[02353] Finite volume coupled with finite element scheme for the chemotaxis-fluid model
Session Time & Room : 4D (Aug.24, 15:30-17:10) @D401
Type : Contributed Talk
Abstract : We propose a linear decoupled positivity-preserving scheme for the chemotaxis-fluid system modeling the mutual interaction of the swimming aerobic bacteria with the surrounding fluid flow. The scheme consists of 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 can be regarded as the piecewise constant functions on the dual mesh $($or equivalently, the piecewise linear functions on the triangulation in the mass-lumping sense$)$, which are obtained by the upwind finite volume approximation satisfying the positivity-preserving and mass conservation laws.
The numerical velocity is computed by the finite element method in the triangulation and is utilized to define the upwind-type numerical flux in the dual mesh. We examine 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 establish the optimal error estimates for chemical concentration, cellular density and velocity field in $(l^\infty(W^{1,p}), l^\infty(L^p),l^\infty(W^{1,p}))$-norms. Several numerical examples are presented to confirm the theoretical results.
