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Abstract

We investigate the performance of various discrete Hodge star operators for discrete exterior calculus (DEC) using circumcentric and barycentric dual meshes. The performance is evaluated through the DEC solution of Darcy and incompressible Navier–Stokes flows over surfaces. While the circumcentric Hodge operators may be favorable due to their diagonal structure, the barycentric (geometric) and the Galerkin Hodge operators have the advantage of admitting arbitrary simplicial meshes. Numerical experiments reveal that the barycentric and the Galerkin Hodge operators retain the numerical convergence order attained through the circumcentric (diagonal) Hodge operators. Furthermore, when the barycentric or the Galerkin Hodge operators are employed, a super-convergence behavior is observed for the incompressible flow solution over unstructured simplicial surface meshes generated by successive subdivision of coarser meshes. Insofar as the computational cost is concerned, the Darcy flow solutions exhibit a moderate increase in the solution time when using the barycentric or the Galerkin Hodge operators due to a modest decrease in the linear system sparsity. On the other hand, for the incompressible flow simulations, both the solution time and the linear system sparsity do not change for either the circumcentric or the barycentric and the Galerkin Hodge operators.

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.