Multiscale Finite Element Coarse Spaces for the Analysis of Linear Elastic Composites

  • In this work we extend the multiscale finite element method (MsFEM) as formulated by Hou and Wu in [14] to the PDE system of linear elasticity. The application, motivated from the multiscale analysis of highly heterogeneous composite materials, is twofold. Resolving the heterogeneities on the finest scale, we utilize the linear MsFEM basis for the construction of robust coarse spaces in the context of two-level overlapping Domain Decomposition preconditioners. We motivate and explain the construction and present numerical results validating the approach. Under the assumption that the material jumps are isolated, that is they occur only in the interior of the coarse grid elements, our experiments show uniform convergence rates independent of the contrast in the Young's modulus within the heterogeneous material. Elsewise, if no restrictions on the position of the high coefficient inclusions are imposed, robustness can not be guaranteed any more. These results justify expectations to obtain coefficient-explicit condition number bounds for the PDE system of linear elasticity similar to existing ones for scalar elliptic PDEs as given in the work of Graham, Lechner and Scheichl [12]. Furthermore, we numerically observe the properties of the MsFEM coarse space for linear elasticity in an upscaling framework. Therefore, we present experimental results showing the approximation errors of the multiscale coarse space w.r.t. the fine-scale solution.

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Metadaten
Author:M. Buck, O. Iliev, H. Andrä
URN (permanent link):urn:nbn:de:hbz:386-kluedo-29884
Serie (Series number):Berichte des Fraunhofer-Instituts für Techno- und Wirtschaftsmathematik (ITWM Report) (212)
Document Type:Report
Language of publication:English
Publication Date:2012/04/18
Year of Publication:2012
Publishing Institute:Fraunhofer-Institut für Techno- und Wirtschaftsmathematik
Number of page:44
Faculties / Organisational entities:Fraunhofer (ITWM)
DDC-Cassification:510 Mathematik

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