235 research outputs found
Spaces of finite element differential forms
We discuss the construction of finite element spaces of differential forms
which satisfy the crucial assumptions of the finite element exterior calculus,
namely that they can be assembled into subcomplexes of the de Rham complex
which admit commuting projections. We present two families of spaces in the
case of simplicial meshes, and two other families in the case of cubical
meshes. We make use of the exterior calculus and the Koszul complex to define
and understand the spaces. These tools allow us to treat a wide variety of
situations, which are often treated separately, in a unified fashion.Comment: To appear in: Analysis and Numerics of Partial Differential
Equations, U. Gianazza, F. Brezzi, P. Colli Franzone, and G. Gilardi, eds.,
Springer 2013. v2: a few minor typos corrected. v3: a few more typo
correction
Computer Algebra meets Finite Elements: an Efficient Implementation for Maxwell's Equations
We consider the numerical discretization of the time-domain Maxwell's
equations with an energy-conserving discontinuous Galerkin finite element
formulation. This particular formulation allows for higher order approximations
of the electric and magnetic field. Special emphasis is placed on an efficient
implementation which is achieved by taking advantage of recurrence properties
and the tensor-product structure of the chosen shape functions. These
recurrences have been derived symbolically with computer algebra methods
reminiscent of the holonomic systems approach.Comment: 16 pages, 1 figure, 1 table; Springer Wien, ISBN 978-3-7091-0793-
A Gauge field Induced by the Global Gauge Invariance of Action Integral
As a general rule, it is considered that the global gauge invariance of an
action integral does not cause the occurrence of gauge field. However, in this
paper we demonstrate that when the so-called localized assumption is excluded,
the gauge field will be induced by the global gauge invariance of the action
integral. An example is given to support this conclusion.Comment: 13 pages. Some typing errors are corrected and the format is update
Semilinear mixed problems on Hilbert complexes and their numerical approximation
Arnold, Falk, and Winther recently showed [Bull. Amer. Math. Soc. 47 (2010),
281-354] that linear, mixed variational problems, and their numerical
approximation by mixed finite element methods, can be studied using the
powerful, abstract language of Hilbert complexes. In another recent article
[arXiv:1005.4455], we extended the Arnold-Falk-Winther framework by analyzing
variational crimes (a la Strang) on Hilbert complexes. In particular, this gave
a treatment of finite element exterior calculus on manifolds, generalizing
techniques from surface finite element methods and recovering earlier a priori
estimates for the Laplace-Beltrami operator on 2- and 3-surfaces, due to Dziuk
[Lecture Notes in Math., vol. 1357 (1988), 142-155] and later Demlow [SIAM J.
Numer. Anal., 47 (2009), 805-827], as special cases. In the present article, we
extend the Hilbert complex framework in a second distinct direction: to the
study of semilinear mixed problems. We do this, first, by introducing an
operator-theoretic reformulation of the linear mixed problem, so that the
semilinear problem can be expressed as an abstract Hammerstein equation. This
allows us to obtain, for semilinear problems, a priori solution estimates and
error estimates that reduce to the Arnold-Falk-Winther results in the linear
case. We also consider the impact of variational crimes, extending the results
of our previous article to these semilinear problems. As an immediate
application, this new framework allows for mixed finite element methods to be
applied to semilinear problems on surfaces.Comment: 22 pages; v2: major revision, particularly sharpening of error
estimates in Section
On Bogovski\u{\i} and regularized Poincar\'e integral operators for de Rham complexes on Lipschitz domains
We study integral operators related to a regularized version of the classical
Poincar\'e path integral and the adjoint class generalizing Bogovski\u{\i}'s
integral operator, acting on differential forms in . We prove that these
operators are pseudodifferential operators of order -1. The Poincar\'e-type
operators map polynomials to polynomials and can have applications in finite
element analysis. For a domain starlike with respect to a ball, the special
support properties of the operators imply regularity for the de Rham complex
without boundary conditions (using Poincar\'e-type operators) and with full
Dirichlet boundary conditions (using Bogovski\u{\i}-type operators). For
bounded Lipschitz domains, the same regularity results hold, and in addition we
show that the cohomology spaces can always be represented by
functions.Comment: 23 page
Time evolution in linear response: Boltzmann equations and beyond
In this work a perturbative linear response analysis is performed for the
time evolution of the quasi-conserved charge of a scalar field. One can find
two regimes, one follows exponential damping, where the damping rate is shown
to come from quantum Boltzmann equations. The other regime (coming from
multiparticle cuts and products of them) decays as power law. The most
important, non-oscillating contribution in our model comes from a 4-particle
intermediate state and decays as 1/t^3. These results may have relevance for
instance in the context of lepton number violation in the Early Universe.Comment: 19 page
One-Loop Calculations for a Translation Invariant Non-Commutative Gauge Model
In this paper we discuss one-loop results for the translation invariant
non-commutative gauge field model we recently introduced in arXiv:0804.1914.
This model relies on the addition of some carefully chosen extra terms in the
action which mix long and short scales in order to circumvent the infamous
UV/IR mixing, and were motivated by the renormalizable non-commutative scalar
model of Gurau et al. (cf. arXiv:0802.0791).Comment: 18 pages, v2: minor correction
Meshfree finite differences for vector Poisson and pressure Poisson equations with electric boundary conditions
We demonstrate how meshfree finite difference methods can be applied to solve
vector Poisson problems with electric boundary conditions. In these, the
tangential velocity and the incompressibility of the vector field are
prescribed at the boundary. Even on irregular domains with only convex corners,
canonical nodal-based finite elements may converge to the wrong solution due to
a version of the Babuska paradox. In turn, straightforward meshfree finite
differences converge to the true solution, and even high-order accuracy can be
achieved in a simple fashion. The methodology is then extended to a specific
pressure Poisson equation reformulation of the Navier-Stokes equations that
possesses the same type of boundary conditions. The resulting numerical
approach is second order accurate and allows for a simple switching between an
explicit and implicit treatment of the viscosity terms.Comment: 19 pages, 7 figure
Discrete exterior calculus (DEC) for the surface Navier-Stokes equation
We consider a numerical approach for the incompressible surface Navier-Stokes
equation. The approach is based on the covariant form and uses discrete
exterior calculus (DEC) in space and a semi-implicit discretization in time.
The discretization is described in detail and related to finite difference
schemes on staggered grids in flat space for which we demonstrate second order
convergence. We compare computational results with a vorticity-stream function
approach for surfaces with genus 0 and demonstrate the interplay between
topology, geometry and flow properties. Our discretization also allows to
handle harmonic vector fields, which we demonstrate on a torus.Comment: 21 pages, 9 figure
Interaction of N solitons in the massive Thirring model and optical gap system: the Complex Toda Chain Model
Using the Karpman-Solov''ev quasiparticle approach for soliton-soliton
interaction I show that the train propagation of N well separated solitons of
the massive Thirring model is described by the complex Toda chain with N nodes.
For the optical gap system a generalised (non-integrable) complex Toda chain is
derived for description of the train propagation of well separated gap
solitons. These results are in favor of the recently proposed conjecture of
universality of the complex Toda chain.Comment: RevTex, 23 pages, no figures. Submitted to Physical Review
- …