Reduced Basis Multiscale Finite Element Methods for Elliptic Problems

Jan S Hesthaven; Shun Zhang; Xueyu Zhu

doi:10.1137/140955070

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Reduced Basis Multiscale Finite Element Methods for Elliptic Problems

Journal article

Peer reviewed

Reduced Basis Multiscale Finite Element Methods for Elliptic Problems

Jan S Hesthaven, Shun Zhang and Xueyu Zhu

Multiscale modeling & simulation, Vol.13(1), pp.316-337

01/01/2015

DOI: 10.1137/140955070

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https://infoscience.epfl.ch/handle/20.500.14299/100273View

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Abstract

In this paper, we propose reduced basis multiscale finite element methods (RB-MsFEMs) for elliptic problems with highly oscillating coefficients. The method is based on MsFEMs with local test functions that encode the oscillatory behavior (see [G. Allaire and R. Brizzi, Multiscale Model. Simul., 4 (2005), pp. 790-812, J. S. Hesthaven, S. Zhang, and X. Zhu, Multiscale Model. Simul., 12 (2014), pp. 650-666]). For uniform rectangular meshes, the local oscillating test functions are represented by a reduced basis method (RBM), parameterizing the center of the elements. For triangular elements, we introduce a slightly different approach. By exploring oversampling of the oscillating test functions, initially introduced to recover a better approximation of the global harmonic coordinate map, we first build the reduced basis on uniform rectangular elements containing the original triangular elements and then restrict the oscillating test function to the triangular elements. These techniques are also generalized to the case where the coefficients dependent on additional independent parameters. The analysis of the proposed methods is supported by various numerical results, obtained on regular and unstructured grids.

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Details

Title: Subtitle: Reduced Basis Multiscale Finite Element Methods for Elliptic Problems
Creators: Jan S Hesthaven - Ecole Polytech Fed Lausanne, Math Inst Computat Sci & Engn MATHICSE, Chair Computat Math & Simulat Sci MCSS, CH-1015 Lausanne, Switzerland
Shun Zhang - City Univ Hong Kong, Dept Math, Kowloon, Hong Kong, Peoples R China
Xueyu Zhu
Resource Type: Journal article
Publication Details: Multiscale modeling & simulation, Vol.13(1), pp.316-337
DOI: 10.1137/140955070
ISSN: 1540-3459
eISSN: 1540-3467
Publisher: SIAM PUBLICATIONS
Number of pages: 22
Grant note: FA9550-09-1-0613 / OSD/AFOSR 11303914; CityU 9042090 / Research Grants Council of the Hong Kong SAR, China under GRF
Language: English
Date published: 01/01/2015
Academic Unit: Mathematics
Record Identifier: 9984240877902771

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