9 research outputs found
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
Can DtN and GenEO coarse spaces be sufficiently robust for heterogeneous Helmholtz problems?
Numerical solutions of heterogeneous Helmholtz problems present various computational challenges, with descriptive theory remaining out of reach for many popular approaches. Robustness and scalability are key for practical and reliable solvers in large-scale applications, especially for large wave number problems. In this work, we explore the use of a GenEO-type coarse space to build a two-level additive Schwarz method applicable to highly indefinite Helmholtz problems. Through a range of numerical tests on a 2D model problem, discretised by finite elements on pollution-free meshes, we observe robust convergence, iteration counts that do not increase with the wave number, and good scalability of our approach. We further provide results showing a favourable comparison with the DtN coarse space. Our numerical study shows promise that our solver methodology can be effective for challenging heterogeneous applications
Domain decomposition preconditioning for the high-frequency time-harmonic Maxwell equations with absorption
This paper rigorously analyses preconditioners for the timeharmonic Maxwell equations with absorption, where the PDE is discretised using curl-conforming finite-element methods of fixed, arbitrary order and the preconditioner is constructed using additive Schwarz domain decomposition methods. The theory developed here shows that if the absorption is large enough, and if the subdomain and coarse mesh diameters and overlap are chosen appropriately, then the classical two-level overlapping additive Schwarz preconditioner (with PEC boundary conditions on the subdomains) performs optimally-in the sense that GMRES converges in a wavenumber-independent number of iterations-for the problem with absorption. An important feature of the theory is that it allows the coarse space to be built from low-order elements even if the PDE is discretised using high-order elements. It also shows that additive methods with minimal overlap can be robust. Numerical experiments are given that illustrate the theory and its dependence on various parameters. These experiments motivate some extensions of the preconditioners which have better robustness for problems with less absorption, including the propagative case. At the end of the paper we illustrate the performance of these on two substantial applications; the first (a problem with absorption arising from medical imaging) shows the empirical robustness of the preconditioner against heterogeneity, and the second (scattering by a COBRA cavity) shows good scalability of the preconditioner with up to 3,000 processors.</p
Domain decomposition preconditioning for the high-frequency time-harmonic Maxwell equations with absorption
This paper rigorously analyses preconditioners for the time-harmonic Maxwell
equations with absorption, where the PDE is discretised using curl-conforming
finite-element methods of fixed, arbitrary order and the preconditioner is
constructed using Additive Schwarz domain decomposition methods. The theory
developed here shows that if the absorption is large enough, and if the
subdomain and coarse mesh diameters and overlap are chosen appropriately, then
the classical two-level overlapping Additive Schwarz preconditioner (with PEC
boundary conditions on the subdomains) performs optimally -- in the sense that
GMRES converges in a wavenumber-independent number of iterations -- for the
problem with absorption. An important feature of the theory is that it allows
the coarse space to be built from low-order elements even if the PDE is
discretised using high-order elements. It also shows that additive methods with
minimal overlap can be robust. Numerical experiments are given that illustrate
the theory and its dependence on various parameters. These experiments motivate
some extensions of the preconditioners which have better robustness for
problems with less absorption, including the propagative case. At the end of
the paper we illustrate the performance of these on two substantial
applications; the first (a problem with absorption arising from medical
imaging) shows the empirical robustness of the preconditioner against
heterogeneity, and the second (scattering by a COBRA cavity) shows good
scalability of the preconditioner with up to 3,000 processors
Domain decomposition preconditioning for the Helmholtz equation: a coarse space based on local Dirichlet-to-Neumann maps
In this thesis, we present a two-level domain decomposition method for the iterative solution of the heterogeneous Helmholtz equation. The Helmholtz equation governs wave propagation and scattering phenomena arising in a wide range of engineering applications. Its discretization with piecewise linear finite elements results in typically large, ill-conditioned, indefinite, and non- Hermitian linear systems of equations, for which standard iterative and direct methods encounter convergence problems. Therefore, especially designed methods are needed. The inherently parallel domain decomposition methods constitute a promising class of preconditioners, as they subdivide the large problems into smaller subproblems and are hence able to cope with many degrees of freedom. An essential element of these methods is a good coarse space. Here, the Helmholtz equation presents a particular challenge, as even slight deviations from the optimal choice can be fatal. We develop a coarse space that is based on local eigenproblems involving the Dirichlet-to-Neumann operator. Our construction is completely automatic, ensuring good convergence rates without the need for parameter tuning. Moreover, it naturally respects local variations in the wave number and is hence suited also for heterogeneous Helmholtz problems. Apart from the question of how to design the coarse space, we also investigate the question of how to incorporate the coarse space into the method. Also here the fact that the stiffness matrix is non-Hermitian and indefinite constitutes a major challenge. The resulting method is parallel by design and its efficiency is investigated for two- and three-dimensional homogeneous and heterogeneous numerical examples