On a Completely Discrete Discontinuous Galerkin Method for Incompressible Chemotaxis-Navier-Stokes Equations

TL;DR

This paper introduces a fully discrete discontinuous Galerkin finite element scheme for the incompressible Chemotaxis-Navier-Stokes equations, providing optimal error estimates and numerical validation.

Contribution

It develops a new fully discrete scheme with optimal error bounds for all variables in the system, including pressure, using a novel projection method.

Findings

Optimal error estimates in $L^2$ and $H^1$ norms for density, concentration, and velocity.

Optimal $L^2$ error bound for fluid pressure.

Numerical simulations confirm theoretical error bounds.

Abstract

This paper deals with a fully discrete numerical scheme for the incompressible Chemotaxis(Keller-Segel)-Navier-Stokes system. Based on a discontinuous Galerkin finite element scheme in the spatial directions, a semi-implicit first-order finite difference method in the temporal direction is applied to derive a completely discrete scheme. With the help of a new projection, optimal error estimates in $L^2$ and $H^1$-norms for the cell density, the concentration of chemical substances and the fluid velocity are derived. Further, optimal error bound in $L^2$-norm for the fluid pressure is obtained. Finally, some numerical simulations are performed, whose results confirm the theoretical findings.