2,463 research outputs found
Homogenization of Parabolic Equations with an Arbitrary Number of Scales in Both Space and Time
The main contribution of this paper is the homogenization of the linearparabolic equationtu (x, t) − ·axq1, ...,xqn,tr1, ...,trmu (x, t)= f(x, t)exhibiting an arbitrary finite number of both spatial and temporal scales.We briefly recall some fundamentals of multiscale convergence and providea characterization of multiscale limits for gradients in an evolution settingadapted to a quite general class of well-separated scales, which we nameby jointly well-separated scales (see Appendix for the proof). We proceedwith a weaker version of this concept called very weak multiscale convergence.We prove a compactness result with respect to this latter typefor jointly well-separated scales. This is a key result for performing thehomogenization of parabolic problems combining rapid spatial and temporaloscillations such as the problem above. Applying this compactnessresult together with a characterization of multiscale limits of sequences ofgradients we carry out the homogenization procedure, where we togetherwith the homogenized problem obtain n local problems, i.e. one for eachspatial microscale. To illustrate the use of the obtained result we apply itto a case with three spatial and three temporal scales with q1 = 1, q2 = 2and 0 < r1 < r2.MSC: 35B27; 35K10Publ online Dec 2013</p
Localized bases for finite dimensional homogenization approximations with non-separated scales and high-contrast
We construct finite-dimensional approximations of solution spaces of
divergence form operators with -coefficients. Our method does not
rely on concepts of ergodicity or scale-separation, but on the property that
the solution space of these operators is compactly embedded in if source
terms are in the unit ball of instead of the unit ball of .
Approximation spaces are generated by solving elliptic PDEs on localized
sub-domains with source terms corresponding to approximation bases for .
The -error estimates show that -dimensional spaces
with basis elements localized to sub-domains of diameter (with ) result in an
accuracy for elliptic, parabolic and hyperbolic
problems. For high-contrast media, the accuracy of the method is preserved
provided that localized sub-domains contain buffer zones of width
where the contrast of the medium
remains bounded. The proposed method can naturally be generalized to vectorial
equations (such as elasto-dynamics).Comment: Accepted for publication in SIAM MM
Homogenisation of Monotone Parabolic Problems with Several Temporal Scales: The Detailed arXiv e-Print Version
In this paper we homogenise monotone parabolic problems with two spatial
scales and finitely many temporal scales. Under a certain well-separatedness
assumption on the spatial and temporal scales as explained in the paper, we
show that there is an H-limit defined by at most four distinct sets of local
problems corresponding to slow temporal oscillations, slow resonant spatial and
temporal oscillations (the "slow" self-similar case), rapid temporal
oscillations, and rapid resonant spatial and temporal oscillations (the "rapid"
self-similar case), respectively.Comment: 57 pages, no figures. Intended as a detailed version of a journal
paper. Revised with some added reference
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
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