Comparison of discrete Hodge star operators for surfaces. (September 2016)

• Record Type: Journal Article
• Authors: Mohamed, Mamdouh S.; Hirani, Anil N.; Samtaney, Ravi
• Is Part Of: Computer aided design. Volume 78(2016)
• Page Start: 118
• Page End: 125
• Publication Date: 2016-09
• Subjects: Discrete exterior calculus (DEC) -- Hodge star -- Circumcentric dual -- Barycentric dual; Computer-aided design -- Periodicals; Engineering design -- Data processing -- Periodicals; Computer graphics -- Periodicals; Conception technique -- Informatique -- Périodiques; Infographie -- Périodiques; Computer graphics; Engineering design -- Data processing; Periodicals; Electronic journals; 620.00420285
• DOI: 10.1016/j.cad.2016.05.002
• Languages: English
• ISSNs: 0010-4485

Abstract

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. Highlights: Various definitions for discrete Hodge star operators are investigated. Darcy and incompressible Navier–Stokes flows are used for numerical experiments. Barycentric Hodge star reproduces the convergence rate of the circumcentric Hodge. The difference in the computational cost is generally minor.