Differential forms and exterior calculus

2018 ◽  
Author(s):  
Eduardo Nahmad-Achar
Keyword(s):  
Differential Forms ◽  
Exterior Calculus

scholarly journals A PROPOSAL FOR A DIFFERENTIAL CALCULUS IN QUANTUM MECHANICS

1994 ◽  
Vol 09 (13) ◽  
pp. 2191-2227 ◽  
Author(s):  
E. GOZZI ◽  
M. REUTER
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Hamiltonian System ◽  
Differential Calculus ◽  
Cotangent Bundle ◽  
Differential Forms ◽  
Lie Derivative ◽  
Exterior Derivative ◽  
Tensor Calculus ◽  
Exterior Calculus

In this paper, using, the Weyl-Wigner-Moyal formalism for quantum mechanics, we develop a quantum-deformed exterior calculus on the phase space of an arbitrary Hamiltonian system. Introducing additional bosonic and fermionic coordinates, we construct a supermanifold which is closely related to the tangent and cotangent bundle over phase space. Scalar functions on the supermanifold become equivalent to differential forms on the standard phase space. The algebra of these functions is equipped with a Moyal superstar product which deforms the pointwise product of the classical tensor calculus. We use the Moyal bracket algebra to derive a set of quantum-deformed rules for the exterior derivative, Lie derivative, contraction, and similar operations of the Cartan calculus.

scholarly journals Finite element exterior calculus, homological techniques, and applications

Acta Numerica ◽  
2006 ◽  
Vol 15 ◽  
pp. 1-155 ◽  
Author(s):  
Douglas N. Arnold ◽  
Richard S. Falk ◽  
Ragnar Winther
Keyword(s):  
Finite Element ◽  
Algebraic Topology ◽  
Eigenvalue Problems ◽  
Differential Forms ◽  
Element Discretization ◽  
Hodge Laplacian ◽  
Finite Element Exterior Calculus ◽  
Exterior Calculus ◽  
Elliptic Eigenvalue Problems ◽  
Elliptic Complex

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 to bear tools from differential geometry, algebraic topology, and homological algebra to develop discretizations which are compatible with the geometric, topological, and algebraic structures which underlie well-posedness of the PDE problem being solved. In the finite element exterior calculus, many finite element spaces are revealed as spaces of piecewise polynomial differential forms. These connect to each other in discrete subcomplexes of elliptic differential complexes, and are also related to the continuous elliptic complex through projections which commute with the complex differential. Applications are made to the finite element discretization of a variety of problems, including the Hodge Laplacian, Maxwell’s equations, the equations of elasticity, and elliptic eigenvalue problems, and also to preconditioners.

scholarly journals GAUGE FIELDS ON RIEMANN-CARTAN SPACE-TIMES

1994 ◽  
Vol 09 (11) ◽  
pp. 971-981 ◽  
Author(s):  
ALBERTO SAA
Keyword(s):  
Invariant Theory ◽  
Differential Forms ◽  
Space Time ◽  
Gauge Fields ◽  
Riemannian Structure ◽  
Minimal Coupling ◽  
Gauge Invariant ◽  
Exterior Calculus ◽  
Coupling Procedure ◽  
Action Formulation

Gauge fields are described on a Riemann-Cartan space-time by means of tensor-valued differential forms and exterior calculus. It is shown that minimal coupling procedure leads to a gauge invariant theory where gauge fields interact with torsion, and that consistency conditions for the gauge fields impose restrictions in the non-Riemannian structure of space-time. The new results differ from the well established ones obtained by using minimal coupling procedure at the action formulation. The sources of these differences are pointed out and discussed.

scholarly journals Construction of Scalar and Vector Finite Element Families on Polygonal and Polyhedral Meshes

2016 ◽  
Vol 16 (4) ◽  
pp. 667-683 ◽  
Author(s):  
Andrew Gillette ◽  
Alexander Rand ◽  
Chandrajit Bajaj
Keyword(s):  
Finite Element ◽  
Differential Forms ◽  
Convex Polytope ◽  
Basis Functions ◽  
Barycentric Coordinates ◽  
Convex Polygons ◽  
Finite Element Exterior Calculus ◽  
Exterior Calculus ◽  
Vector Finite Element ◽  
Generalized Barycentric Coordinates

AbstractWe combine theoretical results from polytope domain meshing, generalized barycentric coordinates, and finite element exterior calculus to construct scalar- and vector-valued basis functions for conforming finite element methods on generic convex polytope meshes in dimensions 2 and 3. Our construction recovers well-known bases for the lowest order Nédélec, Raviart–Thomas, and Brezzi–Douglas–Marini elements on simplicial meshes and generalizes the notion of Whitney forms to non-simplicial convex polygons and polyhedra. We show that our basis functions lie in the correct function space with regards to global continuity and that they reproduce the requisite polynomial differential forms described by finite element exterior calculus. We present a method to count the number of basis functions required to ensure these two key properties.

scholarly journals Local finite element approximation of Sobolev differential forms 

2021 ◽  
Author(s):  
Evan Gawlik ◽  
Michael Holst ◽  
Martin Werner Licht
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Differential Forms ◽  
Mesh Size ◽  
Finite Element Approximation ◽  
Local Error ◽  
Element Approximation ◽  
Exterior Calculus ◽  
Vector Valued ◽  
Theoretical Results

We address fundamental aspects in the approximation theory of vector-valued finite element methods, using finite element exterior calculus as a unifying framework. We generalize the Clément interpolant and the Scott-Zhang interpolant to finite element differential forms, and we derive a broken Bramble-Hilbert lemma. Our interpolants require only minimal smoothness assumptions and respect partial boundary conditions. This permits us to state local error estimates in terms of the mesh size. Our theoretical results apply to curl-conforming and divergence-conforming finite element methods over simplicial triangulations.

scholarly journals Differential Forms in Lattice Field Theories: An Overview

2013 ◽  
Vol 2013 ◽  
pp. 1-16 ◽  
Author(s):  
F. L. Teixeira
Keyword(s):  
Ab Initio ◽  
Differential Forms ◽  
Field Theories ◽  
Exterior Calculus ◽  
Lattice Field ◽  
Random Lattices

We provide an overview on the application of the exterior calculus of differential forms to the ab initio formulation of lattice field theories, with a focus on irregular or “random” lattices.

Sign in / Sign up

Export Citation Format

Share Document