The abstract Hodge-Dirac operator and its stable discretization

Abstract

This paper adapts the techniques of finite element exterior calculus to study and discretize the abstract Hodge-Dirac operator, which is a square root of the abstract Hodge-Laplace operator considered by Arnold, Falk, and Winther [Bull. Amer. Math. Soc. 47 (2010), 281-354]. Dirac-type operators are central to the field of Clifford analysis, where recently there has been considerable interest in their discretization. We prove a priori stability and convergence estimates, and show that several of the results in finite element exterior calculus can be recovered as corollaries of these new estimates.

21 pages, 1 figure; v2: minor revisions

Cited by in corpus (5)

• Finite element approximation of the Levi-Civita connection and its curvature in two dimensions
• An arbitrary order and pointwise divergence-free finite element scheme for the incompressible 3D Navier-Stokes equations
• First-Kind Boundary Integral Equations for the Dirac Operator in 3D Lipschitz Domains
• Galerkin Formulation of Path Integrals in Lattice Field Theory
• 2D discrete Hodge-Dirac operator on the torus