Computing homogenized coefficients via multiscale representation and hierarchical hybrid grids. (English) Zbl 1481.65255

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

ESAIM, Math. Model. Numer. Anal. 55, Suppl., 149-185 (2021).

Summary: 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

•  Wikipedia
•  Wolfram MathWorld

, 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.

MSC:

65N55	Multigrid methods; domain decomposition for boundary value problems involving PDEs
65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35B27	Homogenization in context of PDEs; PDEs in media with periodic structure

Keywords:

homogenization; multiscale method; hierarchical hybrid grids

References:

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