# An offline-online strategy for multiscale problems with random defects

Målqvist, Axel; Verfürth, Barbara

##### Abstract:
In this paper, we propose an offline-online strategy based on the Localized Orthogonal Decomposition (LOD) method for elliptic multiscale problems with randomly perturbed diffusion coefficient. We consider a periodic deterministic coefficient with local defects that occur with probability $p$. The offline phase pre-computes entries to global LOD stiffness matrices on a single reference element (exploiting the periodicity) for a selection of defect configurations. Given a sample of the perturbed diffusion the corresponding LOD stiffness matrix is then computed by taking linear combinations of the pre-computed entries, in the online phase. Our computable error estimates show that this yields a good coarse-scale approximation of the solution for small $p$. Moreover, extensive numerical experiments illustrate that relative errors of a few percent are achieved up to at least $p = 0.1$. This makes the proposed technique attractive already for moderate sample sizes in a Monte Carlo simulation.

 Zugehörige Institution(en) am KIT Institut für Angewandte und Numerische Mathematik (IANM)Sonderforschungsbereich 1173 (SFB 1173) Publikationstyp Forschungsbericht/Preprint Publikationsmonat/-jahr 02.2021 Sprache Englisch Identifikator ISSN: 2365-662X KITopen-ID: 1000129328 Verlag Karlsruher Institut für Technologie (KIT) Umfang 22 S. Serie CRC 1173 Preprint ; 2021/7 Projektinformation SFB 1173/2 (DFG, DFG KOORD, SFB 1173/2 2019) Externe Relationen Siehe auch Schlagwörter numerical homogenization, multiscale method, finite elements, random perturbations
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