Computing homogenized coefficients<i>via</i>multiscale representation and hierarchical hybrid grids

Hannukainen, Antti; Mourrat, Jean-Christophe; Stoppels, Harmen T.

doi:10.1051/m2an/2020024

Cited by 5 publications

(3 citation statements)

References 60 publications

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“…The method proposed here also requires that the user computes a beforehand. An efficient method for doing so was presented in [23,29]; see also [11,17] and references therein for previous work on this problem. Moreover, one can check that in order to guarantee the contraction property of the iteration described in Theorem 1.1, say by a factor of 1/2, a coarse approximation of a, which may be off by a small but fixed positive amount, suffices.…”

Section: Statement Of the Main Resultsmentioning

confidence: 99%

An iterative method for elliptic problems with rapidly oscillating coefficients

et al. 2021

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We introduce a new iterative method for computing solutions of elliptic equations with random rapidly oscillating coefficients. Similarly to a multigrid method, each step of the iteration involves different computations meant to address different length scales. However, we use here the homogenized equation on all scales larger than a fixed multiple of the scale of oscillation of the coefficients. While the performance of standard multigrid methods degrades rapidly under the regime of large scale separation that we consider here, we show an explicit estimate on the contraction factor of our method which is independent of the size of the domain. We also present numerical experiments which confirm the effectiveness of the method, with openly available source code.

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Section: Statement Of the Main Resultsmentioning

confidence: 99%

An iterative method for elliptic problems with rapidly oscillating coefficients

et al. 2021

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“…Representative volume method, a popular approach used by engineers, are systematically analyzed in [Fis19,KKO20]. Iterative multigrid methods have been studied in [Mou19,HMS21,AHKM21]. There have also been abundant research in numerical homogenization where the coefficients do not necessarily arise from random setting.…”

Section: Related Workmentioning

confidence: 99%

Optimal Artificial Boundary Condition for Random Elliptic Media

Otto

2021

Found Comput Math

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We are interested in numerical algorithms for computing the electrical field generated by a charge distribution localized on scale in an infinite heterogeneous medium, in a situation where the medium is only known in a box of diameter L around the support of the charge. We propose a boundary condition that with overwhelming probability is (near) optimal with respect to scaling in terms of and L, in the setting where the medium is a sample from a stationary ensemble with a finite range of dependence (set to be unity and with the assumption that 1). The boundary condition is motivated by quantitative stochastic homogenization that allows for a multipole expansion [BGO20].This work extends [LO21] from two to three dimensions, and thus we need to take quadrupoles, next to dipoles, into account. This in turn relies on stochastic estimates of second-order, next to first-order, correctors. These estimates are provided for finite range ensembles under consideration, based on an extension of the semi-group approach of [GO15].

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“…Several interesting approaches have recently been proposed to reduce the above mentioned boundary error, see [2,3,4,5,9,11,23,24,27], where the core idea relies on modifying the corrector problem (6) so that the artificial boundary conditions imposed on the boundary ∂K R has a minor influence on the accuracy of the approximate homogenized coefficient. In the remaining part of the introduction, we briefly mention two successful approaches from [3,23], which use elliptic corrector problems.…”

Section: A Naive Elliptic Approachmentioning

confidence: 99%

Analytical and numerical study of a modified cell problem for the numerical homogenization of multiscale random fields

Abdulle¹,

Arjmand²,

Paganoni³

2020

Preprint

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A central question in numerical homogenization of partial differential equations with multiscale coefficients is the accurate computation of effective quantities, such as the homogenized coefficients. Computing homogenized coefficients requires solving local corrector problems followed by upscaling relevant local data. The most naive way of computing homogenized coefficients is by solving a local elliptic problem, which is known to suffer from the so-called resonance error dominating all other errors inherent in multiscale computations. A far more efficient modelling strategy, based on adding an exponential correction term to the standard local elliptic problem, has recently been proved to result in exponentially decaying error bounds with respect to the size of the local geometry. The questions in relation with the accuracy and computational efficiency of this approach has been previously addressed in the context of periodic homogenization. The present article concerns the extension of mathematical and numerical study of this modified elliptic corrector problem to stochastic homogenization problems. In particular, we assume a stationary, ergodic micro-structure and i) establish the well-posedness of the corrector equation, ii) analyse the bias (or the systematic error) originating from additional exponential correction term in the model. Numerical results corroborating our theoretical findings are presented.

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Computing homogenized coefficientsviamultiscale representation and hierarchical hybrid grids

Cited by 5 publications

References 60 publications

An iterative method for elliptic problems with rapidly oscillating coefficients

An iterative method for elliptic problems with rapidly oscillating coefficients

Optimal Artificial Boundary Condition for Random Elliptic Media

Analytical and numerical study of a modified cell problem for the numerical homogenization of multiscale random fields

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