54,929 research outputs found
The localization effect for eigenfunctions of the mixed boundary value problem in a thin cylinder with distorted ends
A simple sufficient condition on curved end of a straight cylinder is found
that provides a localization of the principal eigenfunction of the mixed
boundary value for the Laplace operator with the Dirichlet conditions on the
lateral side. Namely, the eigenfunction concentrates in the vicinity of the
ends and decays exponentially in the interior. Similar effects are observed in
the Dirichlet and Neumann problems, too.Comment: 25 pages, 10 figure
A fresh look at midpoint singularities in the algebra of string fields
In this paper we study the midpoint structure of the algebra of open strings
from the standpoint of the operator/Moyal formalism. We construct a split
string description for the continuous Moyal product of hep-th/0202087, study
the breakdown of associativity in the star algebra, and identify in infinite
sequence of new (anti)commutative coordinates for the star product in in the
complex plane. We also explain how poles in the open string
non(anti)commutativity parameter correspond to certain ``null'' operators which
annihilate the vertex, implying that states proportional to such operators tend
to have vanishing star product with other string fields. The existence of such
poles, we argue, presents an obstruction to realizing a well-defined
formulation of the theory in terms of a Moyal product. We also comment on the
interesting, but singular, representation which has appeared prominently
in the recent studies of Bars {\it et al}.Comment: 40 pages, 5 figures. Version to be submitted to JHEP. Some
interesting and previouusly unpublished results are included here. These
include both an interpretation of poles in the open string noncommutativity
parameter as corresponding to null operators in the algebra, and an
identification of an infinite sequence of new commutative and null
coordinates in the complex plan
On the Singular Neumann Problem in Linear Elasticity
The Neumann problem of linear elasticity is singular with a kernel formed by
the rigid motions of the body. There are several tricks that are commonly used
to obtain a non-singular linear system. However, they often cause reduced
accuracy or lead to poor convergence of the iterative solvers. In this paper,
different well-posed formulations of the problem are studied through
discretization by the finite element method, and preconditioning strategies
based on operator preconditioning are discussed. For each formulation we derive
preconditioners that are independent of the discretization parameter.
Preconditioners that are robust with respect to the first Lam\'e constant are
constructed for the pure displacement formulations, while a preconditioner that
is robust in both Lam\'e constants is constructed for the mixed formulation. It
is shown that, for convergence in the first Sobolev norm, it is crucial to
respect the orthogonality constraint derived from the continuous problem. Based
on this observation a modification to the conjugate gradient method is proposed
that achieves optimal error convergence of the computed solution
A Finite Element Method With Singularity Reconstruction for Fractional Boundary Value Problems
We consider a two-point boundary value problem involving a Riemann-Liouville
fractional derivative of order \al\in (1,2) in the leading term on the unit
interval . Generally the standard Galerkin finite element method can
only give a low-order convergence even if the source term is very smooth due to
the presence of the singularity term x^{\al-1} in the solution
representation. In order to enhance the convergence, we develop a simple
singularity reconstruction strategy by splitting the solution into a singular
part and a regular part, where the former captures explicitly the singularity.
We derive a new variational formulation for the regular part, and establish
that the Galerkin approximation of the regular part can achieve a better
convergence order in the , H^{\al/2}(0,1) and -norms
than the standard Galerkin approach, with a convergence rate for the recovered
singularity strength identical with the error estimate. The
reconstruction approach is very flexible in handling explicit singularity, and
it is further extended to the case of a Neumann type boundary condition on the
left end point, which involves a strong singularity x^{\al-2}. Extensive
numerical results confirm the theoretical study and efficiency of the proposed
approach.Comment: 23 pp. ESAIM: Math. Model. Numer. Anal., to appea
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