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Numerische Mathematik

, Volume 141, Issue 2, pp 569–604 | Cite as

New hybridized mixed methods for linear elasticity and optimal multilevel solvers

  • Shihua Gong
  • Shuonan Wu
  • Jinchao XuEmail author
Article
  • 107 Downloads

Abstract

In this paper, we present a family of new mixed finite element methods for linear elasticity for both spatial dimensions \(n=2,3\), which yields a conforming and strongly symmetric approximation for stress. Applying \(\mathcal {P}_{k+1}-\mathcal {P}_k\) as the local approximation for the stress and displacement, the mixed methods achieve the optimal order of convergence for both the stress and displacement when \(k \ge n\). For the lower order case \((n-2\le k<n)\), the stability and convergence still hold on some special grids. The proposed mixed methods are efficiently implemented by hybridization, which imposes the inter-element normal continuity of the stress by a Lagrange multiplier. Then, we develop and analyze multilevel solvers for the Schur complement of the hybridized system in the two dimensional case. Provided that no nearly singular vertex on the grids, the proposed solvers are proved to be uniformly convergent with respect to both the grid size and Poisson’s ratio. Numerical experiments are provided to validate our theoretical results.

Mathematics Subject Classification

65N30 65N55 

Notes

Acknowledgements

The work of the first and third authors was supported in part by National Natural Science Foundation of China (NSFC) (Grant Nos. 91430215, 41390452) and by Beijing International Center for Mathematical Research of Peking University, China. The work of the second and third authors was supported in part by the DOE Grant DE-SC0009249 as part of the Collaboratory on Mathematics for Mesoscopic Modeling of Materials and by DOE Grant DE-SC0014400 and NSF Grant DMS-1522615.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Beijing International Center for Mathematical ResearchPeking UniversityBeijingPeople’s Republic of China
  2. 2.Department of MathematicsPennsylvania State UniversityUniversity ParkUSA

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