Domain Decomposition Preconditioners for Mixed Finite-Element Discretization of High-Contrast Elliptic Problems

doi:10.1007/s42967-019-0005-z

Communications on Applied Mathematics and Computation ›› 2019, Vol. 1 ›› Issue (1): 141-165.doi: 10.1007/s42967-019-0005-z

• ORIGINAL PAPERS • 上一篇

Domain Decomposition Preconditioners for Mixed Finite-Element Discretization of High-Contrast Elliptic Problems

Hui Xie¹, Xuejun Xu^2,3

1. Institute of Applied Physics and Computational Mathematics, Beijing 100094, China;
2. LSEC, Institute of Computational Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, P. O. Box 2719, Beijing 100190, China;
3. School of Mathematical Sciences, Tongji University, Shanghai 200092, China

收稿日期:2018-06-12 修回日期:2018-11-15 出版日期:2019-03-20 发布日期:2019-05-11
通讯作者: Xuejun Xu, Hui Xie E-mail:xxj@lsec.cc.ac.cn;xiehui@lsec.cc.ac.cn
基金资助:
The authors would like to thank the editor and anonymous referees who made many helpful comments and suggestions which lead to an improved presentation of this paper.The work of the second author was supported by the National Natural Science Foundation of China (No.11671302).

Domain Decomposition Preconditioners for Mixed Finite-Element Discretization of High-Contrast Elliptic Problems

Hui Xie¹, Xuejun Xu^2,3

1. Institute of Applied Physics and Computational Mathematics, Beijing 100094, China;
2. LSEC, Institute of Computational Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, P. O. Box 2719, Beijing 100190, China;
3. School of Mathematical Sciences, Tongji University, Shanghai 200092, China

Received:2018-06-12 Revised:2018-11-15 Online:2019-03-20 Published:2019-05-11
Contact: Xuejun Xu, Hui Xie E-mail:xxj@lsec.cc.ac.cn;xiehui@lsec.cc.ac.cn
Supported by:
The authors would like to thank the editor and anonymous referees who made many helpful comments and suggestions which lead to an improved presentation of this paper.The work of the second author was supported by the National Natural Science Foundation of China (No.11671302).

摘要/Abstract

摘要： In this paper, we design an efficient domain decomposition (DD) preconditioner for the saddle-point problem resulting from the mixed finite-element discretization of multiscale elliptic problems. By proper equivalent algebraic operations, the original saddle-point system can be transformed to another saddle-point system which can be preconditioned by a block-diagonal matrix efficiently. Actually, the first block of this block-diagonal matrix corresponds to a multiscale H(div) problem, and thus, the direct inverse of this block is unpractical and unstable for the large-scale problem. To remedy this issue, a two-level overlapping DD preconditioner is proposed for this H(div) problem. Our coarse space consists of some velocities obtained from mixed formulation of local eigenvalue problems on the coarse edge patches multiplied by the partition of unity functions and the trivial coarse basis (e.g., Raviart-Thomas element) on the coarse grid. The condition number of our preconditioned DD method for this multiscale H(div) system is bounded by C(1 + (H²/δ²))(1 + log⁴(H/h)), where δ denotes the width of overlapping region, and H, h are the typical sizes of the subdomain and fine mesh. Numerical examples are presented to confirm the validity and robustness of our DD preconditioner.

关键词: High contrast, Mixed FEM, DD preconditioner, Spectral coarse space

Abstract: In this paper, we design an efficient domain decomposition (DD) preconditioner for the saddle-point problem resulting from the mixed finite-element discretization of multiscale elliptic problems. By proper equivalent algebraic operations, the original saddle-point system can be transformed to another saddle-point system which can be preconditioned by a block-diagonal matrix efficiently. Actually, the first block of this block-diagonal matrix corresponds to a multiscale H(div) problem, and thus, the direct inverse of this block is unpractical and unstable for the large-scale problem. To remedy this issue, a two-level overlapping DD preconditioner is proposed for this H(div) problem. Our coarse space consists of some velocities obtained from mixed formulation of local eigenvalue problems on the coarse edge patches multiplied by the partition of unity functions and the trivial coarse basis (e.g., Raviart-Thomas element) on the coarse grid. The condition number of our preconditioned DD method for this multiscale H(div) system is bounded by C(1 + (H²/δ²))(1 + log⁴(H/h)), where δ denotes the width of overlapping region, and H, h are the typical sizes of the subdomain and fine mesh. Numerical examples are presented to confirm the validity and robustness of our DD preconditioner.

Key words: High contrast, Mixed FEM, DD preconditioner, Spectral coarse space

中图分类号:

Hui Xie, Xuejun Xu. Domain Decomposition Preconditioners for Mixed Finite-Element Discretization of High-Contrast Elliptic Problems[J]. Communications on Applied Mathematics and Computation, 2019, 1(1): 141-165.

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Domain Decomposition Preconditioners for Mixed Finite-Element Discretization of High-Contrast Elliptic Problems

Domain Decomposition Preconditioners for Mixed Finite-Element Discretization of High-Contrast Elliptic Problems

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