[petsc-dev] finite element energy functions

[Geoffrey Irving]

On Mon, Dec 2, 2013 at 3:06 PM, Jed Brown <jedbrown at mcs.anl.gov> wrote:
> 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.

Yes, a routine to evaluate the energy only makes sense in contexts
where the residuals are integrable/conservative.  Obviously the user
has to ensure this, possibly with the help of consistency checking
routines (which I would write anyways if they aren't immediately
available).

>> 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.

Of course.  In this case it would be useful as a consistency check but
not as an objective to minimize.

Geoffrey