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Research Papers
- Metric Based Upscaling
(Communications on Pure and Applied Mathematics, Vol. 60, Issue 5, p 675-723. Also as arXiv, math.NA/0505223). Correction
We consider divergence form elliptic operators in dimension $n\geq 2$ with
$L^{\infty}$ coefficients. Although solutions of these operators are only
Holder continuous, we show that they are differentiable $C^{1,\alpha}$ with
respect to harmonic coordinates. It follows that numerical homogenization can
be extended to situations where the medium has no ergodicity at small scales
and is characterized by a continuum of scales. This new numerical
homogenization method is based on the transfer a new metric in addition to
traditional averaged (homogenized) quantities from subgrid scales into
computational scales. Error bounds can be given and this method can also be
used as a compression tool for differential operators.
- Homogenization of Parabolic Equations with a Continuum Space and Time Scales
(arXiv, math.AP/0512504, accepted by SINUM)
This paper addresses the issue of 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 $\Omega\in\mathbb{R}^n$ with
$L^{\infty}(\Omega\times(0,T)))$-coefficients. It appears that the inverse
operator maps the unit ball of $L^2(\Omega\times(0,T))$ into a space of
functions which at small (time and space) scales are close in $H^1$ norm to a
functional space of dimension n. It follows that once one has solved these
equations at least n-times it is possible to homogenize them both in space and
in time, reducing the number of operations counts necessary to obtain further
solutions. In practice we show that 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 $L^2$
(instead of $H^{-1}$ with Euclidean coordinates). If the medium is time
independent then it is sufficient to solve n times the associated elliptic
equation in order to homogenize the parabolic equation.
- Homogenization of the Accoustic Wave Equation with a Continuum Scales
(preprint, math.AP/0604380)
We consider the acoustic wave equation in dimension $n$ in
situations where the bulk modulus and the density of the medium are
only bounded. We show that under a Cordes type condition the second
order derivatives of the solution with respect to harmonic
coordinates are in $L^2$ (instead of $H^{-1}$ with respect to
Euclidean coordinates) and the solution itself is in
$L^{\infty}(0,T,H^2(\Omega))$ (instead of $L^{\infty}(0,T,H^1(\Omega))$ with
respect to Euclidean coordinates). It follows that it is possible to
homogenize the wave equation without assumptions of scale separation or
ergodicity by pre-computing $n$ solutions of the associated elliptic equation.
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