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Convergence and Stability of Discrete Exterior Calculus for the Hodge Laplace Problem in Two Dimensions

Chengbin Zhu, Snorre H. Christiansen, Kaibo Hu, Anil N. Hirani

TL;DR

This work proves convergence and stability of discrete exterior calculus (DEC) for the two-dimensional Hodge-Laplace problem on meshes that are non-degenerate Delaunay and shape-regular. By relating DEC to the lowest-order FEEC discretization, it derives a discrete Poincaré inequality and a discrete inf-sup condition, and establishes norm equivalence between DEC and FEEC norms under various geometric mesh conditions. The analysis first handles uniformly acute triangulations and then extends to uniformly Delaunay, boundary-acute, and curvature-bounded meshes, showing that only one direction of norm equivalence suffices for stability and convergence. The results yield concrete error estimates between DEC and FEEC solutions, guaranteeing convergence of DEC to the FEEC solution and to the true Hodge-Laplace solution as the mesh is refined, with constants depending only on mesh regularity parameters. This advances the theoretical foundation of DEC in 2D by connecting it rigorously to FEEC and providing robust stability guarantees for a broad class of meshes.

Abstract

We prove convergence and stability of the discrete exterior calculus (DEC) solutions for the Hodge-Laplace problems in two dimensions for families of meshes that are non-degenerate Delaunay and shape regular. We do this by relating the DEC solutions to the lowest order finite element exterior calculus (FEEC) solutions. A Poincaré inequality and a discrete inf-sup condition for DEC are part of this proof. We also prove that under appropriate geometric conditions on the mesh the DEC and FEEC norms are equivalent. Only one side of the norm equivalence is needed for proving stability and convergence and this allows us to relax the conditions on the meshes.

Convergence and Stability of Discrete Exterior Calculus for the Hodge Laplace Problem in Two Dimensions

TL;DR

This work proves convergence and stability of discrete exterior calculus (DEC) for the two-dimensional Hodge-Laplace problem on meshes that are non-degenerate Delaunay and shape-regular. By relating DEC to the lowest-order FEEC discretization, it derives a discrete Poincaré inequality and a discrete inf-sup condition, and establishes norm equivalence between DEC and FEEC norms under various geometric mesh conditions. The analysis first handles uniformly acute triangulations and then extends to uniformly Delaunay, boundary-acute, and curvature-bounded meshes, showing that only one direction of norm equivalence suffices for stability and convergence. The results yield concrete error estimates between DEC and FEEC solutions, guaranteeing convergence of DEC to the FEEC solution and to the true Hodge-Laplace solution as the mesh is refined, with constants depending only on mesh regularity parameters. This advances the theoretical foundation of DEC in 2D by connecting it rigorously to FEEC and providing robust stability guarantees for a broad class of meshes.

Abstract

We prove convergence and stability of the discrete exterior calculus (DEC) solutions for the Hodge-Laplace problems in two dimensions for families of meshes that are non-degenerate Delaunay and shape regular. We do this by relating the DEC solutions to the lowest order finite element exterior calculus (FEEC) solutions. A Poincaré inequality and a discrete inf-sup condition for DEC are part of this proof. We also prove that under appropriate geometric conditions on the mesh the DEC and FEEC norms are equivalent. Only one side of the norm equivalence is needed for proving stability and convergence and this allows us to relax the conditions on the meshes.
Paper Structure (18 sections, 24 theorems, 96 equations, 3 figures)

This paper contains 18 sections, 24 theorems, 96 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Summary of results
  3. Background and setup
  4. Triangulations and DEC spaces
  5. Hodge-Laplace problem
  6. DEC and FEEC inner products and norms
  7. DEC results implied by finite element results
  8. Previous work
  9. Norm equivalence for uniformly acute triangulations
  10. Norm of Whitney 1-forms for a single triangle
  11. Norm of Whitney 0-forms for a single triangle
  12. From single triangle to triangulation
  13. Norm equivalence for uniformly Delaunay triangulations
  14. Norm of Whitney 1-forms
  15. Norm of Whitney 0-forms and 2-forms
  16. ...and 3 more sections

Key Result

Lemma 3.2

Let $T$ be a triangle and $\omega_i \in \mathcal{P}_1^-\Lambda^{k}(T)$ the basis Whitney form corresponding to edge $i$. Then there exist constants $c_1(\delta_T), c_2(\delta_T)$ such that

Figures (3)

  • Figure 1: For an acute triangle the dual area of a vertex is bounded above by triangle area and below by half of that. Here $O$ is the circumcenter of $v_0v_1v_2$. The dual of vertex $v_0$ is the quadrilateral $v_0M_1OM_2$. Its area is bounded below by the area of triangle $v_0M_1M_2$ and above by the half the area of $v_0v_1v_2$.
  • Figure 2: In a Delaunay pair of triangles the sum of angles $C_1+C_2$ opposite to a shared edge $c$ should satisfy $C_1 + C_2\le \pi$. DEC requires nondegenerate Delaunay condition $C_1 + C_2 < \pi$ and norm equivalence requires uniform Delaunay condition $C_1 + C_2 \le \pi - \delta_\pi$ for some fixed $0 < \delta_\pi < \pi$ for all such angle pairs.
  • Figure 3: Two examples of meshes (A) $v_0$ internal; and (B) $v_0$ on the boundary.

Theorems & Definitions (55)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 3.1
  • Lemma 3.2: Norm inequalities for basis forms
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 45 more