Stabilisation methods

When using a continuous Galerkin discretisation in advection-dominated problems, it may be necessary to stabilise the advection term in the momentum equation.

The implementation of the stabilisation methods can be found in the file

Streamline upwind

This method adds some upwind diffusion in the direction of the streamlines. The term is given by

\[\int_{\Omega} \frac{\bar{k}}{||\mathbf{u}||^2}(\mathbf{u}\cdot\nabla\mathbf{w})(\mathbf{u}\cdot\nabla\mathbf{u})\]

which is added to the LHS of the momentum equation. The term \(\bar{k}\) takes the form

\[\bar{k} = \frac{1}{2}\left(\frac{1}{\tanh(\mathrm{Pe})} - \frac{1}{\mathrm{Pe}}\right)||\mathbf{u}||\Delta x\]


\[\mathrm{Pe} = \frac{||\mathbf{u}||\Delta x}{2\nu}\]

is the Peclet number, and \(\Delta x\) is the size of each element.