37 research outputs found
Higher-order FEM and CIP-FEM for Helmholtz equation with high wave number and perfectly matched layer truncation
The high-frequency Helmholtz equation on the entire space is truncated into a
bounded domain using the perfectly matched layer (PML) technique and
subsequently, discretized by the higher-order finite element method (FEM) and
the continuous interior penalty finite element method (CIP-FEM). By formulating
an elliptic problem involving a linear combination of a finite number of
eigenfunctions related to the PML differential operator, a wave-number-explicit
decomposition lemma is proved for the PML problem, which implies that the PML
solution can be decomposed into a non-oscillating elliptic part and an
oscillating but analytic part. The preasymptotic error estimates in the energy
norm for both the -th order CIP-FEM and FEM are proved to be under the mesh condition that
is sufficiently small, where is the wave number, is the mesh size, and
is the PML truncation error which is exponentially small. In
particular, the dependences of coefficients on the source are
improved. Numerical experiments are presented to validate the theoretical
findings, illustrating that the higher-order CIP-FEM can greatly reduce the
pollution errors
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
The -FEM applied to the Helmholtz equation with PML truncation does not suffer from the pollution effect
We consider approximation of the variable-coefficient Helmholtz equation in
the exterior of a Dirichlet obstacle using perfectly-matched-layer (PML)
truncation; it is well known that this approximation is exponentially accurate
in the PML width and the scaling angle, and the approximation was recently
proved to be exponentially accurate in the wavenumber in [Galkowski,
Lafontaine, Spence, 2021].
We show that the -FEM applied to this problem does not suffer from the
pollution effect, in that there exist such that if
and then the Galerkin solutions are quasioptimal (with
constant independent of ), under the following two conditions (i) the
solution operator of the original Helmholtz problem is polynomially bounded in
(which occurs for "most" by [Lafontaine, Spence, Wunsch, 2021]), and
(ii) either there is no obstacle and the coefficients are smooth or the
obstacle is analytic and the coefficients are analytic in a neighbourhood of
the obstacle and smooth elsewhere.
This -FEM result is obtained via a decomposition of the PML solution into
"high-" and "low-frequency" components, analogous to the decomposition for the
original Helmholtz solution recently proved in [Galkowski, Lafontaine, Spence,
Wunsch, 2022]. The decomposition is obtained using tools from semiclassical
analysis (i.e., the PDE techniques specifically designed for studying Helmholtz
problems with large ).Comment: arXiv admin note: text overlap with arXiv:2102.1308
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
Perfectly-matched-layer truncation is exponentially accurate at high frequency
We consider a wide variety of scattering problems including scattering by Dirichlet, Neumann, and penetrable obstacles. We consider a radial perfectly-matched layer (PML) and show that for any PML width and a steep-enough scaling angle, the PML solution is exponentially close, both in frequency and the tangent of the scaling angle, to the true scattering solution. Moreover, for a fixed scaling angle and large enough PML width, the PML solution is exponentially close to the true scattering solution in both frequency and the PML width. In fact, the exponential bound holds with rate of decay c (omicrontanθ − C)k where omicron is the PML width and θ is the scaling angle. More generally, the results of the paper hold in the framework of black-box scattering under the assumption of an exponential bound on the norm of the cutoff resolvent, thus including problems with strong trapping. These are the first results on the exponential accuracy of PML at high-frequency with non-trivial scatterers