1,078 research outputs found
Weak Form of Stokes-Dirac Structures and Geometric Discretization of Port-Hamiltonian Systems
We present the mixed Galerkin discretization of distributed parameter
port-Hamiltonian systems. On the prototypical example of hyperbolic systems of
two conservation laws in arbitrary spatial dimension, we derive the main
contributions: (i) A weak formulation of the underlying geometric
(Stokes-Dirac) structure with a segmented boundary according to the causality
of the boundary ports. (ii) The geometric approximation of the Stokes-Dirac
structure by a finite-dimensional Dirac structure is realized using a mixed
Galerkin approach and power-preserving linear maps, which define minimal
discrete power variables. (iii) With a consistent approximation of the
Hamiltonian, we obtain finite-dimensional port-Hamiltonian state space models.
By the degrees of freedom in the power-preserving maps, the resulting family of
structure-preserving schemes allows for trade-offs between centered
approximations and upwinding. We illustrate the method on the example of
Whitney finite elements on a 2D simplicial triangulation and compare the
eigenvalue approximation in 1D with a related approach.Comment: Copyright 2018. This manuscript version is made available under the
CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0
Short-time asymptotics of the regularizing effect for semigroups generated by quadratic operators
We study accretive quadratic operators with zero singular spaces. These
degenerate non-selfadjoint differential operators are known to be hypoelliptic
and to generate contraction semigroups which are smoothing in the Schwartz
space for any positive time. In this work, we study the short-time asymptotics
of the regularizing effect induced by these semigroups. We show that these
short-time asymptotics of the regularizing effect depend on the directions of
the phase space, and that this dependence can be nicely understood through the
structure of the singular space. As a byproduct of these results, we derive
sharp subelliptic estimates for accretive quadratic operators with zero
singular spaces pointing out that the loss of derivatives with respect to the
elliptic case also depends on the phase space directions according to the
structure of the singular space. Some applications of these results are then
given to the study of degenerate hypoelliptic Ornstein-Uhlenbeck operators and
degenerate hypoelliptic Fokker-Planck operators.Comment: 46 pages. arXiv admin note: text overlap with arXiv:1411.622
Local Finite Element Approximation of Sobolev Differential Forms
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 Cl\'ement 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.Comment: 22 pages. Comments welcom
Symplectic Techniques for Semiclassical Completely Integrable Systems
This article is a survey of classical and quantum completely integrable
systems from the viewpoint of local ``phase space'' analysis. It advocates the
use of normal forms and shows how to get global information from glueing local
pieces. Many crucial phenomena such as monodromy or eigenvalue concentration
are shown to arise from the presence of non-degenerate critical points.Comment: 32 pages, 7 figures. Review articl
p-Adic Mathematical Physics
A brief review of some selected topics in p-adic mathematical physics is
presented.Comment: 36 page
Isometric Immersions and the Waving of Flags
In this article we propose a novel geometric model to study the motion of a
physical flag. In our approach a flag is viewed as an isometric immersion from
the square with values into satisfying certain boundary
conditions at the flag pole. Under additional regularity constraints we show
that the space of all such flags carries the structure of an infinite
dimensional manifold and can be viewed as a submanifold of the space of all
immersions. The submanifold result is then used to derive the equations of
motion, after equipping the space of isometric immersions with its natural
kinetic energy. This approach can be viewed in a similar spirit as Arnold's
geometric picture for the motion of an incompressible fluid.Comment: 25 pages, 1 figur
- …