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

Bikram Bir, Harsha Hutridurga, Amiya K. Pani

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.

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

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 and -norms for the cell density, the concentration of chemical substances and the fluid velocity are derived. Further, optimal error bound in -norm for the fluid pressure is obtained. Finally, some numerical simulations are performed, whose results confirm the theoretical findings.

Paper Structure

This paper contains 13 sections, 23 theorems, 156 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Discrete Formulation
  4. Semidiscrete dG Formulation
  5. Fully Discrete dG Formulation
  6. A Priori Error Analysis
  7. $L^2$-Error Estimates
  8. $H^1$-Error Estimates
  9. Error estimate for fluid pressure
  10. Numerical Experiments
  11. Convergence test
  12. Movement of cells guided by the concentration
  13. Conclusion

Key Result

Lemma 2.1

For each element $E\in\mathcal{E}_h$ with diameter $h_E$, there exists a constant $C$ independent of $h_E$ such that the followings hold: where $e$ is an edge of the element $E$. These results are also valid for vector-valued function $\hbox{\boldmath $\phi$}\in {\bm V}$.

Figures (11)

  • Figure 1: Numerical errors with $(\bm{P}_1,P_0,P_1,P_1)$ pairs for Example \ref{['exm1']}.
  • Figure 2: Numerical errors with $(\bm{P}_2,P_1,P_2,P_2)$ pairs for Example \ref{['exm1']}.
  • Figure 3: Numerical errors with $(\bm{P}_3,P_2,P_3,P_3)$ pairs for Example \ref{['exm1']}.
  • Figure 4: Numerical errors with respect to time variable for Example \ref{['exm1']}.
  • Figure 5: Numerical errors with $(\bm{P}_1,P_0,P_1,P_1)$ pairs for Example \ref{['exm3d']}.
  • ...and 6 more figures

Theorems & Definitions (37)

  • Lemma 2.1
  • Lemma 3.1: Discrete coercivityDE11
  • Lemma 3.2
  • Lemma 3.3: Skew-symmetric property
  • Lemma 3.4: Boundedness of trilinear terms
  • Lemma 3.5: Boundedness of chemotactic term
  • Lemma 3.6: Optimal approximation property
  • Lemma 3.7
  • Lemma 3.8
  • Lemma 3.9
  • ...and 27 more