A new multiscale finite element method for the 2D transient Navier–Stokes equations

ABSTRACT

A new multiscale finite element method for the two-dimensional (2D) transient Navier–Stokes equations is proposed in this paper. This new method is based on multiscale enrichment with the lowest equal-order finite element pair $P_1/P_1$. Under certain regularity assumptions, the optimal error estimates in $H^1$-norm for velocity and $L^2$-norm for pressure are obtained. Especially, via applying a new dual problem for the transient Navier–Stokes problem and some techniques in the proof process, we establish the convergence of the optimal order in $L^2$-norm for the velocity. Finally, a numerical example confirms our theory analysis and validates the high effectiveness of this new method.

KEYWORDS:

• Multiscale finite element method
• $P_1/P_1$ elements
• error analysis
• the transient Navier–Stokes equations
• numerical analysis

2010 AMS SUBJECT CLASSIFICATIONS:

• 35Q10
• 65N30
• 76D05

Acknowledgements

We thank the anonymous referees for their careful reading of the manuscript and suggestions that greatly helped to improve the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by the Basic Research Priorities Program of Shaanxi Province [grant number 2017JQ1003], the National Natural Science Foundation of China [grant numbers 11771348, 11571275], and the Major Research and Development Program of China [grant number 2016YFB0200901].