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Removing the cell resonance error in the multiscale finite element method via a Petrov-Galerkin formulation

Hou, Thomas Y. and Wu, Xiao-Hui and Zhang, Yu (2004) Removing the cell resonance error in the multiscale finite element method via a Petrov-Galerkin formulation. Communications in Mathematical Sciences, 2 (2). pp. 185-205. ISSN 1539-6746.

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We continue the study of the nonconforming multiscale finite element method (Ms- FEM) introduced in 17, 14 for second order elliptic equations with highly oscillatory coefficients. The main difficulty in MsFEM, as well as other numerical upscaling methods, is the scale resonance effect. It has been show that the leading order resonance error can be effectively removed by using an over-sampling technique. Nonetheless, there is still a secondary cell resonance error of O(Є^2/h^2). Here, we introduce a Petrov-Galerkin MsFEM formulation with nonconforming multiscale trial functions and linear test functions. We show that the cell resonance error is eliminated in this formulation and hence the convergence rate is greatly improved. Moreover, we show that a similar formulation can be used to enhance the convergence of an immersed-interface finite element method for elliptic interface problems.

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Zhang, Yu0000-0001-8938-1927
Additional Information:© 2004 International Press. Received: February 11, 2004; accepted (in revised version): March 17, 2004. Communicated by Shi Jin. This work is supported in part by NSF under the grant DMS-0073916 and ITR Grant ACI-0204932. The authors thank Prof. Yalchin Efendiev for helpful discussions on this work and Dr. Andrew Westhead for providing the report on his special MsFEM method.
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Subject Keywords:finite element, nonconforming, homogenization, resonance, interface
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Zentralblatt MATH identifier1085.65109
Mathematical Reviews number (MathSciNet) MR2119937
Issue or Number:2
Classification Code:AMS subject classifications: 65F10, 65F30
Record Number:CaltechAUTHORS:20111014-070953819
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Official Citation:Removing the Cell Resonance Error in the Multiscale Finite Element Method via a Petrov-Galerkin Formulation Thomas Y. Hou, Xiao-Hui Wu and Yu Zhang; 185-205
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:27219
Deposited By: Ruth Sustaita
Deposited On:14 Oct 2011 14:39
Last Modified:09 Mar 2020 13:19

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