A Positivity Preserving Moving Mesh Finite Element Method for the Keller–Segel Chemotaxis Model

Sulman, Mohamed H. M.; Nguyen, Truong

doi:10.1007/s10915-019-00951-0

Cited by 15 publications

(11 citation statements)

References 32 publications

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“…where Ω = [−0.5, −0.5] 2 . In [11], the estimated blow-up time is reported around 1.21e − 4. Given a set of N 0 = 400 and N b = 200 training data, we try to train the wavelet network growing up to ten centers with q = 4.…”

Section: A One-dimensional Examplementioning

confidence: 99%

“…However, many of existed approximating methods have some sort of limitations that can only solve the problem under certain conditions. The finite element method (FEM) [11] and the finite volume method (FVM) [12] are two classical numerical schemes, both of which are domain-dependent techniques and need a mesh over the domain. Because the chemotaxis models with blow-up are convection-dominated diffusion equations, employing the standard FEM leads to spurious oscillations over the domain, and therefore, it is a challenge to capture the blow-up of solutions of the chemotaxis models.…”

Section: Introductionmentioning

confidence: 99%

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A novel growing wavelet neural network algorithm for solving chemotaxis systems with blow‐up

Mostajeran

Hosseini

2023

Math Methods in App Sciences

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In this study, we introduce a new growing neural network algorithm that is based on wavelet neural networks and call our algorithm a growing wavelet neural network (GWNN) method. We apply our proposed scheme to train a wavelet neural network to solve chemotaxis problems with blow‐up. These problems are highly nonlinear time‐dependent systems of partial differential equations, and it is a challenge to get the pattern of the solution accurately. The proposed structure is partial retraining of the network, which increases its capacity to catch the spiky pattern of the solution. Our neural network‐based algorithm allows us to solve the nonlinear chemotaxis problems without the use of linearization techniques and regularization techniques, most of which reduce the accuracy of the model. This mesh‐free‐based method can manage a variety of blow‐up models with curved boundaries without imposing an extra cost. By proving the consistency and stability of the method, we show the convergence of GWNN solutions to analytical solutions of the chemotaxis problem. Several illustrative examples and simulation results are provided to demonstrate the correctness of the results and the robust performance of the presented algorithm. Moreover, to illustrate the effectiveness of the GWNN method, we make a comparison with two other network‐based methods.

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Section: A One-dimensional Examplementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A novel growing wavelet neural network algorithm for solving chemotaxis systems with blow‐up

Mostajeran

Hosseini

2023

Math Methods in App Sciences

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“…Nevertheless, Algorithm 2 does use a mass lumping technique for some terms in equations ( 20) and (21). The stabilization of equation ( 20) is based on the new discretization (18) of the chemotaxis term, which obligates to redesign the stabilizing term B u 2 in (22) and the shock detector ᾱi in (23) as well. Remark 2.1.…”

Section: Algorithmmentioning

confidence: 99%

“…Remark 2.1. In principle, there is no drawback with applying the shock detector α i defined in (16), which acts on both maxima and minima to limit the action of the stabilizing term (22). The reason for not doing so is that the nonlinear solver used to obtain the solution of each time step does not work properly for some numerical tests.…”

Section: Algorithmmentioning

confidence: 99%

Bound-preserving finite element approximations of the Keller-Segel equations

Badia¹,

Bonilla²,

Gutiérrez-Santacreu³

2022

Preprint

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This paper aims to develop numerical approximations of the Keller-Segel equations that mimic at the discrete level the lower bounds and the energy law of the continuous problem. We solve these equations for two unknowns: the organism (or cell) density, which is a positive variable, and the chemoattractant density, which is a nonnegative variable. We propose two algorithms, which combine a stabilized finite element method and a semi-implicit time integration. The stabilization consists of a nonlinear artificial diffusion that employs a graph-Laplacian operator and a shock detector that localizes local extrema. As a result, both algorithms turn out to be nonlinear. Both algorithms can generate cell and chemoattractant numerical densities fulfilling lower bounds. However, the first algorithm requires a suitable constraint between the space and time discrete parameters, whereas the second one does not. We design the latter to attain a discrete energy law on acute meshes. We report some numerical experiments to validate the theoretical results on blowup and non-blowup phenomena. In the blowup setting, we identify a locking phenomenon that relates the L ∞ (Ω)-norm to the L 1 (Ω)-norm limiting the growth of the singularity when supported on a macroelement.

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“…Numerical algorithms are mainly designed so as to keep lower bounds and to be mass-preserving. We refer the reader to [11][12][13][14][15][16]. We were pointed out by a referee the paper [17].…”

Section: The Keller-segel Equationsmentioning

confidence: 99%

Analysis of a fully discrete approximation for the classical Keller–Segel model: Lower and a priori bounds

Gutiérrez-Santacreu

Galván

2021

Computers & Mathematics with Applications

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A Positivity Preserving Moving Mesh Finite Element Method for the Keller–Segel Chemotaxis Model

Cited by 15 publications

References 32 publications

A novel growing wavelet neural network algorithm for solving chemotaxis systems with blow‐up

A novel growing wavelet neural network algorithm for solving chemotaxis systems with blow‐up

Bound-preserving finite element approximations of the Keller-Segel equations

Analysis of a fully discrete approximation for the classical Keller–Segel model: Lower and a priori bounds

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