , σ n ε ) of the scheme USε is unique.Proof.…”