Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Framework for Analysis of DEC Approximations to Hodge-Laplacian Problems using Generalized Whitney Forms

Johnny Guzmán, Pratyush Potu

TL;DR

This work develops a unifying FEEC-based framework for analyzing Discrete Exterior Calculus (DEC) approximations to Hodge-Laplacian problems across general differential $k$-forms. By identifying cochains on primal and dual meshes with Whitney and generalized Whitney forms, the authors recast DEC in a form-language setting, enabling rigorous a priori error analysis and optimal convergence rates on well-centered, contractible domains. The paper extends known $0$-form results to general $k$, derives commuting projections and a diamond-cell based kernel analysis to bound the $\Pi-J$ discrepancy, and provides a detailed treatment of superconvergence under symmetry, supported by comprehensive numerical experiments. The framework thus offers a robust, dimension-agnostic toolkit for structure-preserving discretizations of exterior calculus problems with practical implications for accuracy and stability in computational geometry and physics simulations.

Abstract

We provide a framework for interpreting Discrete Exterior Calculus (DEC) numerical schemes in terms of Finite Element Exterior Calculus (FEEC). We demonstrate the equivalence of cochains on primal and dual meshes with Whitney and generalized Whitney forms which allows us to analyze DEC approximations using tools from FEEC. We demonstrate the applicability of our framework by rigorously proving convergence with rates for the Hodge-Laplacian problem in full $k$-form generality on well-centered meshes on contractible domains. We also provide numerical results illustrating optimality of our derived convergence rates. Moreover, we demonstrate how superconvergence phenomena can be explained in our framework with corresponding numerical results.

A Framework for Analysis of DEC Approximations to Hodge-Laplacian Problems using Generalized Whitney Forms

TL;DR

This work develops a unifying FEEC-based framework for analyzing Discrete Exterior Calculus (DEC) approximations to Hodge-Laplacian problems across general differential -forms. By identifying cochains on primal and dual meshes with Whitney and generalized Whitney forms, the authors recast DEC in a form-language setting, enabling rigorous a priori error analysis and optimal convergence rates on well-centered, contractible domains. The paper extends known -form results to general , derives commuting projections and a diamond-cell based kernel analysis to bound the discrepancy, and provides a detailed treatment of superconvergence under symmetry, supported by comprehensive numerical experiments. The framework thus offers a robust, dimension-agnostic toolkit for structure-preserving discretizations of exterior calculus problems with practical implications for accuracy and stability in computational geometry and physics simulations.

Abstract

We provide a framework for interpreting Discrete Exterior Calculus (DEC) numerical schemes in terms of Finite Element Exterior Calculus (FEEC). We demonstrate the equivalence of cochains on primal and dual meshes with Whitney and generalized Whitney forms which allows us to analyze DEC approximations using tools from FEEC. We demonstrate the applicability of our framework by rigorously proving convergence with rates for the Hodge-Laplacian problem in full -form generality on well-centered meshes on contractible domains. We also provide numerical results illustrating optimality of our derived convergence rates. Moreover, we demonstrate how superconvergence phenomena can be explained in our framework with corresponding numerical results.
Paper Structure (21 sections, 24 theorems, 170 equations, 10 figures)

This paper contains 21 sections, 24 theorems, 170 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Continuous Setting
  3. Discrete Setting and DEC
  4. Primal Mesh
  5. Dual Mesh
  6. DEC and the Numerical Scheme
  7. Whitney Form Setting
  8. Whitney forms on the Primal Mesh
  9. Generalized Whitney Forms on the Dual Mesh
  10. DEC and the Numerical Scheme on Forms
  11. Equivalence Between Operators on Cochains and Forms
  12. Discrete Framework on Forms
  13. Error Estimates
  14. Interpolant that commutes with the discrete codifferential
  15. The difference $\Pi-J$
  16. ...and 6 more sections

Key Result

Proposition 4.1

There exists a sequence of spaces $W_h^k \subset H \Lambda^k(\Omega)$ which forms the discrete complex: Moreover, each $\omega_h \in W_h^k$ is uniquely defined by the following dofs The dual bases of these dofs are the generalized Whitney forms $\{\phi_{\tau}\}_{\tau\in {{\Delta}}^{\hbox{$\pentagon$}}_k}$.

Figures (10)

  • Figure 1: Examples of our definition of polyhedral cells with explicit simplicial decomposition.
  • Figure 2: 2d example of the oriented dual mesh following Desbrun05Hirani03.
  • Figure 3: 2d illustration of a primal mesh, dual mesh, and diamond cell mesh for $k=1$. The primal edges are highlighted in blue. The dual edges are highlighted in red. Each diamond cell contains exactly one primal edge and its dual.
  • Figure 4: The two meshes we use for the numerical experiments.
  • Figure 5: Convergence of the DEC scheme for $k=0$ on the symmetric mesh.
  • ...and 5 more figures

Theorems & Definitions (56)

  • Definition 3.1: Polyhedral cells
  • Definition 3.2: Polyhedral Cell Complex
  • Definition 3.3: Orthogonal Dual Mesh
  • Definition 3.4: Dual Set
  • Definition 3.5: Discrete Exterior Derivative on Cochains
  • Definition 3.6: Discrete Hodge Star on Cochains
  • Definition 3.7: Discrete Codifferential on Cochains
  • Definition 3.8: Discrete Hodge-Laplacian on Cochains
  • Proposition 4.1
  • Definition 4.2: Discrete Hodge Star on Forms
  • ...and 46 more