5. Conclusions
The paper presented a comparison between the circumcentric (diagonal) Hodge operator versus the Galerkin and the barycentric (geometric) Hodge operators on surface simplicial meshes. While the circumcentric Hodge operator works properly only on Delaunay meshes, the Galerkin and the barycentric Hodge operators admit arbitrary simplicial meshes. This provides more flexibility to the meshing process and facilitates any subsequent mesh subdivision, which would require edge flips or remeshing in case only Delaunay simplicial meshes were allowed. The Galerkin Hodge operator definition is based on the integration of Whitney forms over the triangles. The barycentric Hodge operator is defined also through Whitney forms, and can be considered to correspond to a one-point quadrature version of the Galerkin Hodge operator with quadrature points at the triangles barycenters. Both definitions resulted in a sparse, nondiagonal, definition for the Hodge star operator ∗1. Such a sparse representation complicates the development of a sparse barycentric or Galerkin definitions for the inverse operator ∗ −1 1 . The barycentric and the Galerkin definitions for ∗1 are, however, advantageous for many physical problems that do not require ∗ −1 1 ; e.g. the Poisson and incompressible Navier–Stokes equations. In regard to the three-dimensional case, although the existence of barycentric and Galerkin definitions for the ∗1 and ∗2 operators is known, it is not known yet how a sparse representation for ∗ −1 1 and ∗ −1 2 can be defined. The operator ∗ −1 1 is essential for the 3D DEC discretization of incompressible Navier–Stokes flows.
