Adaptive mixed GMsFEM for flows in heterogeneous media

Ho Yuen Chan Tsz Shun Eric CHUNG Yalchin Efendiev

Numerical Analysis and Scientific Computing mathscidoc:1910.43489

Numerical Mathematics: Theory, Methods and Applications, 9, (4), 497-527, 2016.11
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
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@inproceedings{ho2016adaptive,
  title={Adaptive mixed GMsFEM for flows in heterogeneous media},
  author={Ho Yuen Chan, Tsz Shun Eric CHUNG, and Yalchin Efendiev},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020180457231126018},
  booktitle={Numerical Mathematics: Theory, Methods and Applications},
  volume={9},
  number={4},
  pages={497-527},
  year={2016},
}
Ho Yuen Chan, Tsz Shun Eric CHUNG, and Yalchin Efendiev. Adaptive mixed GMsFEM for flows in heterogeneous media. 2016. Vol. 9. In Numerical Mathematics: Theory, Methods and Applications. pp.497-527. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020180457231126018.
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