The Positivity-Preserving Finite Volume Coupled with Finite Element Method for the Keller-Segel-Navier-Stokes Model (Communications in Computational Physics, 2024)
We propose a linear decoupled positivity-preserving scheme for thechemotaxis-fluid system, which models the interaction between aerobic bacteria andthe fluid flow surrounding them. This scheme comprises the finite element method(FEM) for the fluid equations on a regular triangulation and an upwind finite volumemethod (FVM) for the chemotaxis system on two types of dual mesh. The discretecellular density and chemical concentration are represented as the piecewise constantfunctions 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 satisfyingthe laws of positivity preservation and mass conservation. The finite element methodis used to compute the numerical velocity in the triangulation, which is then usedto 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 andthe positivity-preserving property. By using the $L^p$-estimate of the discrete Laplaceoperators, 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.