44 research outputs found
A simple proof that the -FEM does not suffer from the pollution effect for the constant-coefficient full-space Helmholtz equation
In dimensions, approximating an arbitrary function oscillating with
frequency requires degrees of freedom. A numerical
method for solving the Helmholtz equation (with wavenumber ) suffers from
the pollution effect if, as , the total number of degrees of
freedom needed to maintain accuracy grows faster than this natural threshold.
While the -version of the finite element method (FEM) (where accuracy is
increased by decreasing the meshwidth and keeping the polynomial degree
fixed) suffers from the pollution effect, the celebrated papers [Melenk, Sauter
2010], [Melenk, Sauter 2011], [Esterhazy, Melenk 2012], and [Melenk, Parsania,
Sauter 2013] showed that the -FEM (where accuracy is increased by
decreasing the meshwidth and increasing the polynomial degree ) applied
to a variety of constant-coefficient Helmholtz problems does not suffer from
the pollution effect.
The heart of the proofs of these results is a PDE result splitting the
solution of the Helmholtz equation into "high" and "low" frequency components.
In this expository paper we prove this splitting for the constant-coefficient
Helmholtz equation in full space (i.e., in ) using only
integration by parts and elementary properties of the Fourier transform; this
is in contrast to the proof for this set-up in [Melenk, Sauter 2010] which uses
somewhat-involved bounds on Bessel and Hankel functions. The proof in this
paper is motivated by the recent proof in [Lafontaine, Spence, Wunsch 2022] of
this splitting for the variable-coefficient Helmholtz equation in full space;
indeed, the proof in [Lafontaine, Spence, Wunsch 2022] uses more-sophisticated
tools that reduce to the elementary ones above for constant coefficients
Can coercive formulations lead to fast and accurate solution of the Helmholtz equation?
A new, coercive formulation of the Helmholtz equation was introduced in
[Moiola, Spence, SIAM Rev. 2014]. In this paper we investigate -version
Galerkin discretisations of this formulation, and the iterative solution of the
resulting linear systems. We find that the coercive formulation behaves
similarly to the standard formulation in terms of the pollution effect (i.e. to
maintain accuracy as , must decrease with at the same rate
as for the standard formulation). We prove -explicit bounds on the number of
GMRES iterations required to solve the linear system of the new formulation
when it is preconditioned with a prescribed symmetric positive-definite matrix.
Even though the number of iterations grows with , these are the first such
rigorous bounds on the number of GMRES iterations for a preconditioned
formulation of the Helmholtz equation, where the preconditioner is a symmetric
positive-definite matrix.Comment: 27 pages, 7 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
Analysis of a Helmholtz preconditioning problem motivated by uncertainty quantification
This paper analyses the following question: let , be
the Galerkin matrices corresponding to finite-element discretisations of the
exterior Dirichlet problem for the heterogeneous Helmholtz equations
. How small must and be (in terms of -dependence) for
GMRES applied to either or
to converge in a -independent number of
iterations for arbitrarily large ? (In other words, for to be
a good left- or right-preconditioner for ?). We prove results
answering this question, give theoretical evidence for their sharpness, and
give numerical experiments supporting the estimates.
Our motivation for tackling this question comes from calculating quantities
of interest for the Helmholtz equation with random coefficients and .
Such a calculation may require the solution of many deterministic Helmholtz
problems, each with different and , and the answer to the question above
dictates to what extent a previously-calculated inverse of one of the Galerkin
matrices can be used as a preconditioner for other Galerkin matrices
Sharp bounds on Helmholtz impedance-to-impedance maps and application to overlapping domain decomposition
We prove sharp bounds on certain impedance-to-impedance maps (and their compositions) for the Helmholtz equation with large wavenumber (i.e., at high frequency) using semiclassical defect measures. Gong et al. (Numer. Math. 152:2 (2022), 259–306) recently showed that the behaviour of these impedance-to-impedance maps (and their compositions) dictates the convergence of the parallel overlapping Schwarz domain-decomposition method with impedance boundary conditions on the subdomain boundaries. For a model decomposition with two subdomains and sufficiently large overlap, the results of this paper combined with those of Gong et al. show that the parallel Schwarz method is power contractive, independent of the wavenumber. For strip-type decompositions with many subdomains, the results of this paper show that the composite impedance-to-impedance maps, in general, behave “badly” with respect to the wavenumber; nevertheless, by proving results about the composite maps applied to a restricted class of data, we give insight into the wavenumber-robustness of the parallel Schwarz method observed in the numerical experiments of Gong et al.</p
Sharp preasymptotic error bounds for the Helmholtz -FEM
In the analysis of the -version of the finite-element method (FEM), with
fixed polynomial degree , applied to the Helmholtz equation with wavenumber
, the is when is
sufficiently small and the sequence of Galerkin solutions are quasioptimal;
here is the norm of the Helmholtz solution operator, normalised
so that for nontrapping problems. In the
, one expects that if is
sufficiently small, then (for physical data) the relative error of the Galerkin
solution is controllably small. In this paper, we prove the natural error
bounds in the preasymptotic regime for the variable-coefficient Helmholtz
equation in the exterior of a Dirichlet, or Neumann, or penetrable obstacle (or
combinations of these) and with the radiation condition
realised exactly using the Dirichlet-to-Neumann map on the boundary of a ball
approximated either by a radial perfectly-matched layer (PML) or
an impedance boundary condition. Previously, such bounds for were only
available for Dirichlet obstacles with the radiation condition approximated by
an impedance boundary condition. Our result is obtained via a novel
generalisation of the "elliptic-projection" argument (the argument used to
obtain the result for ) which can be applied to a wide variety of abstract
Helmholtz-type problems