28 research outputs found
Fourier Method for Approximating Eigenvalues of Indefinite Stekloff Operator
We introduce an efficient method for computing the Stekloff eigenvalues
associated with the Helmholtz equation. In general, this eigenvalue problem
requires solving the Helmholtz equation with Dirichlet and/or Neumann boundary
condition repeatedly. We propose solving the related constant coefficient
Helmholtz equation with Fast Fourier Transform (FFT) based on carefully
designed extensions and restrictions of the equation. The proposed Fourier
method, combined with proper eigensolver, results in an efficient and clear
approach for computing the Stekloff eigenvalues.Comment: 12 pages, 4 figure
Domain Decomposition preconditioning for high-frequency Helmholtz problems with absorption
In this paper we give new results on domain decomposition preconditioners for
GMRES when computing piecewise-linear finite-element approximations of the
Helmholtz equation , with
absorption parameter . Multigrid approximations of
this equation with are commonly used as preconditioners
for the pure Helmholtz case (). However a rigorous theory for
such (so-called "shifted Laplace") preconditioners, either for the pure
Helmholtz equation, or even the absorptive equation (), is
still missing. We present a new theory for the absorptive equation that
provides rates of convergence for (left- or right-) preconditioned GMRES, via
estimates of the norm and field of values of the preconditioned matrix. This
theory uses a - and -explicit coercivity result for the
underlying sesquilinear form and shows, for example, that if , then classical overlapping additive Schwarz will perform optimally for
the absorptive problem, provided the subdomain and coarse mesh diameters are
carefully chosen. Extensive numerical experiments are given that support the
theoretical results. The theory for the absorptive case gives insight into how
its domain decomposition approximations perform as preconditioners for the pure
Helmholtz case . At the end of the paper we propose a
(scalable) multilevel preconditioner for the pure Helmholtz problem that has an
empirical computation time complexity of about for
solving finite element systems of size , where we have
chosen the mesh diameter to avoid the pollution effect.
Experiments on problems with , i.e. a fixed number of grid points
per wavelength, are also given
A short note on a fast and high-order hybridizable discontinuous Galerkin solver for the 2D high-frequency Helmholtz equation
The method of polarized traces provides the first documented algorithm with truly scalable complexity for the highfrequency Helmholtz equation, i.e., with a runtime sublinear in the number of volume unknowns in a parallel environment. However, previous versions of this method were either restricted to a low order of accuracy, or suffered from computationally unfavorable boundary reduction to Ï(p) interfaces in the p-th order case. In this note we rectify this issue by proposing a high-order method of polarized traces with compact reduction to two, rather than Ï(p), interfaces. This method is based on a primal Hybridizable Discontinuous Galerkin (HDG) discretization in a domain decomposition setting. In addition, HDG is a welcome upgrade for the method of polarized traces, since it can be made to work with flexible meshes that align with discontinuous coefficients, and it allows for adaptive refinement in h and p. High order of accuracy is very important for attenuation of the pollution error, even in settings when the medium is not smooth. We provide some examples to corroborate the convergence and complexity claims. Keywords: finite element; frequency-domain; numerical; acoustic; wave equatio
Overlapping domains for topology optimization of large-area metasurfaces
We introduce an overlapping-domain approach to large-area metasurface design,
in which each simulated domain consists of a unit cell and overlapping regions
from the neighboring cells plus PML absorbers. We show that our approach
generates greatly improved metalens quality compared to designs produced using
a locally periodic approximation, thanks to better accuracy
with similar computational cost. We use the new approach with topology
optimization to design large-area () high-NA (0.71) multichrome and
broadband achromatic lenses with high focusing efficiency (),
greatly improving upon previously reported works
Additive Sweeping Preconditioner for the Helmholtz Equation
We introduce a new additive sweeping preconditioner for the Helmholtz
equation based on the perfect matched layer (PML). This method divides the
domain of interest into thin layers and proposes a new transmission condition
between the subdomains where the emphasis is on the boundary values of the
intermediate waves. This approach can be viewed as an effective approximation
of an additive decomposition of the solution operator. When combined with the
standard GMRES solver, the iteration number is essentially independent of the
frequency. Several numerical examples are tested to show the efficiency of this
new approach.Comment: 27 page
An overlapping splitting double sweep method for the Helmholtz equation
We consider the domain decomposition method approach to solve the Helmholtz
equation. Double sweep based approaches for overlapping decompositions are
presented. In particular, we introduce an overlapping splitting double sweep
(OSDS) method valid for any type of interface boundary conditions. Despite the
fact that first order interface boundary conditions are used, the OSDS method
demonstrates good stability properties with respect to the number of subdomains
and the frequency even for heterogeneous media. In this context, convergence is
improved when compared to the double sweep methods in Nataf et al. (1997) and
Vion et al. (2014, 2016} for all of our test cases: waveguide, open cavity and
wedge problems