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 . Under certain regularity assumptions, the optimal error estimates in -norm for velocity and -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 -norm for the velocity. Finally, a numerical example confirms our theory analysis and validates the high effectiveness of this new method.

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