24 research outputs found
On the linearization of Regge calculus
We study the linearization of three dimensional Regge calculus around
Euclidean metric. We provide an explicit formula for the corresponding
quadratic form and relate it to the curlTcurl operator which appears in the
quadratic part of the Einstein-Hilbert action and also in the linear elasticity
complex. We insert Regge metrics in a discrete version of this complex,
equipped with densely defined and commuting interpolators. We show that the
eigenpairs of the curlTcurl operator, approximated using the quadratic part of
the Regge action on Regge metrics, converge to their continuous counterparts,
interpreting the computation as a non-conforming finite element method.Comment: 26 page
A simplicial gauge theory
We provide an action for gauge theories discretized on simplicial meshes,
inspired by finite element methods. The action is discretely gauge invariant
and we give a proof of consistency. A discrete Noether's theorem that can be
applied to our setting, is also proved.Comment: 24 pages. v2: New version includes a longer introduction and a
discrete Noether's theorem. v3: Section 4 on Noether's theorem has been
expanded with Proposition 8, section 2 has been expanded with a paragraph on
standard LGT. v4: Thorough revision with new introduction and more background
materia
Complexes of Discrete Distributional Differential Forms and their Homology Theory
Complexes of discrete distributional differential forms are introduced into
finite element exterior calculus. Thus we generalize a notion of Braess and
Sch\"oberl, originally studied for a posteriori error estimation. We construct
isomorphisms between the simplicial homology groups of the triangulation, the
discrete harmonic forms of the finite element complex, and the harmonic forms
of the distributional finite element complexes. As an application, we prove
that the complexes of finite element exterior calculus have cohomology groups
isomorphic to the de Rham cohomology, including the case of partial boundary
conditions. Poincar\'e-Friedrichs-type inequalities will be studied in a
subsequent contribution.Comment: revised preprint, 26 page
Nonlinear elasticity complex and a finite element diagram chase
In this paper, we present a nonlinear version of the linear elasticity
(Calabi, Kr\"oner, Riemannian deformation) complex which encodes isometric
embedding, metric, curvature and the Bianchi identity. We reformulate the
rigidity theorem and a fundamental theorem of Riemannian geometry as the
exactness of this complex. Then we generalize an algebraic approach for
constructing finite elements for the Bernstein-Gelfand-Gelfand (BGG) complexes.
In particular, we discuss the reduction of degrees of freedom with injective
connecting maps in the BGG diagrams. We derive a strain complex in two space
dimensions with a diagram chase.Comment: Manuscript prepared for proceedings of the INdAM conference
"Approximation Theory and Numerical Analysis meet Algebra, Geometry,
Topology'', which was held in September 2022 at Cortona, Ital
Extended Regge complex for linearized Riemann-Cartan geometry and cohomology
We show that the cohomology of the Regge complex in three dimensions is
isomorphic to , the
infinitesimal-rigid-body-motion-valued de~Rham cohomology. Based on an
observation that the twisted de~Rham complex extends the elasticity (Riemannian
deformation) complex to the linearized version of coframes, connection 1-forms,
curvature and Cartan's torsion, we construct a discrete version of linearized
Riemann-Cartan geometry on any triangulation and determine its cohomology.Comment: 24 page