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Volume 35, Issue 4
The Positivity-Preserving Finite Volume Coupled with Finite Element Method for the Keller-Segel-Navier-Stokes Model

Ping Zeng & Guanyu Zhou

Commun. Comput. Phys., 35 (2024), pp. 1073-1119.

Published online: 2024-05

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  • Abstract

We propose a linear decoupled positivity-preserving scheme for the chemotaxis-fluid system, which models the interaction between aerobic bacteria and the fluid flow surrounding them. This scheme comprises the finite element method (FEM) for the fluid equations on a regular triangulation and an upwind finite volume method (FVM) for the chemotaxis system on two types of dual mesh. The discrete cellular density and chemical concentration are represented as the piecewise constant functions on the dual mesh. They can also be equivalently expressed as the piecewise linear functions on the triangulation in the sense of mass-lumping. These discrete solutions are obtained by the upwind finite volume approximation satisfying the laws of positivity preservation and mass conservation. The finite element method is used to compute the numerical velocity in the triangulation, which is then used to determine the upwind-style numerical flux in the dual mesh. We analyze the $M$-property of the matrices from the discrete system and prove the well-posedness and the positivity-preserving property. By using the $L^p$-estimate of the discrete Laplace operators, semigroup analysis, and induction method, we are able to establish the optimal error estimates for chemical concentration, cellular density, and velocity field in $(l^∞(W^{1,p}), l^∞(L^p),l^∞(W^{1,p}))$-norm. Several numerical examples are presented to verify the theoretical results.

  • AMS Subject Headings

65M08, 76M10, 76D05, 35Q92, 92C17

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-35-1073, author = {Zeng , Ping and Zhou , Guanyu}, title = {The Positivity-Preserving Finite Volume Coupled with Finite Element Method for the Keller-Segel-Navier-Stokes Model}, journal = {Communications in Computational Physics}, year = {2024}, volume = {35}, number = {4}, pages = {1073--1119}, abstract = {

We propose a linear decoupled positivity-preserving scheme for the chemotaxis-fluid system, which models the interaction between aerobic bacteria and the fluid flow surrounding them. This scheme comprises the finite element method (FEM) for the fluid equations on a regular triangulation and an upwind finite volume method (FVM) for the chemotaxis system on two types of dual mesh. The discrete cellular density and chemical concentration are represented as the piecewise constant functions on the dual mesh. They can also be equivalently expressed as the piecewise linear functions on the triangulation in the sense of mass-lumping. These discrete solutions are obtained by the upwind finite volume approximation satisfying the laws of positivity preservation and mass conservation. The finite element method is used to compute the numerical velocity in the triangulation, which is then used to determine the upwind-style numerical flux in the dual mesh. We analyze the $M$-property of the matrices from the discrete system and prove the well-posedness and the positivity-preserving property. By using the $L^p$-estimate of the discrete Laplace operators, semigroup analysis, and induction method, we are able to establish the optimal error estimates for chemical concentration, cellular density, and velocity field in $(l^∞(W^{1,p}), l^∞(L^p),l^∞(W^{1,p}))$-norm. Several numerical examples are presented to verify the theoretical results.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2023-0309}, url = {http://global-sci.org/intro/article_detail/cicp/23095.html} }
TY - JOUR T1 - The Positivity-Preserving Finite Volume Coupled with Finite Element Method for the Keller-Segel-Navier-Stokes Model AU - Zeng , Ping AU - Zhou , Guanyu JO - Communications in Computational Physics VL - 4 SP - 1073 EP - 1119 PY - 2024 DA - 2024/05 SN - 35 DO - http://doi.org/10.4208/cicp.OA-2023-0309 UR - https://global-sci.org/intro/article_detail/cicp/23095.html KW - Finite volume method, finite element method, chemotaxis-fluid system, conservation law, positivity preserving, error estimates. AB -

We propose a linear decoupled positivity-preserving scheme for the chemotaxis-fluid system, which models the interaction between aerobic bacteria and the fluid flow surrounding them. This scheme comprises the finite element method (FEM) for the fluid equations on a regular triangulation and an upwind finite volume method (FVM) for the chemotaxis system on two types of dual mesh. The discrete cellular density and chemical concentration are represented as the piecewise constant functions on the dual mesh. They can also be equivalently expressed as the piecewise linear functions on the triangulation in the sense of mass-lumping. These discrete solutions are obtained by the upwind finite volume approximation satisfying the laws of positivity preservation and mass conservation. The finite element method is used to compute the numerical velocity in the triangulation, which is then used to determine the upwind-style numerical flux in the dual mesh. We analyze the $M$-property of the matrices from the discrete system and prove the well-posedness and the positivity-preserving property. By using the $L^p$-estimate of the discrete Laplace operators, semigroup analysis, and induction method, we are able to establish the optimal error estimates for chemical concentration, cellular density, and velocity field in $(l^∞(W^{1,p}), l^∞(L^p),l^∞(W^{1,p}))$-norm. Several numerical examples are presented to verify the theoretical results.

Zeng , Ping and Zhou , Guanyu. (2024). The Positivity-Preserving Finite Volume Coupled with Finite Element Method for the Keller-Segel-Navier-Stokes Model. Communications in Computational Physics. 35 (4). 1073-1119. doi:10.4208/cicp.OA-2023-0309
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