On a Completely Discrete Discontinuous Galerkin Method for Incompressible Chemotaxis-Navier-Stokes Equations
This work provides rigorous error analysis for a numerical method applied to a complex coupled PDE system, which is incremental for computational mathematics.
The authors developed a fully discrete discontinuous Galerkin scheme for the Chemotaxis-Navier-Stokes equations and derived optimal error estimates in L2 and H1 norms for density, concentration, and velocity, and L2 norm for pressure, confirmed by numerical simulations.
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.