Newton algorithm to solve the stationary Navier-Stokes equations in 2D.

Solve:

## Variational form

$\displaystyle{ \mu\int_{\Omega}{\nabla\mathbf{u}:\nabla\mathbf{v} - p\nabla\cdot\mathbf{v}} - \int_{\partial\Omega}{\left(\nu\frac{\partial\mathbf{u}}{\partial\mathbf{n}}-p\mathbf{n}\right)\cdot\mathbf{v}} = \int_{\Omega}{\mathbf{f}\cdot\mathbf{v}} }$

## Algorithms

Start with a guess $\mathbf{u}^k$ and iteratively solve the linearized Navier-Stokes at $(\mathbf{u}^k, p^k)$. $\left\{ \begin{array}{rccl} -\nu \Delta \mathbf{u}^{k+1} + (\mathbf{u}^k \cdot\nabla)\mathbf{u}^{k+1}+(\mathbf{u}^{k+1} \cdot\nabla)\mathbf{u}^{k} +\nabla p^{k+1} = (\mathbf{u}^k\cdot\nabla)\mathbf{u}^k\\ \nabla\cdot\mathbf{u}^{k+1}=0 \end{array} \right.$

## FreeFEM code

Result - velocity (top) and pressure (bottom)

TODO

## References

This example is inspired from the official FreeFEM documentation, “Newton Method for the Steady Navier-Stokes equations”.

Florian Omnès