1,350 research outputs found
Near-optimal perfectly matched layers for indefinite Helmholtz problems
A new construction of an absorbing boundary condition for indefinite
Helmholtz problems on unbounded domains is presented. This construction is
based on a near-best uniform rational interpolant of the inverse square root
function on the union of a negative and positive real interval, designed with
the help of a classical result by Zolotarev. Using Krein's interpretation of a
Stieltjes continued fraction, this interpolant can be converted into a
three-term finite difference discretization of a perfectly matched layer (PML)
which converges exponentially fast in the number of grid points. The
convergence rate is asymptotically optimal for both propagative and evanescent
wave modes. Several numerical experiments and illustrations are included.Comment: Accepted for publication in SIAM Review. To appear 201
Scalar problems in junctions of rods and a plate. II. Self-adjoint extensions and simulation models
In this work we deal with a scalar spectral mixed boundary value problem in a
spacial junction of thin rods and a plate. Constructing asymptotics of the
eigenvalues, we employ two equipollent asymptotic models posed on the skeleton
of the junction, that is, a hybrid domain. We, first, use the technique of
self-adjoint extensions and, second, we impose algebraic conditions at the
junction points in order to compile a problem in a function space with detached
asymptotics. The latter problem is involved into a symmetric generalized Green
formula and, therefore, admits the variational formulation. In comparison with
a primordial asymptotic procedure, these two models provide much better
proximity of the spectra of the problems in the spacial junction and in its
skeleton. However, they exhibit the negative spectrum of finite multiplicity
and for these "parasitic" eigenvalues we derive asymptotic formulas to
demonstrate that they do not belong to the service area of the developed
asymptotic models.Comment: 31 pages, 2 figur
Sharp high-frequency estimates for the Helmholtz equation and applications to boundary integral equations
We consider three problems for the Helmholtz equation in interior and
exterior domains in R^d (d=2,3): the exterior Dirichlet-to-Neumann and
Neumann-to-Dirichlet problems for outgoing solutions, and the interior
impedance problem. We derive sharp estimates for solutions to these problems
that, in combination, give bounds on the inverses of the combined-field
boundary integral operators for exterior Helmholtz problems.Comment: Version 3: 42 pages; improved exposition in response to referee
comments and added several reference
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