# How to implementthe penalty FEM for solving the steady stokes equations

Hello everyone,
When using the penalty method for solving Stokes equation, we can decouple the computation of velocity field with the pressure. But how to implement projection operator?

I think that the only proper way to implement the projection term is to use the original coupled formulation

Otherwise a simplified way could be to use a lumped projection, but I do not know how to do that.

Nevertheless a particular case which is clean is if the spaces X_h, Q_h are such that \mathop{\rm div}X_h\subset Q_h, because then I_h is identity and you can directly solve
(\nabla u_{\varepsilon h},\nabla v_h)+\frac{1}{\varepsilon}(\nabla\cdot u_{\varepsilon h},\nabla\cdot v_h)=(f,v_h),\quad\forall v_h\in X_h.
Not that the space Q_h disappears then.
The couple of spaces X_h, Q_h need of course to satisfy an \inf\sup condition. This leads to so called Scott-Vogelius type finite elements.
Two simple cases exist:

1. X_h=P2, Q_h=P1dc, with a Hsieh–Clough–Tocher mesh. Such a mesh Th can be obtained by a subdivision from an arbitrary mesh Th0 by the command