Volume 56, Number 1, January-February 2022
|Page(s)||237 - 260|
|Published online||10 February 2022|
An offline-online strategy for multiscale problems with random defects
Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, 412 96 Göteborg, Sweden
2 Institut für Angewandte und Numerische Mathematik, Karlsruher Institut für Technologie, Englerstr. 2, 76131 Karlsruhe, Germany
* Corresponding author: email@example.com
Accepted: 10 January 2022
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 approximation of the solution for small p, which is illustrated by extensive numerical experiments. This makes the proposed technique attractive already for moderate sample sizes in a Monte Carlo simulation.
Mathematics Subject Classification: 65N30 / 65N12 / 65N15 / 35J15
Key words: Numerical homogenization / multiscale method / finite elements / random perturbations
© The authors. Published by EDP Sciences, SMAI 2022
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