The Hodge Star Operator and the Dual Mesh¶
On an -dimensional Riemanian manifold the continuous Hodge Star operator establishes an isomorphism between -forms and -forms.
It satisfies the following identity
where is the identity map between -forms (it maps the form to itself)
The hodge star operators is its own inverse up to a sign. Let us consider only the positive case for now (an analogous argument will hold for the negative case).
We want to study under what conditions the discrete Hodge star operator satisfies the discrete counterpart of the above equation
The discrete Hodge star operator acting on primal forms is given by:
The discrete Hodge star acting on dual forms, on the other hand, is given by
Let us assume that Eq. (1) holds. Then in terms of the basis functions it becomes
The basis functions by their own definition must satisfy
Additionally, we desire the dual basis functions to be linear combinations of the hodge-star of the primal ones, therefore
where the matrix represents the transform between the two basis sets. The hodge star is required in order for the dual basis functions to correspond to the correct simplex dimensions. For example if we take the basis functions for 0-forms in 2d corresponding to vertices, the dual basis functions must correspond to the corresponding elements on the dual mesh wich are faces, and therefore must be 2-forms.
The following sequence of equations are equivalent
and for Eq. (2)
In matrix form
and since is of full rank, this implies that
We have shown that Eq. (1) (the Hodge star and its dual are exact inverses) and Eq. (2) (dual basis functions are linear combinations of primal ones) are completely consistent with each other (can it be shown that one implies the other?).
Therefore, the primal basis functions and the dual simplices completely determine the dual basis functions, and they are implicitly given by