4,110 research outputs found
On the negative spectrum of the Robin Laplacian in corner domains
For a bounded corner domain , we consider the Robin Laplacian in
with large Robin parameter. Exploiting multiscale analysis and a
recursive procedure, we have a precise description of the mechanism giving the
ground state of the spectrum. It allows also the study of the bottom of the
essential spectrum on the associated tangent structures given by cones. Then we
obtain the asymptotic behavior of the principal eigenvalue for this singular
limit in any dimension, with remainder estimates. The same method works for the
Schr\"odinger operator in with a strong attractive
delta-interaction supported on . Applications to some Erhling's
type estimates and the analysis of the critical temperature of some
superconductors are also provided
Accurate computation of surface stresses and forces with immersed boundary methods
Many immersed boundary methods solve for surface stresses that impose the
velocity boundary conditions on an immersed body. These surface stresses may
contain spurious oscillations that make them ill-suited for representing the
physical surface stresses on the body. Moreover, these inaccurate stresses
often lead to unphysical oscillations in the history of integrated surface
forces such as the coefficient of lift. While the errors in the surface
stresses and forces do not necessarily affect the convergence of the velocity
field, it is desirable, especially in fluid-structure interaction problems, to
obtain smooth and convergent stress distributions on the surface. To this end,
we show that the equation for the surface stresses is an integral equation of
the first kind whose ill-posedness is the source of spurious oscillations in
the stresses. We also demonstrate that for sufficiently smooth delta functions,
the oscillations may be filtered out to obtain physically accurate surface
stresses. The filtering is applied as a post-processing procedure, so that the
convergence of the velocity field is unaffected. We demonstrate the efficacy of
the method by computing stresses and forces that converge to the physical
stresses and forces for several test problems
hp-version time domain boundary elements for the wave equation on quasi-uniform meshes
Solutions to the wave equation in the exterior of a polyhedral domain or a
screen in exhibit singular behavior from the edges and corners.
We present quasi-optimal -explicit estimates for the approximation of the
Dirichlet and Neumann traces of these solutions for uniform time steps and
(globally) quasi-uniform meshes on the boundary. The results are applied to an
-version of the time domain boundary element method. Numerical examples
confirm the theoretical results for the Dirichlet problem both for screens and
polyhedral domains.Comment: 41 pages, 11 figure
A moving boundary problem motivated by electric breakdown: I. Spectrum of linear perturbations
An interfacial approximation of the streamer stage in the evolution of sparks
and lightning can be written as a Laplacian growth model regularized by a
`kinetic undercooling' boundary condition. We study the linear stability of
uniformly translating circles that solve the problem in two dimensions. In a
space of smooth perturbations of the circular shape, the stability operator is
found to have a pure point spectrum. Except for the zero eigenvalue for
infinitesimal translations, all eigenvalues are shown to have negative real
part. Therefore perturbations decay exponentially in time. We calculate the
spectrum through a combination of asymptotic and series evaluation. In the
limit of vanishing regularization parameter, all eigenvalues are found to
approach zero in a singular fashion, and this asymptotic behavior is worked out
in detail. A consideration of the eigenfunctions indicates that a strong
intermediate growth may occur for generic initial perturbations. Both the
linear and the nonlinear initial value problem are considered in a second
paper.Comment: 37 pages, 6 figures, revised for Physica
Direct and Inverse Computational Methods for Electromagnetic Scattering in Biological Diagnostics
Scattering theory has had a major roll in twentieth century mathematical
physics. Mathematical modeling and algorithms of direct,- and inverse
electromagnetic scattering formulation due to biological tissues are
investigated. The algorithms are used for a model based illustration technique
within the microwave range. A number of methods is given to solve the inverse
electromagnetic scattering problem in which the nonlinear and ill-posed nature
of the problem are acknowledged.Comment: 61 pages, 5 figure
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