A Consistent Discrete 3D Hodge-type Decomposition: implementation and practical evaluation

TL;DR

This paper introduces a new discretization method for 3D Hodge decomposition that maintains mathematical properties and is computationally efficient for large volumetric meshes, with practical implementation insights.

Contribution

It presents a consistent discretization approach combining edge-based Nedelec and face-based Crouzeix-Raviart elements for 3D Hodge decomposition on volumetric meshes.

Findings

Method is stable under noisy data and varying mesh resolutions.

Achieves good performance on large models.

Provides pseudocode and implementation insights.

Abstract

The Hodge decomposition provides a very powerful mathematical method for the analysis of 2D and 3D vector fields. It states roughly that any vector field can be $L^2$-orthogonally decomposed into a curl-free, divergence-free, and a harmonic field. The harmonic field itself can be further decomposed into three components, two of which are closely tied to the topology of the underlying domain. For practical computations it is desirable to find a discretization which preserves as many aspects inherent to the smooth theory as possible while at the same time remains computationally tractable, in particular on large-sized models. The correctness and convergence of such a discretization depends strongly on the choice of ansatz spaces defined on the surface or volumetric mesh to approximate infinite dimensional subspaces. This paper presents a consistent discretization of Hodge-type decomposition for piecewise constant vector fields on volumetric meshes. Our approach is based on a careful interplay between edge-based \textNedelec elements and face-based Crouzeix-Raviart elements resulting in a very simple formulation. The method is stable under noisy vector field and mesh resolution, and has a good performance for large sized models. We give pseudocodes for a possible implementation of the method together with some insights on how the Hodge decomposition could answer some central question in computational fluid.