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Duality-based adaptivity of model and discretization in multiscale finite-element methods

Maier, Matthias Sebastian

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Abstract

This thesis develops strategies for a posteriori error control of discretization and model errors, as well as adaptation strategies, in the context of multiscale finite-element methods. This is done within the general methodology of the DualWeighted Residual Method (DWR). In particular, a reformulation of the Heterogeneous Multiscale Method(HMM) as an abstract model-adaptation framework is introduced that explicitly decouples discretization and model parameters. Based on the framework a samplingadaptation strategy is proposed that allows for simultaneous control of discretization and model errors with the help of classical refinement strategies for mesh and sampling regions. Further, a model-adaptation approach is derived that interprets model adaptivity as a minimization problem of a local model-error indicator.

This allows for the formulation of an efficient post-processing strategy that lifts the requirement of strict a priori knowledge about applicability and quality of effective models. The proposed framework is tested on an elliptic model problem with heterogeneous coefficients, as well as on an advection-diffusion problem with dominant microscopic transport.

Dokumententyp: Dissertation
Erstgutachter: Rannacher, Prof. Dr. Rolf
Tag der Prüfung: 12 Juni 2015
Erstellungsdatum: 01 Jul. 2015 11:17
Erscheinungsjahr: 2015
Institute/Einrichtungen: Fakultät für Mathematik und Informatik > Institut für Mathematik
DDC-Sachgruppe: 510 Mathematik
Normierte Schlagwörter: Numerical Mathematics, Dual Weighted Residual Method, Multiscale Methods, Model Adaptivity
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