22,240 research outputs found
A Simple Method for Computing Singular or Nearly Singular Integrals on Closed Surfaces
We present a simple, accurate method for computing singular or nearly
singular integrals on a smooth, closed surface, such as layer potentials for
harmonic functions evaluated at points on or near the surface. The integral is
computed with a regularized kernel and corrections are added for regularization
and discretization, which are found from analysis near the singular point. The
surface integrals are computed from a new quadrature rule using surface points
which project onto grid points in coordinate planes. The method does not
require coordinate charts on the surface or special treatment of the
singularity other than the corrections. The accuracy is about , where
is the spacing in the background grid, uniformly with respect to the point
of evaluation, on or near the surface. Improved accuracy is obtained for points
on the surface. The treecode of Duan and Krasny for Ewald summation is used to
perform sums. Numerical examples are presented with a variety of surfaces.Comment: to appear in Commun. Comput. Phy
Hadamard Regularization
Motivated by the problem of the dynamics of point-particles in high
post-Newtonian (e.g. 3PN) approximations of general relativity, we consider a
certain class of functions which are smooth except at some isolated points
around which they admit a power-like singular expansion. We review the concepts
of (i) Hadamard ``partie finie'' of such functions at the location of singular
points, (ii) the partie finie of their divergent integral. We present and
investigate different expressions, useful in applications, for the latter
partie finie. To each singular function, we associate a partie-finie (Pf)
pseudo-function. The multiplication of pseudo-functions is defined by the
ordinary (pointwise) product. We construct a delta-pseudo-function on the class
of singular functions, which reduces to the usual notion of Dirac distribution
when applied on smooth functions with compact support. We introduce and analyse
a new derivative operator acting on pseudo-functions, and generalizing, in this
context, the Schwartz distributional derivative. This operator is uniquely
defined up to an arbitrary numerical constant. Time derivatives and partial
derivatives with respect to the singular points are also investigated. In the
course of the paper, all the formulas needed in the application to the physical
problem are derived.Comment: 50 pages, to appear in Journal of Mathematical Physic
An Extension of the Faddeev-Jackiw Technique to Fields in Curved Spacetimes
The Legendre transformation on singular Lagrangians, e.g. Lagrangians
representing gauge theories, fails due to the presence of constraints. The
Faddeev-Jackiw technique, which offers an alternative to that of Dirac, is a
symplectic approach to calculating a Hamiltonian paired with a well-defined
initial value problem when working with a singular Lagrangian. This phase space
coordinate reduction was generalized by Barcelos-Neto and Wotzasek to simplify
its application. We present an extension of the Faddeev-Jackiw technique for
constraint reduction in gauge field theories and non-gauge field theories that
are coupled to a curved spacetime that is described by General Relativity. A
major difference from previous formulations is that we do not explicitly
construct the symplectic matrix, as that is not necessary. We find that the
technique is a useful tool that avoids some of the subtle complications of the
Dirac approach to constraints. We apply this formulation to the Ginzburg-Landau
action and provide a calculation of its Hamiltonian and Poisson brackets in a
curved spacetime.Comment: 30 pages, updated to reflect published versio
On a Poincare lemma for foliations
In this paper we revisit a Poincare lemma for foliated forms, with respect to
a regular foliation, and compute the foliated cohomology for local models of
integrable systems with singularities of nondegenerate type. A key point in
this computation is the use of some analytical tools for integrable systems
with nondegenerate singularities, including a Poincare lemma for the
deformation complex associated to this singular foliation.Comment: 23 pages, an error in proposition 5.1 in the former version was
detected and the paper has been rewritten accordingl
The Fourier Singular Complement Method for the Poisson problem. Part I: prismatic domains
This is the first part of a threefold article, aimed at solving numerically
the Poisson problem in three-dimensional prismatic or axisymmetric domains. In
this first part, the Fourier Singular Complement Method is introduced and
analysed, in prismatic domains. In the second part, the FSCM is studied in
axisymmetric domains with conical vertices, whereas, in the third part,
implementation issues, numerical tests and comparisons with other methods are
carried out. The method is based on a Fourier expansion in the direction
parallel to the reentrant edges of the domain, and on an improved variant of
the Singular Complement Method in the 2D section perpendicular to those edges.
Neither refinements near the reentrant edges of the domain nor cut-off
functions are required in the computations to achieve an optimal convergence
order in terms of the mesh size and the number of Fourier modes used
ZZ-type aposteriori error estimators for adaptive boundary element methods on a curve
In the context of the adaptive finite element method (FEM), ZZ-error
estimators named after Zienkiewicz and Zhu are mathematically well-established
and widely used in practice. In this work, we propose and analyze ZZ-type error
estimators for the adaptive boundary element method (BEM). We consider
weakly-singular and hyper-singular integral equations and prove, in particular,
convergence of the related adaptive mesh-refining algorithms
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