Abstract: For transient Naiver-Stokes problems, a stabilized nonconforming finite element method is presented, focusing on two pairs inf-sup unstable finite element spaces, i.e., PNC1 /PNC1 triangular and PNQ1 /PNQ1 quadrilateral finite element spaces. The semiand full-discrete schemes of the stabilized method are studied based on the pressure projection and a variational multi-scale method. It has some attractive features: avoiding higher-order derivatives and edge-based data structures, adding a discrete velocity term only on the fine scale, being effective for high Reynolds number fluid flows, and avoiding increased computation cost. For the full-discrete scheme, it has second-order estimations of time and is unconditionally stable. The presented numerical results agree well with the theoretical results.

Key words: transient Naiver-Stokes problem, nonconforming finite element method, pressure projection, variational multiscale method, functionally graded material, semi-analytical
method, complex shape, 3D analysis, hole