Fully Discrete Analysis of a Discontinuous Finite Element Method for the Keller-Segel Chemotaxis Model

Epshteyn, Yekaterina; Izmirlioglu, Ahmet

doi:10.1007/s10915-009-9281-5

Cited by 51 publications

(23 citation statements)

References 35 publications

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“…We recall that Π h : C(Ω) → U h is the Lagrange interpolation operator, and the discrete semiinner product (·, ·) h was defined in (8). Once the scheme US is solved, given v n−1 ε ∈ V h , we can…”

Section: Scheme Usmentioning

confidence: 99%

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

2019

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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.

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Section: Scheme Usmentioning

confidence: 99%

“…A mixed FE approximation is studied in [14]. In [8], some error estimates are proved for a fully discrete discontinuous FE method. In the case where the chemotaxis occurs in heterogeneous medium, in [6] the convergence of a combined finite volume-nonconforming finite element scheme is studied, and some discrete properties are proved.…”

Section: Introductionmentioning

confidence: 99%

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

2019

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“…Several numerical methods have already been developed to solve the Patlak-Keller-Segel model for chemotaxis using finite element methods [17], finite volume methods [8,9,13], and the references therein. Other models have also been investigated numerically [11,12,21].…”

Section: Kinetic Models For Bacterial Chemotaxismentioning

confidence: 99%

Numerical Simulations of Kinetic Models for Chemotaxis

Filbet

Yang

2014

SIAM J. Sci. Comput.

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Abstract. We present a new algorithm based on a Cartesian mesh for the numerical approximation of kinetic models for chemosensitive movements set in an arbitrary geometry. We investigate the influence of the geometry on the collective behavior of bacteria described by a kinetic equation interacting with nutrients and chemoattractants. Numerical simulations are performed to verify accuracy and stability of the scheme and its ability to exhibit aggregation of cells and wave propagations. Finally some comparisons with experiments show the robustness and accuracy of such kinetic models.

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“…Let us briefly review some of the recent numerical methods that have been proposed for the "parabolic-parabolic" coupling of chemotaxis models in the literature. High-order discontinuous Galerkin methods have been proposed in [15,16] and [30] for chemotaxis models in 2D rectangular domains. A flux corrected finite element method is designed in 2D in [43], and is extended to chemotaxis models on stationary surface domains and cylindrical domains in [42,44].…”

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confidence: 99%

Efficient Numerical Algorithms Based on Difference Potentials for Chemotaxis Systems in 3D

Epshteyn

Xia

2019

J Sci Comput

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In this work, we propose efficient and accurate numerical algorithms based on Difference Potentials Method for numerical solution of chemotaxis systems and related models in 3D. The developed algorithms handle 3D irregular geometry with the use of only Cartesian meshes and employ Fast Poisson Solvers. In addition, to further enhance computational efficiency of the methods, we design a Difference-Potentialsbased domain decomposition approach which allows mesh adaptivity and easy parallelization of the algorithm in space. Extensive numerical experiments are presented to illustrate the accuracy, efficiency and robustness of the developed numerical algorithms.In this work we develop efficient and accurate numerical algorithms based on Difference Potentials Method (DPM) for the Patlak-Keller-Segel (PKS) chemotaxis model and related problems in 3D. The proposed methods handle irregular geometry with the use of only Cartesian meshes, employ Fast Poisson Solvers, allow easy parallelization and adaptivity in space.Chemotaxis refers to mechanisms by which cellular motion occurs in response to an external stimulus, for instance a chemical one. Chemotaxis is an essential process in many medical and biological applications, including cell aggregation and pattern formation mechanisms and tumor growth. Modeling of chemotaxis dates back to the pioneering work by Patlak, Keller and Segel [28,29,38]. The PKS chemotaxis model consists of coupled convection-diffusion and reaction-diffusion partial differential equations:subject to zero Neumann boundary conditions and the initial conditions on the cell density ρ(x, y, z, t) and on the chemoattractant concentration c(x, y, z, t). In the system (1), the coefficient χ is a chemotactic sensitivity constant, γ ρ and γ c are the reaction coefficients. The parameter α is equal to either 1 or 0, which corresponds to the "parabolic-parabolic" or reduced "parabolic-elliptic" coupling, respectively. In this work, we will focus on the PKS chemotaxis model (1) with α = γ ρ = γ c = 1, but the developed methods are not restricted by this assumption and can be easily extended to more general chemotaxis systems and related models.In the past several years, chemotaxis models have been extensively studied (see, e.g., [19,20,21,22, 39] and references below). A known property of many chemotaxis systems is their ability to represent a concentration phenomenon that is mathematically described by fast growth of solutions in small neighborhoods of concentration points/curves. The solutions may blow up or may exhibit a very singular behavior. This blowup represents a mathematical description of a cell concentration phenomenon that occurs in real biological systems, see, e.g., [1,5,6,7,11,12,36,40].Capturing blowing up or spiky solutions is a challenging task numerically, but at the same time design of robust numerical algorithms is crucial for the modeling and analysis of chemotaxis mechanisms. Let us briefly review some of the recent numerical methods that have been proposed for the "parabolic-parabolic" cou...

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Fully Discrete Analysis of a Discontinuous Finite Element Method for the Keller-Segel Chemotaxis Model

Cited by 51 publications

References 35 publications

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

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

Numerical Simulations of Kinetic Models for Chemotaxis

Efficient Numerical Algorithms Based on Difference Potentials for Chemotaxis Systems in 3D

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