Apparently there are many techniques but this is an area of active research.
I'm still trying to learn FEM and work through some of this stuff.
I ran across this paper which suggests that when solutions
oscillate you can pick out real solutions by changing the number of points,

http://am2012.math.cas.cz/proceedings/contributions/madden.pdf

Interested if anyone cares to comment on current status and libmesh facilities
that may be relevant or if this is just a "lmgtfy" case. I gathered the normal
approaches were largely things that modify the apparent diffusivity
which does not seem to be a lot different from anything else you may add to a 
system.
Changing the mesh and comparing points would be a bit more involved I imagine. 

Thanks.


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