An embedded corrector problem for homogenization. Part II: Algorithms and discretization

Cancès, Éric; Ehrlacher, Virginie; Legoll, Frédéric; Stamm, Benjamin; Xiang, Shuyang

doi:10.1016/j.jcp.2020.109254

Cited by 9 publications

(9 citation statements)

References 35 publications

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“…The numerical investigation of the stochastic homogenization problem attracts interest and becomes an active field of research, see the survey and references therein. Recently, the numerical solution of the corrector‐type problem, in the context of homogenization of the diffusion equation with spherical inclusions by using boundary element methods and the fast multipole techniques has been considered in Reference .…”

Section: Introductionmentioning

confidence: 99%

Numerical study in stochastic homogenization for elliptic partial differential equations: Convergence rate in the size of representative volume elements

Khoromskaia

Khoromskij

Otto

2020

Numerical Linear Algebra App

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We describe the numerical scheme for the discretization and solution of 2D elliptic equations with strongly varying piecewise constant coefficients arising in the stochastic homogenization of multiscale composite materials. An efficient stiffness matrix generation scheme based on assembling the local Kronecker product matrices is introduced. The resulting large linear systems of equations are solved by the preconditioned conjugate gradient iteration with a convergence rate that is independent of the grid size and the variation in jumping coefficients (contrast). Using this solver, we numerically investigate the convergence of the representative volume element (RVE) method in stochastic homogenization that extracts the effective behavior of the random coefficient field. Our numerical experiments confirm the asymptotic convergence rate of systematic error and standard deviation in the size of RVE rigorously established in Gloria et al. The asymptotic behavior of covariances of the homogenized matrix in the form of a quartic tensor is also studied numerically. Our approach allows laptop computation of sufficiently large number of stochastic realizations even for large sizes of the RVE.

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Section: Introductionmentioning

confidence: 99%

Numerical study in stochastic homogenization for elliptic partial differential equations: Convergence rate in the size of representative volume elements

Khoromskaia

Khoromskij

Otto

2020

Numerical Linear Algebra App

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“…where the deterministic constant coefficient a 0,T is defined in (12) , and a 0,L,T is obtained by replacing χ T,R in (10) by χ T solving (9) over R d , where the Dirichlet boundary conditions are (naturally) not imposed. These terms are denoted, respectively, by boundary, statistical and systematic errors.…”

Section: An Elliptic Problem With Zero-th Order Regularizationmentioning

confidence: 99%

“…Several methods to improve the decay of the boundary error in the random homogenization setting are currently available. For example, the "embedded" method proposed in [11,12] approximates the homogenized coefficients by solving a variational minimization problem over a domain where a cell with heterogeneous coefficients is embedded into a homogeneous environment. The study of the boundary error for the exponentially regularized corrector problem ( 13) is associated with the decay of the Green's function for the parabolic PDEs on bounded domains, which is reported in a previous paper of the authors, see [3].…”

Section: The Goals Of the Papermentioning

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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“…One alternative method for computing homogenized coefficients, based on the idea of an "embedded corrector problem", is proposed in [22,23]. Well-separated spherical inclusions are considered in the numerical examples.…”

Section: Introductionmentioning

confidence: 99%

Computing homogenized coefficientsviamultiscale representation and hierarchical hybrid grids

2021

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We present an efficient method for the computation of homogenized coefficients of divergence-form operators with random coefficients. The approach is based on a multiscale representation of the homogenized coefficients. We then implement the method numerically using a finite-element method with hierarchical hybrid grids, which is a semi-implicit method allowing for significant gains in memory usage and execution time. Finally, we demonstrate the efficiency of our approach on two-and three-dimensional examples, for piecewise-constant coefficients with corner discontinuities. For moderate ellipticity contrast and for a precision of a few percentage points, our method allows to compute the homogenized coefficients on a laptop computer in a few seconds, in two dimensions, or in a few minutes, in three dimensions.

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An embedded corrector problem for homogenization. Part II: Algorithms and discretization

Cited by 9 publications

References 35 publications

Numerical study in stochastic homogenization for elliptic partial differential equations: Convergence rate in the size of representative volume elements

Numerical study in stochastic homogenization for elliptic partial differential equations: Convergence rate in the size of representative volume elements

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

Computing homogenized coefficientsviamultiscale representation and hierarchical hybrid grids

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