35 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
Flux norm approach to finite dimensional homogenization approximations with non-separated scales and high contrast
We consider divergence-form scalar elliptic equations and vectorial equations
for elasticity with rough (, )
coefficients that, in particular, model media with non-separated scales
and high contrast in material properties. We define the flux norm as the
norm of the potential part of the fluxes of solutions, which is equivalent to
the usual -norm. We show that in the flux norm, the error associated with
approximating, in a properly defined finite-dimensional space, the set of
solutions of the aforementioned PDEs with rough coefficients is equal to the
error associated with approximating the set of solutions of the same type of
PDEs with smooth coefficients in a standard space (e.g., piecewise polynomial).
We refer to this property as the {\it transfer property}.
A simple application of this property is the construction of finite
dimensional approximation spaces with errors independent of the regularity and
contrast of the coefficients and with optimal and explicit convergence rates.
This transfer property also provides an alternative to the global harmonic
change of coordinates for the homogenization of elliptic operators that can be
extended to elasticity equations. The proofs of these homogenization results
are based on a new class of elliptic inequalities which play the same role in
our approach as the div-curl lemma in classical homogenization.Comment: Accepted for publication in Archives for Rational Mechanics and
Analysi
A Toy Model for Testing Finite Element Methods to Simulate Extreme-Mass-Ratio Binary Systems
Extreme mass ratio binary systems, binaries involving stellar mass objects
orbiting massive black holes, are considered to be a primary source of
gravitational radiation to be detected by the space-based interferometer LISA.
The numerical modelling of these binary systems is extremely challenging
because the scales involved expand over several orders of magnitude. One needs
to handle large wavelength scales comparable to the size of the massive black
hole and, at the same time, to resolve the scales in the vicinity of the small
companion where radiation reaction effects play a crucial role. Adaptive finite
element methods, in which quantitative control of errors is achieved
automatically by finite element mesh adaptivity based on posteriori error
estimation, are a natural choice that has great potential for achieving the
high level of adaptivity required in these simulations. To demonstrate this, we
present the results of simulations of a toy model, consisting of a point-like
source orbiting a black hole under the action of a scalar gravitational field.Comment: 29 pages, 37 figures. RevTeX 4.0. Minor changes to match the
published versio