Generalized Multiscale Finite Element Methods with energy minimizing oversampling

Chung, Eric; Efendiev, Yalchin; Leung, Wing Tat

doi:10.1002/nme.5958

Title: Generalized Multiscale Finite Element Methods with energy minimizing oversampling

Abstract

Summary In this paper, we propose a general concept for constructing multiscale basis functions within Generalized Multiscale Finite Element Method, which uses oversampling and stable decomposition. The oversampling refers to using larger regions in constructing multiscale basis functions and stable decomposition allows estimating the local errors. The analysis of multiscale methods involves decomposing the error by coarse regions, where each error contribution is estimated. In this estimate, we often use oversampling techniques to achieve a fast convergence. We demonstrate our concepts in the mixed, the Interior Penalty Discontinuous Galerkin, and Hybridized Discontinuous Galerkin discretizations. One of the important features of the proposed basis functions is that they can be used in online Generalized Multiscale Finite Element Method, where one constructs multiscale basis functions using residuals. In these problems, it is important to achieve a fast convergence, which can be guaranteed if we have a stable decomposition. In our numerical results, we present examples for both offline and online multiscale basis functions. Our numerical results show that one can achieve a fast convergence when using online basis functions. Moreover, we observe that coupling using Hybridized Discontinuous Galerkin provides a better accuracy compared with Interior Penalty Discontinuous Galerkin, which is due tomore » using multiscale glueing functions.« less

Authors:

^[1]; Efendiev, Yalchin ^[2]; Leung, Wing Tat ^[2]

Chinese Univ. of Hong Kong (Hong Kong)
Texas A & M Univ., College Station, TX (United States)

Publication Date:: Wed Sep 19 00:00:00 EDT 2018

Research Org.:: Texas A & M Univ., College Station, TX (United States)

Sponsoring Org.:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); National Science Foundation (NSF); Qatar National Research Fund; Russian Federation Government; Chinese University of Hong Kong

OSTI Identifier:: 1611502

Alternate Identifier(s):: OSTI ID: 1476939

Grant/Contract Number:: SC0010713; FG02-13ER26165; 1620318; 7-1482-1278; N 14.Y26.31.0013; 2017-18

Resource Type:: Accepted Manuscript

Journal Name:: International Journal for Numerical Methods in Engineering

Additional Journal Information:: Journal Volume: 117; Journal Issue: 3; Journal ID: ISSN 0029-5981

Publisher:: Wiley

Country of Publication:: United States

Language:: English

Subject:: 42 ENGINEERING; engineering; mathematics; discontinuous Galerkin; multiscale basis functions; multiscale problems; oversampling

Citation Formats


                    Chung, Eric, Efendiev, Yalchin, and Leung, Wing Tat. Generalized Multiscale Finite Element Methods with energy minimizing oversampling.  United States: N. p., 2018. 
Web.  doi:10.1002/nme.5958.

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                    Chung, Eric, Efendiev, Yalchin, & Leung, Wing Tat. Generalized Multiscale Finite Element Methods with energy minimizing oversampling.  United States.  https://doi.org/10.1002/nme.5958

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                    Chung, Eric, Efendiev, Yalchin, and Leung, Wing Tat. Wed .  
"Generalized Multiscale Finite Element Methods with energy minimizing oversampling".  United States.  https://doi.org/10.1002/nme.5958.  https://www.osti.gov/servlets/purl/1611502.

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@article{osti_1611502,

  title        = {Generalized Multiscale Finite Element Methods with energy minimizing oversampling},

  author       = {Chung, Eric and Efendiev, Yalchin and Leung, Wing Tat},

  abstractNote = {Summary In this paper, we propose a general concept for constructing multiscale basis functions within Generalized Multiscale Finite Element Method, which uses oversampling and stable decomposition. The oversampling refers to using larger regions in constructing multiscale basis functions and stable decomposition allows estimating the local errors. The analysis of multiscale methods involves decomposing the error by coarse regions, where each error contribution is estimated. In this estimate, we often use oversampling techniques to achieve a fast convergence. We demonstrate our concepts in the mixed, the Interior Penalty Discontinuous Galerkin, and Hybridized Discontinuous Galerkin discretizations. One of the important features of the proposed basis functions is that they can be used in online Generalized Multiscale Finite Element Method, where one constructs multiscale basis functions using residuals. In these problems, it is important to achieve a fast convergence, which can be guaranteed if we have a stable decomposition. In our numerical results, we present examples for both offline and online multiscale basis functions. Our numerical results show that one can achieve a fast convergence when using online basis functions. Moreover, we observe that coupling using Hybridized Discontinuous Galerkin provides a better accuracy compared with Interior Penalty Discontinuous Galerkin, which is due to using multiscale glueing functions.},

  doi          = {10.1002/nme.5958},

  journal      = {International Journal for Numerical Methods in Engineering},

  number       = 3,

  volume       = 117,

  place        = {United States},

  year         = {Wed Sep 19 00:00:00 EDT 2018},

  month        = {Wed Sep 19 00:00:00 EDT 2018}

}

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Title: Generalized Multiscale Finite Element Methods with energy minimizing oversampling

Abstract

Citation Formats

A mixed multiscale finite element method for elliptic problems with oscillating coefficients journal, June 2002

Generalized multiscale finite element method. Symmetric interior penalty coupling journal, December 2013

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A Multiscale HDG Method for Second Order Elliptic Equations. Part I. Polynomial and Homogenization-Based Multiscale Spaces journal, January 2015

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A discontinuous enrichment method for the finite element solution of high Péclet advection–diffusion problems journal, March 2009

A localized orthogonal decomposition method for semi-linear elliptic problems journal, August 2014

Mixed Generalized Multiscale Finite Element Methods and Applications journal, January 2015

Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media journal, May 1991

The variational multiscale method—a paradigm for computational mechanics journal, November 1998

Isogeometric variational multiscale modeling of wall-bounded turbulent flows with weakly enforced boundary conditions on unstretched meshes journal, February 2010

Optimal Local Approximation Spaces for Generalized Finite Element Methods with Application to Multiscale Problems preprint, January 2010

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A discontinuous enrichment method for the finite element solution of high Péclet advection–diffusion problems
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A multiscale HDG method for second order elliptic equations. Part I. Polynomial and homogenization-based multiscale spaces
preprint, January 2013

Mixed Generalized Multiscale Finite Element Methods and Applications
preprint, January 2014

Residual-driven online Generalized Multiscale Finite Element Methods
preprint, January 2015