[petsc-dev] finite element energy functions

[Geoffrey Irving]

For weak form PDEs which arise as the gradient of an integrated energy
function, it would be great to have a routine to perform the
appropriate integral.  It would be best to structure this as a generic
"perform an integral" function, evaluating

    E(f) = sum_i w_i f(u(x_i), grad u(x_i))

for a set of FE fields u (plus auxiliary fields) and some quadrature
rule.  The quadrature rule might need to be passed in independently of
the PetscFE objects, since the integral might tie together several
different fields connected to several different PetscFE's, conceivably
with distinct quadrature rules.  The integral is only valid as a
discrete energy if all the quadrature rules match, but there might be
other integral application where this doesn't hold.

Relatedly, how does one ensure that a PetscFE object is Galerkin
(matching primal and dual spaces) so that the energy to be discretely
valid?  Concretely, what needs to be done in a setup like ex12, where
the spaces are configurable via options?

Thanks,
Geoffrey