7,166 research outputs found
Localized orthogonal decomposition method for the wave equation with a continuum of scales
This paper is devoted to numerical approximations for the wave equation with
a multiscale character. Our approach is formulated in the framework of the
Localized Orthogonal Decomposition (LOD) interpreted as a numerical
homogenization with an -projection. We derive explicit convergence rates
of the method in the -, - and
-norms without any assumptions on higher order space
regularity or scale-separation. The order of the convergence rates depends on
further graded assumptions on the initial data. We also prove the convergence
of the method in the framework of G-convergence without any structural
assumptions on the initial data, i.e. without assuming that it is
well-prepared. This rigorously justifies the method. Finally, the performance
of the method is demonstrated in numerical experiments
Variational Multiscale Stabilization and the Exponential Decay of Fine-scale Correctors
This paper addresses the variational multiscale stabilization of standard
finite element methods for linear partial differential equations that exhibit
multiscale features. The stabilization is of Petrov-Galerkin type with a
standard finite element trial space and a problem-dependent test space based on
pre-computed fine-scale correctors. The exponential decay of these correctors
and their localisation to local cell problems is rigorously justified. The
stabilization eliminates scale-dependent pre-asymptotic effects as they appear
for standard finite element discretizations of highly oscillatory problems,
e.g., the poor approximation in homogenization problems or the pollution
effect in high-frequency acoustic scattering
The wave packet propagation using wavelets
It is demonstrated that the wavelets can be used to considerably speed up
simulations of the wave packet propagation in multiscale systems. Extremely
high efficiency is obtained in the representation of both bound and continuum
states. The new method is compared with the fast Fourier algorithm. Depending
on ratios of typical scales of a quantum system in question, the wavelet method
appears to be faster by a few orders of magnitude.Comment: Latex 7 pages, 3 colored figures (Fig1 postscript, Fig2,3 gif) in
files separate from the pape
Localized Orthogonal Decomposition for two-scale Helmholtz-type problems
In this paper, we present a Localized Orthogonal Decomposition (LOD) in
Petrov-Galerkin formulation for a two-scale Helmholtz-type problem. The
two-scale problem is, for instance, motivated from the homogenization of the
Helmholtz equation with high contrast, studied together with a corresponding
multiscale method in (Ohlberger, Verf\"urth. A new Heterogeneous Multiscale
Method for the Helmholtz equation with high contrast, arXiv:1605.03400, 2016).
There, an unavoidable resolution condition on the mesh sizes in terms of the
wave number has been observed, which is known as "pollution effect" in the
finite element literature. Following ideas of (Gallistl, Peterseim. Comput.
Methods Appl. Mech. Engrg. 295:1-17, 2015), we use standard finite element
functions for the trial space, whereas the test functions are enriched by
solutions of subscale problems (solved on a finer grid) on local patches.
Provided that the oversampling parameter , which indicates the size of the
patches, is coupled logarithmically to the wave number, we obtain a
quasi-optimal method under a reasonable resolution of a few degrees of freedom
per wave length, thus overcoming the pollution effect. In the two-scale
setting, the main challenges for the LOD lie in the coupling of the function
spaces and in the periodic boundary conditions.Comment: 20 page
Crossover from the chiral to the standard universality classes in the conductance of a quantum wire with random hopping only
The conductance of a quantum wire with off-diagonal disorder that preserves a
sublattice symmetry (the random hopping problem with chiral symmetry) is
considered. Transport at the band center is anomalous relative to the standard
problem of Anderson localization both in the diffusive and localized regimes.
In the diffusive regime, there is no weak-localization correction to the
conductance and universal conductance fluctuations are twice as large as in the
standard cases. Exponential localization occurs only for an even number of
transmission channels in which case the localization length does not depend on
whether time-reversal and spin rotation symmetry are present or not. For an odd
number of channels the conductance decays algebraically. Upon moving away from
the band center transport characteristics undergo a crossover to those of the
standard universality classes of Anderson localization. This crossover is
calculated in the diffusive regime.Comment: 22 pages, 9 figure
Numerical upscaling for wave equations with time-dependent multiscale coefficients
In this paper, we consider the classical wave equation with time-dependent, spatially multiscale coefficients. We propose a fully discrete computational multiscale method in the spirit of the localized orthogonal decomposition in space with a backward Euler scheme in time. We show optimal convergence rates in space and time beyond the assumptions of spatial periodicity or scale separation of the coefficients. Further, we propose an adaptive update strategy for the time-dependent multiscale basis. Numerical experiments illustrate the theoretical results and showcase the practicability of the adaptive update strategy
- …