Two-scale FEM for homogenization problems

Matache, Ana-Maria; Schwab, Christoph

doi:10.1051/m2an:2002025

Cited by 60 publications

(64 citation statements)

References 9 publications

(13 reference statements)

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“…After unfolding (Cioranescu et al, 2008), it gives rise to the product of the macroscopic physical domain and the periodic microscopic domain of the cell problem; see Matache (2002). For multiple scales, a general product appears here which still can be written as the product of two domains, one containing, for example, the macroscopic scale and the other consisting of the product of the domains of the different microscales (Hoang & Schwab, 2005).…”

Section: Introductionmentioning

confidence: 99%

Approximation of bi-variate functions: singular value decomposition versus sparse grids

Griebel

Harbrecht

2013

IMA Journal of Numerical Analysis

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We compare the cost complexities of two approximation schemes for functions f ∈ H p (Ω 1 × Ω 2 ) which live on the product domain Ω 1 × Ω 2 of sufficiently smooth domains Ω 1 ⊂ R n 1 and Ω 2 ⊂ R n 2 , namely the singular value/Karhunen-Lòeve decomposition and the sparse grid representation. Here, we assume that suitable finite element methods with associated fixed order r of accuracy are given on the domains Ω 1 and Ω 2 . Then, the sparse grid approximation essentially needs only O(ε −q ), with q = max{n 1 , n 2 }/r, unknowns to reach a prescribed accuracy ε, provided that the smoothness of f satisfies p r((n 1 + n 2 )/max{n 1 , n 2 }), which is an almost optimal rate. The singular value decomposition produces this rate only if f is analytical, since otherwise the decay of the singular values is not fast enough. If p < r((n 1 + n 2 )/max{n 1 , n 2 }), then the sparse grid approach gives essentially the rate O(ε −q ) with q = (n 1 + n 2 )/p, while, for the singular value decomposition, we can only prove the rate O(ε −q ) with q = (2 min{r, p} min{n 1 , n 2 } + 2p max{n 1 , n 2 })/(2p − min{n 1 , n 2 }) min{r, p}. We derive the resulting complexities, compare the two approaches and present numerical results which demonstrate that these rates are also achieved in numerical practice.

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Section: Introductionmentioning

confidence: 99%

Approximation of bi-variate functions: singular value decomposition versus sparse grids

Griebel

Harbrecht

2013

IMA Journal of Numerical Analysis

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“…[17,18,19,20,21], as well as for numerical simulations, see e.g. [22,23,24,25,26,27]. For applications of periodic homogenization in physics and engineering, we refer to e.g.…”

Section: Introductionmentioning

confidence: 99%

Error estimates for nonlinear reaction-diffusion systems involving different diffusion length scales

Reichelt

2016

J. Phys.: Conf. Ser.

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Abstract. We derive quantitative error estimates for coupled reaction-diffusion systems, whose coefficient functions are quasi-periodically oscillating modeling the microstructure of the underlying macroscopic domain. The coupling arises via nonlinear reaction terms and we allow for different diffusion length scales, i.e. whereas some species have characteristic diffusion length of order 1 other species may diffuse with the order of the characteristic microstructure-length scale. We consider an effective system, which is rigorously obtained via two-scale convergence, and we derive quantitative error estimates.

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“…We mention for example -methods that supplement oscillatory functions to a coarse FE space, pioneered by Babuška & Osborn [8], generalized through the so-called multiscale finite-element method (MsFEM) [9], developed since then by many authors (MsFEM using harmonic coordinates [10,11], see [12] for a survey and additional references), -methods based on the variational multiscale method (VMM) introduced in [13] and the residual free bubble (RFB) method [14] that are closely related to MsFEM-type strategy for homogenization problems [15], -methods based on the two-scale convergence theory and its generalization [3,16] as proposed in [17] and developed in [18] using sparse tensor product FEM, -projection-based numerical homogenization method based on projecting a fine-scale discretized problem into a low-dimensional space and eliminating successively the fine-scale components [19,20], and -numerical homogenization methods that supplement effective data for coarse FE computation and approximate the fine-scale solution via reconstruction such as the heterogeneous multiscale method (HMM) [21,22] or related micro-macro methods [23][24][25][26].…”

Section: Introductionmentioning

confidence: 99%

Numerical Homogenization

Abdulle¹

2015

Encyclopedia of Applied and Computational Mathematics

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One contribution of 13 to a Theme Issue 'Multi-scale systems in fluids and soft matter: approaches, numerics and applications' . A general framework to combine numerical homogenization and reduced-order modelling techniques for partial differential equations (PDEs) with multiple scales is described. Numerical homogenization methods are usually efficient to approximate the effective solution of PDEs with multiple scales. However, classical numerical homogenization techniques require the numerical solution of a large number of so-called microproblems to approximate the effective data at selected grid points of the computational domain. Such computations become particularly expensive for high-dimensional, time-dependent or nonlinear problems. In this paper, we explain how numerical homogenization method can benefit from reducedorder modelling techniques that allow one to identify offline and online computational procedures. The effective data are only computed accurately at a carefully selected number of grid points (offline stage) appropriately 'interpolated' in the online stage resulting in an online cost comparable to that of a single-scale solver. The methodology is presented for a class of PDEs with multiple scales, including elliptic, parabolic, wave and nonlinear problems. Numerical examples, including wave propagation in inhomogeneous media and solute transport in unsaturated porous media, illustrate the proposed method.

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Two-scale FEM for homogenization problems

Cited by 60 publications

References 9 publications

Approximation of bi-variate functions: singular value decomposition versus sparse grids

Approximation of bi-variate functions: singular value decomposition versus sparse grids

Error estimates for nonlinear reaction-diffusion systems involving different diffusion length scales

Numerical Homogenization

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