11,700 research outputs found
Heat Equation on the Cone and the Spectrum of the Spherical Laplacian
Spectrum of the Laplacian on spherical domains is analyzed from the point of
view of the heat equation on the cone. The series solution to the heat equation
on the cone is known to lead to a study of the Laplacian eigenvalue problem on
domains on the sphere in higher dimensions. It is found that the solution leads
naturally to a spectral function, a `generating function' for the eigenvalues
and multiplicities of the Laplacian, expressible in closed form for certain
domains on the sphere. Analytical properties of the spectral function suggest a
simple scaling procedure for estimating the eigenvalues. Comparison of the
first eigenvalue estimate with the available theoretical and numerical results
for some specific domains shows remarkable agreement.Comment: 16 page
Shape identification in inverse medium scattering problems with a single far-field pattern
Consider time-harmonic acoustic scattering from a bounded penetrable obstacle
embedded in a homogeneous background medium. The index
of refraction characterizing the material inside is supposed to be H\"older
continuous near the corners. If is a convex polygon, we
prove that its shape and location can be uniquely determined by the far-field
pattern incited by a single incident wave at a fixed frequency. In dimensions
, the uniqueness applies to penetrable scatterers of rectangular type
with additional assumptions on the smoothness of the contrast. Our arguments
are motivated by recent studies on the absence of non-scattering wavenumbers in
domains with corners. As a byproduct, we show that the smoothness conditions in
previous corner scattering results are only required near the corners
A multidomain spectral method for solving elliptic equations
We present a new solver for coupled nonlinear elliptic partial differential
equations (PDEs). The solver is based on pseudo-spectral collocation with
domain decomposition and can handle one- to three-dimensional problems. It has
three distinct features. First, the combined problem of solving the PDE,
satisfying the boundary conditions, and matching between different subdomains
is cast into one set of equations readily accessible to standard linear and
nonlinear solvers. Second, touching as well as overlapping subdomains are
supported; both rectangular blocks with Chebyshev basis functions as well as
spherical shells with an expansion in spherical harmonics are implemented.
Third, the code is very flexible: The domain decomposition as well as the
distribution of collocation points in each domain can be chosen at run time,
and the solver is easily adaptable to new PDEs. The code has been used to solve
the equations of the initial value problem of general relativity and should be
useful in many other problems. We compare the new method to finite difference
codes and find it superior in both runtime and accuracy, at least for the
smooth problems considered here.Comment: 31 pages, 8 figure
A domain-decomposition method to implement electrostatic free boundary conditions in the radial direction for electric discharges
At high pressure electric discharges typically grow as thin, elongated
filaments. In a numerical simulation this large aspect ratio should ideally
translate into a narrow, cylindrical computational domain that envelops the
discharge as closely as possible. However, the development of the discharge is
driven by electrostatic interactions and, if the computational domain is not
wide enough, the boundary conditions imposed to the electrostatic potential on
the external boundary have a strong effect on the discharge. Most numerical
codes for electric discharges circumvent this problem by either using a wide
computational domain or by calculating the boundary conditions by integrating
the Green's function of an infinite domain. Here we describe an accurate and
efficient method to impose free boundary conditions for an elongated electric
discharge. To facilitate the use of our method we provide a sample
implementation.Comment: 21 pages, 4 figures, a movie and a sample code in python. A new
Appendix has been adde
Balanced Metric and Berezin Quantization on the Siegel-Jacobi Ball
We determine the matrix of the balanced metric of the Siegel-Jacobi ball and
its inverse. We calculate the scalar curvature, the Ricci form and the
Laplace-Beltrami operator of this manifold. We discuss several geometric
aspects related with Berezin quantization on the Siegel-Jacobi ball
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