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An Online Generalized Multiscale Discontinuous Galerkin Method (GMsDGM) for Flows in Heterogeneous Media

Part of: Partial differential equations, boundary value problems

Published online by Cambridge University Press:  07 February 2017

Eric T. Chung ,
Yalchin Efendiev  and
Wing Tat Leung
Eric T. Chung*
Affiliation:
Department of Mathematics, The Chinese University of Hong Kong, Hong Kong
Yalchin Efendiev*
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA Numerical Porous Media SRI Center, King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Kingdom of Saudi Arabia
Wing Tat Leung*
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA
*
*Corresponding author.Email addresses:tschung@math.cuhk.edu.hk (E. T. Chung), efendiev@math.tamu.edu (Y. Efendiev), leungwt@math.tamu.edu (W. T. Leung)
*Corresponding author.Email addresses:tschung@math.cuhk.edu.hk (E. T. Chung), efendiev@math.tamu.edu (Y. Efendiev), leungwt@math.tamu.edu (W. T. Leung)
*Corresponding author.Email addresses:tschung@math.cuhk.edu.hk (E. T. Chung), efendiev@math.tamu.edu (Y. Efendiev), leungwt@math.tamu.edu (W. T. Leung)
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Abstract

Offline computation is an essential component in most multiscale model reduction techniques. However, there are multiscale problems in which offline procedure is insufficient to give accurate representations of solutions, due to the fact that offline computations are typically performed locally and global information is missing in these offline information. To tackle this difficulty, we develop an online local adaptivity technique for local multiscale model reduction problems. We design new online basis functions within Discontinuous Galerkin method based on local residuals and some optimally estimates. The resulting basis functions are able to capture the solution efficiently and accurately, and are added to the approximation iteratively. Moreover, we show that the iterative procedure is convergent with a rate independent of physical scales if the initial space is chosen carefully. Our analysis also gives a guideline on how to choose the initial space. We present some numerical examples to show the performance of the proposed method.

Keywords

Multiscale methoddiscontinuous Galerkin methodonline basis functionsheterogeneous media

MSC classification

Secondary: 65N12: Stability and convergence of numerical methods 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Type
Research Article
Information
Communications in Computational Physics , Volume 21 , Issue 2 , February 2017 , pp. 401 - 422
Copyright
Copyright © Global-Science Press 2017 

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