petsc-master 2020-10-26
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# PetscDualSpaceTransformType

The type of function transform Notes: These transforms, and their inverses, are used to move functions and functionals between the reference element and real space. Suppose that we have a mapping $\phi$ which maps the reference cell to real space, and its Jacobian $J$. If we want to transform function $F$ on the reference element, so that it acts on real space, we use the pushforward transform $\sigma^*$. The pullback $\sigma_*$ is the inverse transform.

Covariant Piola: $\sigma^*(F) = J^{-T} F \circ \phi^{-1)$
Contravariant Piola: $\sigma^*(F) = 1/|J| J F \circ \phi^{-1)$

Note: For details, please see Rognes, Kirby, and Logg, Efficient Assembly of Hdiv and Hrot Conforming Finite Elements, SISC, 31(6), 4130-4151, arXiv 1205.3085, 2010