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

2025/05/13 by Zhu, Chengbin, Snorre H. Christiansen, Christiansen, Snorre H. +4 · 1 citation
Mathematics · #65N30 #Algebraic and Geometric Analysis #Differential Equations and Boundary Problems #FOS: Mathematics #Numerical Analysis (math.NA) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2505.08966

openalex publication_date 2025/05/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• A Framework for Analysis of DEC Approximations to Hodge-Laplacian Problems using Generalized Whitney Forms
• A finite-element framework for a mimetic finite-difference\n discretization of Maxwell's equations
• Convergence of Discrete Exterior Calculus Approximations for Poisson Problems
• Numerical Calculation of Time-Dependent Viscous Incompressible Flow of Fluid with Free Surface

Cited by

• Convergence of Discrete Exterior Calculus for the Hodge-Dirac Operator

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• A Framework for Analysis of DEC Approximations to Hodge-Laplacian Problems using Generalized Whitney Forms
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• Local coderivatives and approximation of Hodge Laplace problems
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