2,356 research outputs found
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
Sparse Generalized Multiscale Finite Element Methods and their applications
In a number of previous papers, local (coarse grid) multiscale model
reduction techniques are developed using a Generalized Multiscale Finite
Element Method. In these approaches, multiscale basis functions are constructed
using local snapshot spaces, where a snapshot space is a large space that
represents the solution behavior in a coarse block. In a number of applications
(e.g., those discussed in the paper), one may have a sparsity in the snapshot
space for an appropriate choice of a snapshot space. More precisely, the
solution may only involve a portion of the snapshot space. In this case, one
can use sparsity techniques to identify multiscale basis functions. In this
paper, we consider two such sparse local multiscale model reduction approaches.
In the first approach (which is used for parameter-dependent multiscale
PDEs), we use local minimization techniques, such as sparse POD, to identify
multiscale basis functions, which are sparse in the snapshot space. These
minimization techniques use minimization to find local multiscale basis
functions, which are further used for finding the solution. In the second
approach (which is used for the Helmholtz equation), we directly apply
minimization techniques to solve the underlying PDEs. This approach is more
expensive as it involves a large snapshot space; however, in this example, we
can not identify a local minimization principle, such as local generalized SVD
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
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