Analysis of three stabilized finite volume iterative methods for the steady Navier–Stokes equations

Abstract

In this work, three stabilized finite volume iterative schemes for the stationary Navier–Stokes equations are considered. Under the finite volume discretization at each iterative step, the iterative scheme I consists in solving the steady Stokes problem, iterative scheme II consists in solving the stationary linearized Navier–Stokes equations and iterative scheme III consists in solving the steady Oseen equations, respectively. We discuss the stabilities and convergence of three iterative methods. The iterative schemes I and II are stable and convergent under some strong uniqueness conditions, while iterative scheme III is unconditionally stable and convergent under the uniqueness condition. Finally, some numerical results are presented to verify the performance of these iterative schemes.

Keywords:

• steady Navier–Stokes equations
• iterative scheme
• stability
• error estimate
• finite volume method

2010 AMS Subject Classifications:

• 65N08
• 65N30
• 76D05

Acknowledgements

The authors thank the referees very much for helpful comments and suggestions, which led to substantial improvements in the presentation. The first author is very grateful to Professor JinYun Yuan for his kind invitation to visit the Universidade Federal do Paraná, Brazil. This work was partially supported by Chinese NSF (Grant No. 11301157, 11226319), the CAPES and CNPq of Brazil, and the Doctor Fund of Henan Polytechnic University (B2012-098), as well as China.