A new Hodge operator in Discrete Exterior Calculus. Application to fluid mechanics

TL;DR

This paper presents a novel, flexible discrete Hodge operator in Discrete Exterior Calculus that works on arbitrary interior points, improving mesh adaptability for fluid mechanics and thermal transfer simulations.

Contribution

It introduces a general discrete Hodge operator that overcomes mesh limitations and is exact on constant forms without relying on Whitney forms.

Findings

Enables dual meshes based on any interior point.

Demonstrates convergence on various mesh types.

Applicable to fluid mechanics and thermal transfer problems.

Abstract

This article introduces a new and general construction of discrete Hodge operator in the context of Discrete Exterior Calculus (DEC). This discrete Hodge operator enables to circumvent the well-centeredness limitation on the mesh with the popular diagonal Hodge. It allows a dual mesh based on any interior point, such as the incenter or the barycenter. It opens the way towards mesh-optimized discrete Hodge operators. In the particular case of a well-centered triangulation, it reduces to the diagonal Hodge if the dual mesh is circumcentric. Based on an analytical development, this discrete Hodge does not make use of Whitney forms, and is exact on piecewise constant forms, whichever interior point is chosen for the construction of the dual mesh. Numerical tests oriented to the resolution of fluid mechanics problems and thermal transfer are carried out. Convergence on various types of mesh is investigated.