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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
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
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