Finite volume methods for a Keller–Segel system: discrete energy, error estimates and numerical blow-up analysis

Zhou, Guanyu; Saito, Norikazu

doi:10.1007/s00211-016-0793-2

Cited by 33 publications

(20 citation statements)

References 19 publications

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“…by the Hardy inequality [27], and inequalities (46) and (47) imply (44), with a constant C depending only on p.…”

Section: Embedding and Trace Theoremsmentioning

confidence: 99%

“…Moreover, the analytic semigroup theory and its discrete counterparts play important roles in construction and study of numerical schemes for parabolic equations (see e.g. [18,20,21,37,38,46]). Therefore, it is natural to wonder whether a discrete version of CMR is available.…”

mentioning

confidence: 99%

“…Actually, the standard discrete Laplacian has no such property (see [43]). It must be borne in mind that the L q theory for the discrete Laplacian with mass-lumping is of great use in study of nonlinear problems, such as the finite element and finite volume approximation of the Keller-Segel system modelling chemotaxis (see [37,38,46]).…”

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confidence: 99%

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Discrete maximal regularity and the finite element method for parabolic equations

Kemmochi

Saito

2017

Numer. Math.

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Maximal regularity is a fundamental concept in the theory of partial differential equations. In this paper, we establish a fully discrete version of maximal regularity for a parabolic equation. We derive various stability results in L p (0, T ; L q (Ω)) norm, p, q ∈ (1, ∞) for the finite element approximation with the mass-lumping to the linear heat equation. Our method of analysis is an operator theoretical one using pure imaginary powers of operators and might be a discrete version of G. Dore and A. Venni (On the closedness of the sum of two closed operators. Math. Z., 196(2):189-201, 1987). As an application, optimal order error estimates in that norm are proved. Furthermore, we study the finite element approximation for semilinear heat equations with locally Lipschitz continuous nonlinearity and offer a new method for deriving optimal order error estimates. Some interesting auxiliary results including discrete Gagliardo-Nirenberg and Sobolev inequalities are also presented.

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“…by the Hardy inequality [27], and inequalities (46) and (47) imply (44), with a constant C depending only on p.…”

Section: Embedding and Trace Theoremsmentioning

confidence: 99%

mentioning

confidence: 99%

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Discrete maximal regularity and the finite element method for parabolic equations

Kemmochi

Saito

2017

Numer. Math.

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“…Most of the numerical schemes for the Keller-Segel model utilize the implicit Euler method for the time discretization and aim to preserve some properties of the continuous equations, like (local) mass conservation, positivity (more precisely, nonnegativity) preservation, or energy dissipation. These schemes use (semi-implicit) finite-difference methods [12,31,38]; an upwind finite-element discretization [36]; an Eulerian-Lagrangian scheme based on the characteristics method [39]; a Galerkin method with a diminishing flux limiter [42]; and finite-volume methods [19,45]. A finite-volume scheme was also studied in [1] but with a first-order semi-exponentially fitted time discretization.…”

Section: 2mentioning

confidence: 99%

Blow-up of solutions to semi-discrete parabolic-elliptic Keller-Segel models

Jüngel¹,

Leingang²

2017

Discrete &Amp; Continuous Dynamical Systems - B

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The existence of weak solutions and upper bounds for the blow-up time for time-discrete parabolic-elliptic Keller-Segel models for chemotaxis in the two-dimensional whole space are proved. For various time discretizations, including the implicit Euler, BDF, and Runge-Kutta methods, the same bounds for the blow-up time as in the continuous case are derived by discrete versions of the virial argument. The theoretical results are illustrated by numerical simulations using an upwind finite-element method combined with second-order time discretizations.

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“…Some numerical schemes have been studied for chemotaxis models. Existence of discrete solutions, convergence, mass-conservation and error estimates, among other qualitative properties, have been studied in the context of Finite Volume (FV) schemes [13,22,34], Finite Element (FE) approximations [11,25,27,28,33] or combined FV-FE schemes [7].…”

Section: Introductionmentioning

confidence: 99%

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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Finite volume methods for a Keller–Segel system: discrete energy, error estimates and numerical blow-up analysis

Cited by 33 publications

References 19 publications

Discrete maximal regularity and the finite element method for parabolic equations

Discrete maximal regularity and the finite element method for parabolic equations

Blow-up of solutions to semi-discrete parabolic-elliptic Keller-Segel models

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

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