750 research outputs found
Finite Element Approximation of Elliptic Homogenization Problems in Nondivergence-Form
We use uniform estimates to obtain corrector results for periodic
homogenization problems of the form
subject to a homogeneous Dirichlet boundary condition. We propose and
rigorously analyze a numerical scheme based on finite element approximations
for such nondivergence-form homogenization problems. The second part of the
paper focuses on the approximation of the corrector and numerical
homogenization for the case of nonuniformly oscillating coefficients. Numerical
experiments demonstrate the performance of the scheme.Comment: 39 page
Corrector Analysis of a Heterogeneous Multi-scale Scheme for Elliptic Equations with Random Potential
This paper analyzes the random fluctuations obtained by a heterogeneous
multi-scale first-order finite element method applied to solve elliptic
equations with a random potential. We show that the random fluctuations of such
solutions are correctly estimated by the heterogeneous multi-scale algorithm
when appropriate fine-scale problems are solved on subsets that cover the whole
computational domain. However, when the fine-scale problems are solved over
patches that do not cover the entire domain, the random fluctuations may or may
not be estimated accurately. In the case of random potentials with short-range
interactions, the variance of the random fluctuations is amplified as the
inverse of the fraction of the medium covered by the patches. In the case of
random potentials with long-range interactions, however, such an amplification
does not occur and random fluctuations are correctly captured independent of
the (macroscopic) size of the patches.
These results are consistent with those obtained by the authors for more
general equations in the one-dimensional setting and provide indications on the
loss in accuracy that results from using coarser, and hence less
computationally intensive, algorithms
An introduction to the qualitative and quantitative theory of homogenization
We present an introduction to periodic and stochastic homogenization of
ellip- tic partial differential equations. The first part is concerned with the
qualitative theory, which we present for equations with periodic and random
coefficients in a unified approach based on Tartar's method of oscillating test
functions. In partic- ular, we present a self-contained and elementary argument
for the construction of the sublinear corrector of stochastic homogenization.
(The argument also applies to elliptic systems and in particular to linear
elasticity). In the second part we briefly discuss the representation of the
homogenization error by means of a two- scale expansion. In the last part we
discuss some results of quantitative stochastic homogenization in a discrete
setting. In particular, we discuss the quantification of ergodicity via
concentration inequalities, and we illustrate that the latter in combi- nation
with elliptic regularity theory leads to a quantification of the growth of the
sublinear corrector and the homogenization error.Comment: Lecture notes of a minicourse given by the author during the GSIS
International Winter School 2017 on "Stochastic Homogenization and its
applications" at the Tohoku University, Sendai, Japan; This version contains
a correction of Lemma 2.1
A localized orthogonal decomposition method for semi-linear elliptic problems
In this paper we propose and analyze a new Multiscale Method for solving
semi-linear elliptic problems with heterogeneous and highly variable
coefficient functions. For this purpose we construct a generalized finite
element basis that spans a low dimensional multiscale space. The basis is
assembled by performing localized linear fine-scale computations in small
patches that have a diameter of order H |log H| where H is the coarse mesh
size. Without any assumptions on the type of the oscillations in the
coefficients, we give a rigorous proof for a linear convergence of the H1-error
with respect to the coarse mesh size. To solve the arising equations, we
propose an algorithm that is based on a damped Newton scheme in the multiscale
space
Numerical homogenization of H(curl)-problems
If an elliptic differential operator associated with an
-problem involves rough (rapidly varying)
coefficients, then solutions to the corresponding
-problem admit typically very low regularity, which
leads to arbitrarily bad convergence rates for conventional numerical schemes.
The goal of this paper is to show that the missing regularity can be
compensated through a corrector operator. More precisely, we consider the
lowest order N\'ed\'elec finite element space and show the existence of a
linear corrector operator with four central properties: it is computable,
-stable, quasi-local and allows for a correction of
coarse finite element functions so that first-order estimates (in terms of the
coarse mesh-size) in the norm are obtained provided
the right-hand side belongs to . With these four
properties, a practical application is to construct generalized finite element
spaces which can be straightforwardly used in a Galerkin method. In particular,
this characterizes a homogenized solution and a first order corrector,
including corresponding quantitative error estimates without the requirement of
scale separation
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