1,501 research outputs found
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
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
Energy-based comparison between the Fourier--Galerkin method and the finite element method
The Fourier-Galerkin method (in short FFTH) has gained popularity in
numerical homogenisation because it can treat problems with a huge number of
degrees of freedom. Because the method incorporates the fast Fourier transform
(FFT) in the linear solver, it is believed to provide an improvement in
computational and memory requirements compared to the conventional finite
element method (FEM). Here, we systematically compare these two methods using
the energetic norm of local fields, which has the clear physical interpretation
as being the error in the homogenised properties. This enables the comparison
of memory and computational requirements at the same level of approximation
accuracy. We show that the methods' effectiveness relies on the smoothness
(regularity) of the solution and thus on the material coefficients. Thanks to
its approximation properties, FEM outperforms FFTH for problems with jumps in
material coefficients, while ambivalent results are observed for the case that
the material coefficients vary continuously in space. FFTH profits from a good
conditioning of the linear system, independent of the number of degrees of
freedom, but generally needs more degrees of freedom to reach the same
approximation accuracy. More studies are needed for other FFT-based schemes,
non-linear problems, and dual problems (which require special treatment in FEM
but not in FFTH).Comment: 24 pages, 10 figures, 2 table
Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization
We introduce a new variational method for the numerical homogenization of
divergence form elliptic, parabolic and hyperbolic equations with arbitrary
rough () coefficients. Our method does not rely on concepts of
ergodicity or scale-separation but on compactness properties of the solution
space and a new variational approach to homogenization. The approximation space
is generated by an interpolation basis (over scattered points forming a mesh of
resolution ) minimizing the norm of the source terms; its
(pre-)computation involves minimizing quadratic (cell)
problems on (super-)localized sub-domains of size .
The resulting localized linear systems remain sparse and banded. The resulting
interpolation basis functions are biharmonic for , and polyharmonic
for , for the operator -\diiv(a\nabla \cdot) and can be seen as a
generalization of polyharmonic splines to differential operators with arbitrary
rough coefficients. The accuracy of the method ( in energy norm
and independent from aspect ratios of the mesh formed by the scattered points)
is established via the introduction of a new class of higher-order Poincar\'{e}
inequalities. The method bypasses (pre-)computations on the full domain and
naturally generalizes to time dependent problems, it also provides a natural
solution to the inverse problem of recovering the solution of a divergence form
elliptic equation from a finite number of point measurements.Comment: ESAIM: Mathematical Modelling and Numerical Analysis. Special issue
(2013
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