1,353 research outputs found
Adaptive multiscale model reduction with Generalized Multiscale Finite Element Methods
In this paper, we discuss a general multiscale model reduction framework
based on multiscale finite element methods. We give a brief overview of related
multiscale methods. Due to page limitations, the overview focuses on a few
related methods and is not intended to be comprehensive. We present a general
adaptive multiscale model reduction framework, the Generalized Multiscale
Finite Element Method. Besides the method's basic outline, we discuss some
important ingredients needed for the method's success. We also discuss several
applications. The proposed method allows performing local model reduction in
the presence of high contrast and no scale separation
Generalized multiscale finite element methods for wave propagation in heterogeneous media
Numerical modeling of wave propagation in heterogeneous media is important in
many applications. Due to the complex nature, direct numerical simulations on
the fine grid are prohibitively expensive. It is therefore important to develop
efficient and accurate methods that allow the use of coarse grids. In this
paper, we present a multiscale finite element method for wave propagation on a
coarse grid. The proposed method is based on the Generalized Multiscale Finite
Element Method (GMsFEM). To construct multiscale basis functions, we start with
two snapshot spaces in each coarse-grid block where one represents the degrees
of freedom on the boundary and the other represents the degrees of freedom in
the interior. We use local spectral problems to identify important modes in
each snapshot space. These local spectral problems are different from each
other and their formulations are based on the analysis. To our best knowledge,
this is the first time where multiple snapshot spaces and multiple spectral
problems are used and necessary for efficient computations. Using the dominant
modes from local spectral problems, multiscale basis functions are constructed
to represent the solution space locally within each coarse block. These
multiscale basis functions are coupled via the symmetric interior penalty
discontinuous Galerkin method which provides a block diagonal mass matrix, and,
consequently, results in fast computations in an explicit time discretiza-
tion. Our methods' stability and spectral convergence are rigorously analyzed.
Numerical examples are presented to show our methods' performance. We also test
oversampling strategies. In particular, we discuss how the modes from different
snapshot spaces can affect the proposed methods' accuracy
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
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