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Fast auxiliary space preconditioners for linear elasticity in mixed form

Authors: Long Chen, Jun Hu and Xuehai Huang
Journal: Math. Comp. 87 (2018), 1601-1633
MSC (2010): Primary 65N55, 65F10, 65N22, 65N30
Published electronically: November 9, 2017
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Abstract: A block-diagonal preconditioner with the minimal residual method and an approximate block-factorization preconditioner with the generalized minimal residual method are developed for Hu-Zhang mixed finite element methods for linear elasticity. They are based on a new stability result for the saddle point system in mesh-dependent norms. The mesh-dependent norm for the stress corresponds to the mass matrix which is easy to invert while for the displacement it is spectral equivalent to the Schur complement. A fast auxiliary space preconditioner based on the $ H^1$-conforming linear element of the linear elasticity problem is then designed for solving the Schur complement. For both diagonal and triangular preconditioners, it is proved that the conditioning numbers of the preconditioned systems are bounded above by a constant independent of both the crucial Lamé constant and the mesh size. Numerical examples are presented to support theoretical results. As byproducts, a new stabilized low order mixed finite element method is proposed and analyzed and superconvergence results for the Hu-Zhang element are obtained.

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Additional Information

Long Chen
Affiliation: Department of Mathematics, University of California at Irvine, Irvine, California 92697 – and – Beijing Institute for Scientific and Engineering Computing, Beijing University of Technology, Beijing, 100124, People’s Republic of China

Jun Hu
Affiliation: LMAM and School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China

Xuehai Huang
Affiliation: College of Mathematics and Information Science, Wenzhou University, Wenzhou 325035, People’s Republic of China

Received by editor(s): April 8, 2016
Received by editor(s) in revised form: February 12, 2017
Published electronically: November 9, 2017
Additional Notes: The first author was supported by the National Science Foundation (NSF) DMS-1418934 and in part by the Sea Poly Project of Beijing Overseas Talents. This work was finished when the first author visited Peking University in the fall of 2015. He would like to thank Peking University for the support and hospitality, as well as for their exciting research atmosphere.
The second author was supported by the NSFC Projects 11625101, 91430213 and 11421101.
The third author is the corresponding author
The third author was supported by the NSFC Projects 11771338, 11301396 and 11671304, Zhejiang Provincial Natural Science Foundation of China Projects LY17A010010, LY15A010015 and LY15A010016, and Wenzhou Science and Technology Plan Project G20160019.
Article copyright: © Copyright 2017 American Mathematical Society

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