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

Abstract

This work is devoted to study unconditionally energy stable and mass-conservative numerical schemes for the following repulsive-productive chemotaxis model: Find u ≥ 0, the cell density, and v ≥ 0, the chemical concentration, such thatin a bounded domain Ω ⊆ R d , d = 2, 3. By using a regularization technique, we propose three fully discrete Finite Element (FE) approximations. The first one is a nonlinear approximation in the variables (u, v); the second one is another nonlinear approximation obtained by introducing σ = ∇v as an auxiliary variable; and the third one is a linear approximation constructed by mixing the regularization procedure with the energy quadratization technique, in which other auxiliary variables are introduced. In addition, we study the well-posedness of the numerical schemes, proving unconditional existence of solution, but conditional uniqueness (for the nonlinear schemes). Finally, we compare the behavior of such schemes throughout several numerical simulations and provide some conclusions.

Numerical analysis for a chemotaxis-Navier–Stokes system

“…In addition, unconditionally energy stable time-discrete numerical schemes and fully discrete FE schemes for a chemo-repulsion model with quadratic production has been analyzed in [14,15]. Some unconditionally energy stable fully discrete schemes for a parabolic repulsive-productive chemotaxis model (with linear production term) were recently analyzed in [13]. On the other hand, when the interaction with a fluid is assumed, as far as we know, the literature related to the numerical analysis of chemotaxis-Navier-Stokes system is scarce, see [3,22].…”

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

“…On the other hand, the second aim is to design numerical methods for model (3) conserving properties of the continuous problem such as: mass-conservation, energystability and positivity. It is important to mention that approaching chemorepulsion problems by using Finite Element (FE) approximations is not an easy task, because negative (discrete) solutions can be computed (see [17,19,20]). In those cases, some spurious oscillations may appear (see, for instance, [19] for a chemorepulsion model with quadratic production).…”

“…A conditionally energy-stable FV scheme for a chemo-attraction model with an additional cross-diffusion term was analyzed by Bessemoulin and Jüngel [6]. Energy-stability of time-discrete numerical approximations and fully discrete FE schemes for a chemorepulsion model with quadratic production have been analyzed in [17] and [18,19], respectively; while, in the case of linear production, we refer [20]. However, as far as we know, for the chemorepulsion model with production term u p given in (3) there are not works studying energy-stable numerical schemes.…”

“…In [8], Chamoun and collaborators proved a discrete maximum principle for a combined FV-FE scheme approaching a chemotaxis-fluid model. The positivity of only time-discrete schemes and approximated positivity of a fully discrete FE scheme associated with a chemorepulsion model with quadratic signal production were proved in [17] and [19], respectively; while, for the case of linear production, we refer to [20]. Positive numerical methods, using FE techniques, associated with a generalized Keller-Segel model were studied in [10].…”

“…The idea here is to extend the analysis made in [20], although in this case, we need to use two matrix operators (see (51) and (52) below) in order to obtain energystability and approximated positivity. The first one is the operator defined in [2] (and used in [20]); while, the second one, is obtained by constructing regularized functions associated with the test function u p−1 .…”