703 research outputs found
Exponential decay of the resonance error in numerical homogenization via parabolic and elliptic cell problems
This paper presents two new approaches for finding the homogenized
coefficients of multiscale elliptic PDEs. Standard approaches for computing the
homogenized coefficients suffer from the so-called resonance error, originating
from a mismatch between the true and the computational boundary conditions. Our
new methods, based on solutions of parabolic and elliptic cell-problems, result
in an exponential decay of the resonance error
Homogenization of Parabolic Equations with a Continuum of Space and Time Scales
This paper addresses the issue of the homogenization of linear divergence form parabolic operators in situations where no ergodicity and no scale separation in time or space are available. Namely, we consider divergence form linear parabolic operators in with -coefficients. It appears that the inverse operator maps the unit ball of into a space of functions which at small (time and space) scales are close in norm to a functional space of dimension . It follows that once one has solved these equations at least times it is possible to homogenize them both in space and in time, reducing the number of operation counts necessary to obtain further solutions. In practice we show under a Cordes-type condition that the first order time derivatives and second order space derivatives of the solution of these operators with respect to caloric coordinates are in (instead of with Euclidean coordinates). If the medium is time-independent, then it is sufficient to solve times the associated elliptic equation in order to homogenize the parabolic equation
An Equation-Free Approach for Second Order Multiscale Hyperbolic Problems in Non-Divergence Form
The present study concerns the numerical homogenization of second order
hyperbolic equations in non-divergence form, where the model problem includes a
rapidly oscillating coefficient function. These small scales influence the
large scale behavior, hence their effects should be accurately modelled in a
numerical simulation. A direct numerical simulation is prohibitively expensive
since a minimum of two points per wavelength are needed to resolve the small
scales. A multiscale method, under the equation free methodology, is proposed
to approximate the coarse scale behaviour of the exact solution at a cost
independent of the small scales in the problem. We prove convergence rates for
the upscaled quantities in one as well as in multi-dimensional periodic
settings. Moreover, numerical results in one and two dimensions are provided to
support the theory
On the nature of the boundary resonance error in numerical homogenization and its reduction
Numerical homogenization of multiscale equations typically requires taking an
average of the solution to a microscale problem. Both the boundary conditions
and domain size of the microscale problem play an important role in the
accuracy of the homogenization procedure. In particular, imposing naive
boundary conditions leads to a error in the
computation, where is the characteristic size of the microscopic
fluctuations in the heterogeneous media, and is the size of the
microscopic domain. This so-called boundary, or ``cell resonance" error can
dominate discretization error and pollute the entire homogenization scheme.
There exist several techniques in the literature to reduce the error. Most
strategies involve modifying the form of the microscale cell problem. Below we
present an alternative procedure based on the observation that the resonance
error itself is an oscillatory function of domain size . After rigorously
characterizing the oscillatory behavior for one dimensional and quasi-one
dimensional microscale domains, we present a novel strategy to reduce the
resonance error. Rather than modifying the form of the cell problem, the
original problem is solved for a sequence of domain sizes, and the results are
averaged against kernels satisfying certain moment conditions and regularity
properties. Numerical examples in one and two dimensions illustrate the utility
of the approach
Recent advances in the evolution of interfaces: thermodynamics, upscaling, and universality
We consider the evolution of interfaces in binary mixtures permeating
strongly heterogeneous systems such as porous media. To this end, we first
review available thermodynamic formulations for binary mixtures based on
\emph{general reversible-irreversible couplings} and the associated
mathematical attempts to formulate a \emph{non-equilibrium variational
principle} in which these non-equilibrium couplings can be identified as
minimizers.
Based on this, we investigate two microscopic binary mixture formulations
fully resolving heterogeneous/perforated domains: (a) a flux-driven immiscible
fluid formulation without fluid flow; (b) a momentum-driven formulation for
quasi-static and incompressible velocity fields. In both cases we state two
novel, reliably upscaled equations for binary mixtures/multiphase fluids in
strongly heterogeneous systems by systematically taking thermodynamic features
such as free energies into account as well as the system's heterogeneity
defined on the microscale such as geometry and materials (e.g. wetting
properties). In the context of (a), we unravel a \emph{universality} with
respect to the coarsening rate due to its independence of the system's
heterogeneity, i.e. the well-known -behaviour for
homogeneous systems holds also for perforated domains.
Finally, the versatility of phase field equations and their
\emph{thermodynamic foundation} relying on free energies, make the collected
recent developments here highly promising for scientific, engineering and
industrial applications for which we provide an example for lithium batteries
Numerical homogenization of elliptic PDEs with similar coefficients
We consider a sequence of elliptic partial differential equations (PDEs) with
different but similar rapidly varying coefficients. Such sequences appear, for
example, in splitting schemes for time-dependent problems (with one coefficient
per time step) and in sample based stochastic integration of outputs from an
elliptic PDE (with one coefficient per sample member). We propose a
parallelizable algorithm based on Petrov-Galerkin localized orthogonal
decomposition (PG-LOD) that adaptively (using computable and theoretically
derived error indicators) recomputes the local corrector problems only where it
improves accuracy. The method is illustrated in detail by an example of a
time-dependent two-pase Darcy flow problem in three dimensions
Multiscale Finite Element Methods for Nonlinear Problems and their Applications
In this paper we propose a generalization of multiscale finite element methods (Ms-FEM) to nonlinear problems. We study the convergence of the proposed method for nonlinear elliptic equations and propose an oversampling technique. Numerical examples demonstrate that the over-sampling technique greatly reduces the error. The application of MsFEM to porous media flows is considered. Finally, we describe further generalizations of MsFEM to nonlinear time-dependent equations and discuss the convergence of the method for various kinds of heterogeneities
Operator estimates for the crushed ice problem
Let be the Dirichlet Laplacian in the domain
.
Here and is a family of
tiny identical holes ("ice pieces") distributed periodically in
with period . We denote by the
capacity of a single hole. It was known for a long time that
converges to the operator
in strong resolvent sense provided the limit exists and is finite. In the
current contribution we improve this result deriving estimates for the rate of
convergence in terms of operator norms. As an application, we establish the
uniform convergence of the corresponding semi-groups and (for bounded )
an estimate for the difference of the -th eigenvalue of
and . Our proofs
relies on an abstract scheme for studying the convergence of operators in
varying Hilbert spaces developed previously by the second author.Comment: now 24 pages, 3 figures; some typos fixed and references adde
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