Solving the discretized Navier-Stokes equation by penalization and the augmented Lagrangian method
Numerical Solution of the Navier-Stokes Equations by the Finite Element Method

We begin by explaining the penalization method for the Stokes equation. To solve the Navier-Stokes equation, we use a semi-implicit time scheme where, at each time step, we solve a Stokes problem by taking the convective term's value at the previous instant. Thus, knowing u ^k , v ^k and p ^k we determine u ^{k +1} , v ^{k +1} and p ^{k +1} by solving for k = 0, 1, ... the following problem: