[petsc-dev] finite element energy functions

[Jed Brown]

Geoffrey Irving <irving at naml.us> writes:
> I'm not sure what you mean by the weak form being symmetric, unless
> you mean that the primal and dual spaces are the same, which is just a
> restatement of my question (and therefore I'm not sure in the context
> of petsc).  

If my weak form is advection (\int v \cdot \nabla u) then I don't have a
useful energy.  There are lots of things that could cause a stated
"energy" to be nonsense; matching basis functions is just one of them,
and perhaps not even the most common.

> The actual functional in the weak form PDE isn't symmetric except in
> special cases such as Poisson, but presumably that's not what you
> mean.  Indefiniteness is somewhat inevitable (except near well behaved
> extreme solutions), and where it occurs depends on physics.  

That sort of indefiniteness doesn't make the energy meaningless, but add
a Lagrange multiplier and the Lagrangian is no longer an energy.
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