Numerical analysis for a chemotaxis-Navier–Stokes system

Abstract

In this paper we develop a numerical scheme for approximating a $d$-dimensional chemotaxis-Navier-Stokes system, $d=2,3$, modeling cellular swimming in incompressible fluids. This model describes the chemotaxis-fluid interaction in cases where the chemical signal is consumed with a rate proportional to the amount of organisms. We construct numerical approximations based on the Finite Element method and analyze optimal error estimates and convergence towards regular solutions. In order to construct the numerical scheme, we use a splitting technique to deal with the chemo-attraction term in the cell-density equation, leading to introduce a new variable given by the gradient of the chemical concentration. Having the equivalent model, we consider a fully discrete Finite Element approximation which is well-posed and mass-conservative. We obtain uniform estimates and analyze the convergence of the scheme. Finally, we present some numerical simulations to verify the good behavior of our scheme, as well as to check numerically the optimal error estimates proved in our theoretical analysis.

A Diffuse-Domain Based Numerical Method for a Chemotaxis-Fluid Model

“…In [18], a fully decoupled, linear and positivity preserving finite-element method for solving the chemotaxis-Stokes equations has been recently developed for a similar setup. In [12], the chemotaxis-fluid model without the discontinuous oxygen cut-off function r(c) has been considered, for which a finite-element method has been constructed, optimal error estimates have been established, and convergence towards regular solutions has been proved. In [19,20], a generalized chemotaxis-diffusion-convection model, which includes the dynamic free surface and appropriate boundary conditions, has been proposed together with a numerical method, which uses a time dependent grid and incorporates surface tension and a dynamic contact line.…”

An optimal control problem related to a 3D-chemotaxis-Navier-Stokes model

“…Concerning smooth bounded domains and initial data in L p -spaces, results about local and global-in-time existence of solutions were obtained in [17]. A numerical analysis based on the Finite Element method including error estimates and convergence towards regular solutions were performed in [19]. On the other hand, in the whole space R N , N = 2, 3, the local and global-in-time existence of solutions for (1.2) has been studied in [11-13, 30, 56], and some references therein.…”

Finite Element numerical schemes for a chemo-attraction and consumption model

“…For instance, Winkler in [22] has studied the d-dimensional problem (d = 2, 3) of attractive chemotaxis models with consumption of chemical substance, showing that under suitable regularity assumptions on the initial data, the chemotaxis-Navier-Stokes system admits a unique global classical solution (d = 2) and the simplified chemotaxis-Stokes system possesses at least one global weak solution (d = 3). Moreover, in [7] the authors construct numerical approximations for the same type of system. The presented approximations are based on using the Finite Element method, obtaining optimal error estimates and convergence towards regular solutions.…”