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A simple and complete discrete exterior calculus on general polygonal meshes

Lenka Ptackova, Luiz Velho

TL;DR

This paper advances discrete exterior calculus on general polygonal meshes by introducing a polygonal wedge product and a primal-to-primal Hodge star, enabling a full set of DEC-like operators without dual meshes. The approach defines $d$, the wedge product, inner products, contraction $i_X$, Lie derivative $\mathsf{L}_X$, codifferential $\delta$, and Laplacian $\Delta$ directly on primal cells, and provides rigorous convergence analyses showing at least linear convergence for tested forms. It also demonstrates practical applicability through implicit mean curvature flow, Helmholtz–Hodge decomposition, and Lie advection on complex polygonal meshes, with numerical comparisons indicating improved accuracy and robustness relative to prior polygonal DEC methods. Overall, the work broadens DEC to non-simplicial geometries, preserving key algebraic structures while offering new tools for geometry processing and vector-field analysis on polygonal surfaces.$d$,$\wedge$,$\star$ and $\Delta$ are central to the framework, with the Leibniz rule $d(\alpha^k\wedge\beta^l) = d\alpha\wedge\beta^l + (-1)^k \alpha^k\wedge d\beta^l$ preserved in the discrete setting where possible, and $\mathsf{L}_X = i_X d + d i_X$ used for advection tasks across applications.$

Abstract

Discrete exterior calculus (DEC) offers a coordinate-free discretization of exterior calculus especially suited for computations on curved spaces. In this work, we present an extended version of DEC on surface meshes formed by general polygons that bypasses the need for combinatorial subdivision and does not involve any dual mesh. At its core, our approach introduces a new polygonal wedge product that is compatible with the discrete exterior derivative in the sense that it satisfies the Leibniz product rule. Based on the discrete wedge product, we then derive a novel primal-to-primal Hodge star operator. Combining these three `basic operators' we then define new discrete versions of the contraction operator and Lie derivative, codifferential and Laplace operator. We discuss the numerical convergence of each one of these proposed operators and compare them to existing DEC methods. Finally, we show simple applications of our operators on Helmholtz-Hodge decomposition, Laplacian surface fairing, and Lie advection of functions and vector fields on meshes formed by general polygons.

A simple and complete discrete exterior calculus on general polygonal meshes

TL;DR

This paper advances discrete exterior calculus on general polygonal meshes by introducing a polygonal wedge product and a primal-to-primal Hodge star, enabling a full set of DEC-like operators without dual meshes. The approach defines , the wedge product, inner products, contraction , Lie derivative , codifferential , and Laplacian directly on primal cells, and provides rigorous convergence analyses showing at least linear convergence for tested forms. It also demonstrates practical applicability through implicit mean curvature flow, Helmholtz–Hodge decomposition, and Lie advection on complex polygonal meshes, with numerical comparisons indicating improved accuracy and robustness relative to prior polygonal DEC methods. Overall, the work broadens DEC to non-simplicial geometries, preserving key algebraic structures while offering new tools for geometry processing and vector-field analysis on polygonal surfaces.,, and are central to the framework, with the Leibniz rule preserved in the discrete setting where possible, and used for advection tasks across applications.$

Abstract

Discrete exterior calculus (DEC) offers a coordinate-free discretization of exterior calculus especially suited for computations on curved spaces. In this work, we present an extended version of DEC on surface meshes formed by general polygons that bypasses the need for combinatorial subdivision and does not involve any dual mesh. At its core, our approach introduces a new polygonal wedge product that is compatible with the discrete exterior derivative in the sense that it satisfies the Leibniz product rule. Based on the discrete wedge product, we then derive a novel primal-to-primal Hodge star operator. Combining these three `basic operators' we then define new discrete versions of the contraction operator and Lie derivative, codifferential and Laplace operator. We discuss the numerical convergence of each one of these proposed operators and compare them to existing DEC methods. Finally, we show simple applications of our operators on Helmholtz-Hodge decomposition, Laplacian surface fairing, and Lie advection of functions and vector fields on meshes formed by general polygons.
Paper Structure (28 sections, 1 theorem, 38 equations, 20 figures)

This paper contains 28 sections, 1 theorem, 38 equations, 20 figures.

Table of Contents

  1. Introduction
  2. Related Work and Preliminaries
  3. Discrete differential forms and the exterior derivative
  4. The cup product and the wedge product
  5. The Hodge star operator
  6. The Hodge inner product
  7. The codifferential
  8. The Laplace operator
  9. The contraction operator and the Lie derivative
  10. Primal-to-Primal Operators
  11. The Discrete Wedge Product
  12. Numerical evaluation
  13. 1
  14. 2
  15. 3
  16. ...and 13 more sections

Key Result

Lemma 3.1

Let $M$ be a Riemannian $n$--manifold, $X\in \mathfrak{X}(M)$ a vector field, then for the contraction of a differential $k$--form $\alpha$ with a vector field $X$ holds: where $\flat: \mathfrak{X}(M)\rightarrow\Omega(M)$ is the flat operator.

Figures (20)

  • Figure 1: Comparison of implicit mean curvature flows on a general polygonal mesh (29k vertices) after 10 iterations with time step $t = 10^{-4}$. On the far left is the original mesh. Our method (center left) and the algorithm of Alexa2011 with their combinatorially enhanced Laplacian (center right) produce visually well--smoothed meshes. However, their method with purely geometric Laplacian (far right) exhibits some undesirable artifacts on the ears, neck, and tail of the kitten.
  • Figure 2: The wedge product of two 1--forms on a triangle: the product of two 1--forms is a 2--form located on faces (far left).
  • Figure 3: Comparison of the support of the codifferential of 1--forms between the classical DEC (L) and our method (R). The codifferential of a 1--form $\alpha$ is a 0--form located on vertices. The value of $\delta\alpha$ on the red vertex $v$ is a linear combination of values of $\alpha$ on edges colored green. The edge thickness reflects the weight of the corresponding edge values $\alpha$ on $\delta\alpha(v)$.
  • Figure 4: Comparison of the support of the codifferential of 2--forms between the classical DEC (L) and our approach (R). The codifferential of a 2--form $\beta$ is a 1--form $\delta\beta$ located on edges. The value of $\delta\beta$ on the red edge $e$ is a linear combination of the values of $\beta$ on faces colored green. The color intensity of faces reflects their weight of influence on $\delta\beta(e)$.
  • Figure 5: Comparison of support of the Laplacian of 0--forms between the classical DEC (far left) and ours on triangle meshes (center left), Laplacian of Alexa2011 for their $\lambda=0$ (center right) and ours on polygonal meshes (far right). The Laplacian of a 0--form $\alpha$ is a 0--form $\Delta\alpha$ located again on vertices. The value of $\Delta\alpha$ on the red vertex $v$ is a linear combination of values of $\alpha$ on vertices colored green. The support of our Laplacian is always larger, the point size reflects the weight of respective $\alpha$s on $\Delta\alpha(v)$. We also color yellow the faces whose vertices carry the $\alpha$s that enter as variables for $\Delta\alpha(v)$.
  • ...and 15 more figures

Theorems & Definitions (2)

  • Definition 3.1
  • Lemma 3.1