Localization of elliptic multiscale problems
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- by Axel Målqvist and Daniel Peterseim PDF
- Math. Comp. 83 (2014), 2583-2603 Request permission
Abstract:
This paper constructs a local generalized finite element basis for elliptic problems with heterogeneous and highly varying coefficients. The basis functions are solutions of local problems on vertex patches. The error of the corresponding generalized finite element method decays exponentially with respect to the number of layers of elements in the patches. Hence, on a uniform mesh of size $H$, patches of diameter $H\log (1/H)$ are sufficient to preserve a linear rate of convergence in $H$ without pre-asymptotic or resonance effects. The analysis does not rely on regularity of the solution or scale separation in the coefficient. This result motivates new and justifies old classes of variational multiscale methods.References
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Additional Information
- Axel Målqvist
- Affiliation: Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, Chalmers Tvärgata 3, SE-14296 Göteborg, Sweden
- Email: axel@chalmers.se
- Daniel Peterseim
- Affiliation: Rheinische Friedrich-Wilhelms-Universität Bonn, Institute for Numerical Simulation, Wegelerstr. 6, 53115 Bonn, Germany
- MR Author ID: 848711
- Email: peterseim@ins.uni-bonn.de
- Received by editor(s): October 4, 2011
- Received by editor(s) in revised form: March 22, 2012, and October 18, 2012
- Published electronically: June 16, 2014
- Additional Notes: The first author was supported by The Göran Gustafsson Foundation and The Swedish Research Council.
The second author was supported by the Humboldt-Universtät zu Berlin and the DFG Research Center Matheon Berlin through project C33. - © Copyright 2014 American Mathematical Society
- Journal: Math. Comp. 83 (2014), 2583-2603
- MSC (2010): Primary 65N12, 65N30
- DOI: https://doi.org/10.1090/S0025-5718-2014-02868-8
- MathSciNet review: 3246801