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Adaptive Mixed GMsFEM for Flows in Heterogeneous Media

Part of: Partial differential equations, boundary value problems

Published online by Cambridge University Press:  17 November 2016

Ho Yuen Chan ,
Eric Chung  and
Yalchin Efendiev
Ho Yuen Chan*
Affiliation:
Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong SAR
Eric Chung*
Affiliation:
Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong SAR
Yalchin Efendiev*
Affiliation:
Department of Mathematics and Institute for Scientific Computation (ISC), Texas A&M University, College Station, Texas 77843-3368, USA
*
*Corresponding author. Email addresses:hychan@math.cuhk.edu.hk (H. Y. Chan), tschung@math.cuhk.edu.hk (E. Chung), efendiev@math.tamu.edu (Y. Efendiev)
*Corresponding author. Email addresses:hychan@math.cuhk.edu.hk (H. Y. Chan), tschung@math.cuhk.edu.hk (E. Chung), efendiev@math.tamu.edu (Y. Efendiev)
*Corresponding author. Email addresses:hychan@math.cuhk.edu.hk (H. Y. Chan), tschung@math.cuhk.edu.hk (E. Chung), efendiev@math.tamu.edu (Y. Efendiev)
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Abstract

In this paper, we present two adaptive methods for the basis enrichment of the mixed Generalized Multiscale Finite Element Method (GMsFEM) for solving the flow problem in heterogeneous media. We develop an a-posteriori error indicator which depends on the norm of a local residual operator. Based on this indicator, we construct an offline adaptive method to increase the number of basis functions locally in coarse regions with large local residuals. We also develop an online adaptive method which iteratively enriches the function space by adding new functions computed based on the residual of the previous solution and special minimum energy snapshots. We show theoretically and numerically the convergence of the two methods. The online method is, in general, better than the offline method as the online method is able to capture distant effects (at a cost of online computations), and both methods have faster convergence than a uniform enrichment. Analysis shows that the online method should start with a certain number of initial basis functions in order to have the best performance. The numerical results confirm this and show further that with correct selection of initial basis functions, the convergence of the online method can be independent of the contrast of the medium. We consider cases with both very high and very low conducting inclusions and channels in our numerical experiments.

Keywords

Mixed multiscale finite element methodsmultiscale basisadaptivityonline basisflow in heterogeneous media

MSC classification

Secondary: 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65N12: Stability and convergence of numerical methods
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 9 , Issue 4 , November 2016 , pp. 497 - 527
Copyright
Copyright © Global-Science Press 2016 

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