The Abstract Hodge-Dirac Operator and Its Stable Discretization

  title={The Abstract Hodge-Dirac Operator and Its Stable Discretization},
  author={Paul C. Leopardi and Ari Stern},
  journal={SIAM J. Numer. Anal.},
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), pp. 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… 

2D discrete Hodge-Dirac operator on the torus

    V. Sushch
    Mathematics, Computer Science
  • 2022
The goal of this work is to develop a satisfactory discrete model of the de Rham–Hodge theory on manifolds that are homeomorphic to the torus that is compatible with the Hodge decomposition theorem.

Numerical solution of the div-curl problem by finite element exterior calculus

The goal of the paper is to take advantage of the links between usual vector calculus and exterior calculus and show the interest of the exterior calculus framework, without too much prior knowledge of the subject.

First-Kind Boundary Integral Equations for the Dirac Operator in 3-Dimensional Lipschitz Domains

We develop novel first-kind boundary integral equations for Euclidean Dirac operators in 3D Lipschitz domains. They comprise square-integrable potentials and involve only weakly singular kernels.

Finite element approximation of the Levi-Civita connection and its curvature in two dimensions

Finite element approximations of the Levi-Civita connection and its curvature on triangulations of oriented two-dimensional manifolds are constructed and it is shown that these distributional quantities converge in certain dual Sobolev norms to their smooth counterparts under refinement of the triangulation.

An arbitrary order and pointwise divergence-free finite element scheme for the incompressible 3D Navier-Stokes equations

    M. Hanot
    Mathematics, Computer Science
    SIAM Journal on Numerical Analysis
  • 2023
This paper makes use of the Lamb identity to rewrite the Navier-Stokes equations into a vorticity-velocity-pressure form which fits into the de Rham complex of minimal regularity and proposes a discretization on a large class of finite elements, including arbitrary order polynomial spaces readily available in many libraries.

On eigenmode approximation for Dirac equations: Differential forms and fractional Sobolev spaces

Eigenmode convergence is proved, as well as optimal convergence orders, assuming a flat background metric on a periodic domain, in finite element spaces of differential forms.

A unified discrete framework for intrinsic and extrinsic Dirac operators for geometry processing

A unified discretization scheme is introduced that describes both an extrinsic and intrinsic Dirac operator on meshes, based on their continuous counterparts on smooth manifolds, and preserves their key properties from the smooth case.

Modern trends in hypercomplex analysis

In this paper we work in the `split' discrete Cli fford analysis setting, i.e. the m-dimensional function theory concerning null-functions of the discrete Dirac operator d, defi ned on the grid Zm,

A Dirac Operator for Extrinsic Shape Analysis

A new extrinsic differential operator called the relative Dirac operator is introduced, leading to a family of operators with a continuous trade‐off between intrinsic and extrinsics features, and this family spans the entire spectrum.

Rough metrics on manifolds and quadratic estimates

We study the persistence of quadratic estimates related to the Kato square root problem across a change of metric on smooth manifolds by defining a class of “rough” Riemannian-like metrics that are

Finite element exterior calculus: From hodge theory to numerical stability

This article reports on the confluence of two streams of research, one emanating from the fields of numerical analysis and scientific computation, the other from topology and geometry. In it we

Smoothed projections in finite element exterior calculus

The construction of smoothed projections is generalized, such that also non quasi-uniform meshes and essential boundary conditions are covered and the new tool introduced here is a space dependent smoothing operator which commutes with the exterior derivative.

Geometric Variational Crimes: Hilbert Complexes, Finite Element Exterior Calculus, and Problems on Hypersurfaces

This framework extends the work of Arnold, Falk, and Winther to problems that violate their subcomplex assumption, allowing for the extension of finite element exterior calculus to approximate domains, most notably the Hodge–de Rham complex on approximate manifolds.

Quadratic estimates and functional calculi of perturbed Dirac operators

We prove quadratic estimates for complex perturbations of Dirac-type operators, and thereby show that such operators have a bounded functional calculus. As an application we show that spectral

Finite element exterior calculus, homological techniques, and applications

Finite element exterior calculus is an approach to the design and understanding of finite element discretizations for a wide variety of systems of partial differential equations. This approach brings

Principles of Mimetic Discretizations of Differential Operators

This work provides a common framework for mimetic discretizations using algebraic topology to guide the analysis and demonstrates how to apply the framework for compatible discretization for two scalar versions of the Hodge Laplacian.

Local bounded cochain projections

These projections have the properties that they commute with the exterior derivative and are bounded in the HΛk(Ω) norm independent of the mesh size h, and are locally defined in the sense that they are defined by local operators on overlapping macroelements, in the spirit of the Clement interpolant.

Discrete Clifford analysis

This survey is intended as an overview of discrete Clifford analysis and its current developments. Since in the discrete case one has to replace the partial derivative with two difference operators,

Discrete exterior calculus

This thesis presents the beginnings of a theory of discrete exterior calculus (DEC). Our approach is to develop DEC using only discrete combinatorial and geometric operations on a simplicial complex

Hodge Decompositions on Weakly Lipschitz Domains

A theory of nilpotent operators in Hilbert space is derived from the L 2 theory of boundary value problems for exterior and interior derivative operators d and δ on a bounded, weakly Lipschitz domain.