Analysis of the heterogeneous multiscale method for elliptic homogenization problems

By Pingbing Ming, Pingwen Zhang. 2004

340 citations recorded by the source corpus.

The heterogeneous multiscale method (HMM) is applied to various parabolic problems with multiscale coefficients. These problems can be either linear or nonlinear. Optimal estimates are proved for the error between the HMM solution and the homogenized solution.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/36f0512221d05031073d882e8edaf6cc05458a07

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In Computational Mechanics

In Computational Theory and Mathematics

In Mechanics of Materials

Citations in the corpus

Adaptive finite element heterogeneous multiscale method for homogenization problems (2011, methodology): “where vHlin;K‘ ðxÞ 1⁄4 vðxK‘ Þ H þ ðx xK‘ Þ rvðxK‘ Þ ð7Þ is a linearization of the macrofunction v at the integration point xK‘ (see [7,25] for details).”
Adaptive reduced basis finite element heterogeneous multiscale method (2013, background): “the modeling error rMOD can be bounded as follows [26, 29] rMOD = 0 if W (Kδj) = W 1 per(Kδj) and δ/ε ∈ N, 1 (2.”
An optimal variance estimate in stochastic homogenization of discrete elliptic equations (2011, background): “rete setting of i.i.d. coeﬃcients a(e) and for dimension d>2, his result translates into (1.16) h|∇φT −∇φ|2i.T(2−d)/(4+d)+δ, for T ≫1 and some arbitrarily small δ >0, see [21], Theorem 2.1 (and [5], Lemma A.5, for this rephrasing of Yurinskii’s result). Yurinskii derives estimate (1.16) by fai”
Heterogeneous Multiscale Methods: A Review (2007, result): “The mixing condition is stated in [64].”
Reduction of the resonance error---Part 1: Approximation of homogenized coefficients (2011, methodology): “In essence, numerical homogenization techniques aim at computing solutions to partial differential equations whose coefficients depend on a small parameter denoted by ε, without resolving all the details at the scale ε (see for instance [15, 16] for the multiscale finite element method, [6, 5] for t”