### Cheapest stable nonconforming finite elements for Stokes/Navier–Stokes equations

**Monday, June 19, 11 am.**

**Dongwoo Sheen**

Department of Mathematics, Seoul National University, Seoul 08826, Korea

We give a brief review on nonconforming finite elements based on quadrialteral meshes. Then we introduce and analyze a “stable cheapest nonconforming finite element” pair on rectangular grids, with modification on the corner elements adopting the nonconforming finite element method introduced by Cai–Douglas–Ye. Except at these two corner elements, for all other elements we use the simplest P1 nonconforming quadrilateral element for the approximation of each component of velocity fields plus a globally one–dimensional bubble space, while the pressure is approximated by the piecewise constant element. We then apply this stable cheapest nonconforming pairs to approximate the steady–state Navier–Stokes equations in a rectangular cavity. Some numerical and mathematical comparisons ensure the simplicity and superiority in capturing the correct physical properties of cavity flow over other finite element methods. The numerical evidence with this element show simpler and cheaper elements can catch more precise physical characteristics.