7,027 research outputs found
Monte Carlo approximations of the Neumann problem
We introduce Monte Carlo methods to compute the solution of elliptic
equations with pure Neumann boundary conditions. We first prove that the
solution obtained by the stochastic representation has a zero mean value with
respect to the invariant measure of the stochastic process associated to the
equation. Pointwise approximations are computed by means of standard and new
simulation schemes especially devised for local time approximation on the
boundary of the domain. Global approximations are computed thanks to a
stochastic spectral formulation taking into account the property of zero mean
value of the solution. This stochastic formulation is asymptotically perfect in
terms of conditioning. Numerical examples are given on the Laplace operator on
a square domain with both pure Neumann and mixed Dirichlet-Neumann boundary
conditions. A more general convection-diffusion equation is also numerically
studied
On transparent boundary conditions for the high--order heat equation
In this paper we develop an artificial initial boundary value problem for the
high-order heat equation in a bounded domain . It is found an unique
classical solution of this problem in an explicit form and shown that the
solution of the artificial initial boundary value problem is equal to the
solution of the infinite problem (Cauchy problem) in .Comment: 9 page
Boosting the Maxwell double layer potential using a right spin factor
We construct new spin singular integral equations for solving scattering
problems for Maxwell's equations, both against perfect conductors and in media
with piecewise constant permittivity, permeability and conductivity, improving
and extending earlier formulations by the author. These differ in a fundamental
way from classical integral equations, which use double layer potential
operators, and have the advantage of having a better condition number, in
particular in Fredholm sense and on Lipschitz regular interfaces, and do not
suffer from spurious resonances. The construction of the integral equations
builds on the observation that the double layer potential factorises into a
boundary value problem and an ansatz. We modify the ansatz, inspired by a
non-selfadjoint local elliptic boundary condition for Dirac equations
Stable radiation-controlling boundary conditions for the generalized harmonic Einstein equations
This paper is concerned with the initial-boundary value problem for the
Einstein equations in a first-order generalized harmonic formulation. We impose
boundary conditions that preserve the constraints and control the incoming
gravitational radiation by prescribing data for the incoming fields of the Weyl
tensor. High-frequency perturbations about any given spacetime (including a
shift vector with subluminal normal component) are analyzed using the
Fourier-Laplace technique. We show that the system is boundary-stable. In
addition, we develop a criterion that can be used to detect weak instabilities
with polynomial time dependence, and we show that our system does not suffer
from such instabilities. A numerical robust stability test supports our claim
that the initial-boundary value problem is most likely to be well-posed even if
nonzero initial and source data are included.Comment: 27 pages, 4 figures; more numerical results and references added,
several minor amendments; version accepted for publication in Class. Quantum
Gra
Analytic Regularity for Linear Elliptic Systems in Polygons and Polyhedra
We prove weighted anisotropic analytic estimates for solutions of second
order elliptic boundary value problems in polyhedra. The weighted analytic
classes which we use are the same as those introduced by Guo in 1993 in view of
establishing exponential convergence for hp finite element methods in
polyhedra. We first give a simple proof of the known weighted analytic
regularity in a polygon, relying on a new formulation of elliptic a priori
estimates in smooth domains with analytic control of derivatives. The technique
is based on dyadic partitions near the corners. This technique can successfully
be extended to polyhedra, providing isotropic analytic regularity. This is not
optimal, because it does not take advantage of the full regularity along the
edges. We combine it with a nested open set technique to obtain the desired
three-dimensional anisotropic analytic regularity result. Our proofs are global
and do not require the analysis of singular functions.Comment: 54 page
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