Generalized multiscale finite element methods for spacetime heterogeneous parabolic equations

Tsz Shun Eric CHUNG Yalchin Efendiev Wing Tat Leung Shuai Ye

Numerical Analysis and Scientific Computing mathscidoc:1910.43516

Computers & Mathematics with Applications, 76, (2), 419-437, 2018.7
In this paper, we consider local multiscale model reduction for problems with multiple scales in space and time. We developed our approaches within the framework of the Generalized Multiscale Finite Element Method (GMsFEM) using spacetime coarse cells. The main idea of GMsFEM is to construct a local snapshot space and a local spectral decomposition in the snapshot space. Previous research in developing multiscale spaces within GMsFEM focused on constructing multiscale spaces and relevant ingredients in space only. In this paper, our main objective is to develop a multiscale model reduction framework within GMsFEM that uses spacetime coarse cells. We construct spacetime snapshot and offline spaces. We compute these snapshot solutions by solving local problems. A complete snapshot space will use all possible boundary conditions; however, this can be very expensive. We propose using
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@inproceedings{tsz2018generalized,
  title={Generalized multiscale finite element methods for spacetime heterogeneous parabolic equations},
  author={Tsz Shun Eric CHUNG, Yalchin Efendiev, Wing Tat Leung, and Shuai Ye},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020181556741587045},
  booktitle={Computers & Mathematics with Applications},
  volume={76},
  number={2},
  pages={419-437},
  year={2018},
}
Tsz Shun Eric CHUNG, Yalchin Efendiev, Wing Tat Leung, and Shuai Ye. Generalized multiscale finite element methods for spacetime heterogeneous parabolic equations. 2018. Vol. 76. In Computers & Mathematics with Applications. pp.419-437. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020181556741587045.
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