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

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.