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Erschienen in:
2017 | OriginalPaper | Buchkapitel
Direct and Iterative Methods for Numerical Homogenization
verfasst von : Ralf Kornhuber, Joscha Podlesny, Harry Yserentant
Erschienen in: Domain Decomposition Methods in Science and Engineering XXIII
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Abstract
Elliptic problems with oscillating coefficients can be approximated up to arbitrary accuracy by using sufficiently fine meshes, i.e., by resolving the fine scale. Well-known multiscale finite elements (Henning et al. ESAIM: Math. Model. Numer. Anal. 48:1331–1349, 2014; Målqvist and Peterseim, Math. Comput. 83:2583–2603, 2014) can be regarded as direct numerical homogenization methods in the sense that they provide approximations of the corresponding (unfeasibly) large linear systems by much smaller systems while preserving the fine-grid discretization accuracy (model reduction). As an alternative, we present iterative numerical homogenization methods that provide approximations up to fine-grid discretization accuracy and discuss differences and commonalities.