1,959 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
Parallel accelerated cyclic reduction preconditioner for three-dimensional elliptic PDEs with variable coefficients
We present a robust and scalable preconditioner for the solution of
large-scale linear systems that arise from the discretization of elliptic PDEs
amenable to rank compression. The preconditioner is based on hierarchical
low-rank approximations and the cyclic reduction method. The setup and
application phases of the preconditioner achieve log-linear complexity in
memory footprint and number of operations, and numerical experiments exhibit
good weak and strong scalability at large processor counts in a distributed
memory environment. Numerical experiments with linear systems that feature
symmetry and nonsymmetry, definiteness and indefiniteness, constant and
variable coefficients demonstrate the preconditioner applicability and
robustness. Furthermore, it is possible to control the number of iterations via
the accuracy threshold of the hierarchical matrix approximations and their
arithmetic operations, and the tuning of the admissibility condition parameter.
Together, these parameters allow for optimization of the memory requirements
and performance of the preconditioner.Comment: 24 pages, Elsevier Journal of Computational and Applied Mathematics,
Dec 201
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