474 research outputs found
On stability of discretizations of the Helmholtz equation (extended version)
We review the stability properties of several discretizations of the
Helmholtz equation at large wavenumbers. For a model problem in a polygon, a
complete -explicit stability (including -explicit stability of the
continuous problem) and convergence theory for high order finite element
methods is developed. In particular, quasi-optimality is shown for a fixed
number of degrees of freedom per wavelength if the mesh size and the
approximation order are selected such that is sufficiently small and
, and, additionally, appropriate mesh refinement is used near
the vertices. We also review the stability properties of two classes of
numerical schemes that use piecewise solutions of the homogeneous Helmholtz
equation, namely, Least Squares methods and Discontinuous Galerkin (DG)
methods. The latter includes the Ultra Weak Variational Formulation
A Posteriori Error Estimation for Highly Indefinite Helmholtz Problems
We develop a new analysis for residual-type aposteriori error estimation for a class of highly indefinite elliptic boundary value problems by considering the Helmholtz equation at high wavenumber as our model problem. We employ a classical conforming Galerkin discretization by using hp-finite elements. In [Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions, Math. Comp., 79 (2010), pp.1871-1914], Melenk and Sauter introduced an hp-finite element discretization which leads to a stable and pollution-free discretization of the Helmholtz equation under a mild resolution condition which requires only degrees of freedom, where denotes the spatial dimension. In the present paper, we will introduce an aposteriori error estimator for this problem and prove its reliability and efficiency. The constants in these estimates become independent of the, possibly, high wavenumber provided the aforementioned resolution condition for stability is satisfied. We emphasize that, by using the classical theory, the constants in the aposteriori estimates would be amplified by a factor
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
Stable Multiscale Petrov-Galerkin Finite Element Method for High Frequency Acoustic Scattering
We present and analyze a pollution-free Petrov-Galerkin multiscale finite
element method for the Helmholtz problem with large wave number as a
variant of [Peterseim, ArXiv:1411.1944, 2014]. We use standard continuous
finite elements at a coarse discretization scale as trial functions,
whereas the test functions are computed as the solutions of local problems at a
finer scale . The diameter of the support of the test functions behaves like
for some oversampling parameter . Provided is of the order of
and is sufficiently small, the resulting method is stable
and quasi-optimal in the regime where is proportional to . In
homogeneous (or more general periodic) media, the fine scale test functions
depend only on local mesh-configurations. Therefore, the seemingly high cost
for the computation of the test functions can be drastically reduced on
structured meshes. We present numerical experiments in two and three space
dimensions.Comment: The version coincides with v3. We only resized some figures which
were difficult to process for certain printer
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
Mini-Workshop: Efficient and Robust Approximation of the Helmholtz Equation
The accurate and efficient treatment of wave propogation phenomena is still a challenging problem. A prototypical equation is the Helmholtz equation at high wavenumbers. For this equation, Babuška & Sauter showed in 2000 in their seminal SIAM Review paper that standard discretizations must fail in the sense that the ratio of true error and best approximation error has to grow with the frequency. This has spurred the development of alternative, non-standard discretization techniques. This workshop focused on evaluating and comparing these different approaches also with a view to their applicability to more general wave propagation problems
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