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Mathematics of Computation

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Localization of elliptic multiscale problems


Authors: Axel Målqvist and Daniel Peterseim
Journal: Math. Comp. 83 (2014), 2583-2603
MSC (2010): Primary 65N12, 65N30
DOI: https://doi.org/10.1090/S0025-5718-2014-02868-8
Published electronically: June 16, 2014
MathSciNet review: 3246801
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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.


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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
Email: peterseim@ins.uni-bonn.de

DOI: https://doi.org/10.1090/S0025-5718-2014-02868-8
Keywords: Finite element method, a priori error estimate, convergence, multiscale method
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.
Article copyright: © Copyright 2014 American Mathematical Society