Study of a chemo-repulsion model with quadratic production. Part II: Analysis of an unconditionally energy-stable fully discrete scheme

Guillén-González, Francisco; Rodríguez-Bellido, María Ángeles; Rueda-Gómez, Diego Armando

doi:10.1016/j.camwa.2020.04.010

Cited by 28 publications

(37 citation statements)

References 14 publications

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“…Recall that v ε = v(z ε ) is the unique solution of problem (14). From (18) we have that (u ε , v ε ) satisfies the following energy equality:…”

Section: Existence Of Weak-strong Solutions Of (3) Theorem 33 There Exists At Least One (U V) Weak-strong Solution Of Problem (3)mentioning

confidence: 99%

“…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 [18,20,33]). Once problem ( 74) is solved, we can recover v ε from u ε by solving…”

Section: Scheme Usεmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

2021

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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 $$ \left\{ \begin{array}{l} \partial_t u - {\Delta} u - \nabla\cdot (u\nabla v)=0 \ \ \text{ in}\ {\Omega},\ t>0,\\ \partial_t v - {\Delta} v + v = u^p \ \ { in}\ {\Omega},\ t>0, \end{array} \right. $$ ∂ t u − Δ u − ∇ ⋅ ( u ∇ v ) = 0 in Ω , t > 0 , ∂ t v − Δ v + v = u p i n Ω , t > 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.

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“…Recall that v ε = v(z ε ) is the unique solution of problem (14). From (18) we have that (u ε , v ε ) satisfies the following energy equality:…”

Section: Existence Of Weak-strong Solutions Of (3) Theorem 33 There Exists At Least One (U V) Weak-strong Solution Of Problem (3)mentioning

confidence: 99%

Section: Scheme Usεmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

2021

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“…[2]). 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].…”

Section: Introductionmentioning

confidence: 99%

Numerical analysis for a chemotaxis-Navier–Stokes system

et al. 2021

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In this paper we develop a numerical scheme for approximating a $d$-dimensional chemotaxis-Navier-Stokes system, $d=2,3$, modeling cellular swimming in incompressible fluids. This model describes the chemotaxis-fluid interaction in cases where the chemical signal is consumed with a rate proportional to the amount of organisms. We construct numerical approximations based on the Finite Element method and analyze optimal error estimates and convergence towards regular solutions. In order to construct the numerical scheme, we use a splitting technique to deal with the chemo-attraction term in the cell-density equation, leading to introduce a new variable given by the gradient of the chemical concentration. Having the equivalent model, we consider a fully discrete Finite Element approximation which is well-posed and mass-conservative. We obtain uniform estimates and analyze the convergence of the scheme. Finally, we present some numerical simulations to verify the good behavior of our scheme, as well as to check numerically the optimal error estimates proved in our theoretical analysis.

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“…For instance, one related type of models are the ones that focus on repulsive chemotaxis systems with the cell population producing chemical substance, that is, a system like (1) with χ < 0. We refer the reader to [11,12] (and the references therein) where the authors focused on studying unconditionally energy stable and mass-conservative FE numerical schemes, by introducing the gradient of the chemical concentration variable, for chemo-repulsive systems with quadratic (−µu 2 ) and linear production terms (−µu 2 ) in (1) 2 , respectively.…”

Section: Introductionmentioning

confidence: 99%

Finite Element numerical schemes for a chemo-attraction and consumption model

Guillén-González¹,

Tierra²

2021

Preprint

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This work is devoted to design and study efficient and accurate numerical schemes to approximate a chemo-attraction model with consumption effects, which is a nonlinear parabolic system for two variables; the cell density and the concentration of the chemical signal that the cell feel attracted to.We present several finite element schemes to approximate the system, detailing the main properties of each of them, such as conservation of cells, energy-stability and approximated positivity. Moreover, we carry out several numerical simulations to study the efficiency of each of the schemes and to compare them with others classical schemes.

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Study of a chemo-repulsion model with quadratic production. Part II: Analysis of an unconditionally energy-stable fully discrete scheme

Cited by 28 publications

References 14 publications

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

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

Numerical analysis for a chemotaxis-Navier–Stokes system

Finite Element numerical schemes for a chemo-attraction and consumption model

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