There was a previous question as to how one might perform upwind
stabilization in libMesh. One such approach, as demonstrated in the
attached patch to ex9, is the streamline-upwind Petrov-Galerkin (SUPG)
method. In this approach the test function is biased in the upwind
direction, thereby "upwinding" the weak statement in some sense.
This was not previously included as an example because it did not seem
germane to learning how to use the library.
SUPG gets more complicated for systems, but the idea is the same. For more
details check out my dissertation and references therein.
http://www.cfdlab.ae.utexas.edu/~benkirk/dissertation.pdf
-Ben
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