# Numerical analysis for a chemotaxis-Navier-Stokes system

## TL;DR

This paper develops and analyzes a finite element numerical scheme for simulating a chemotaxis-Navier-Stokes system, capturing cellular movement in fluids with error estimates and convergence proofs.

## Contribution

It introduces a novel splitting-based finite element scheme for the chemotaxis-Navier-Stokes system with proven convergence and error estimates.

## Key findings

- The scheme is well-posed and mass-conservative.
- Numerical simulations confirm the scheme's accuracy and stability.
- Error estimates align with theoretical predictions.

## 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 some error estimates and convergence towards weak 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 it is 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 error estimates proved in our theoretical analysis.