Abstract
In this paper, a new stabilized finite volume method is studied and developed for the stationary Navier-Stokes equations. This method is based on a local Gauss integration technique and uses the lowest equal order finite element pair P 1–P 1 (linear functions). Stability and convergence of the optimal order in the H 1-norm for velocity and the L 2-norm for pressure are obtained. A new duality for the Navier-Stokes equations is introduced to establish the convergence of the optimal order in the L 2-norm for velocity. Moreover, superconvergence between the conforming mixed finite element solution and the finite volume solution using the same finite element pair is derived. Numerical results are shown to support the developed convergence theory.
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Communicated by Ragnar Winther.
The first author is partly supported by the NSF of China 10701001 and Natural Science Basic Research Plan in Shaanxi Province of China (Program No. SJ08A14), the second author is partly supported by China NSF Young Scientist Grant 10801101, and the third author is partly supported by NSERC/AERI/Foundation CMG Chair and iCORE Chair Funds in Reservoir Simulation.
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Li, J., Shen, L. & Chen, Z. Convergence and stability of a stabilized finite volume method for the stationary Navier-Stokes equations. Bit Numer Math 50, 823–842 (2010). https://doi.org/10.1007/s10543-010-0277-1
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DOI: https://doi.org/10.1007/s10543-010-0277-1
Keywords
- Navier-Stokes equations
- Finite element method
- Finite volume method
- Inf-sup condition
- Stability
- Convergence
- Superconvergence
- Numerical results