New hybridized mixed methods for linear elasticity and optimal multilevel solvers

Shihua Gong, Shuonan Wu, Jinchao Xu

Research output: Contribution to journalArticle

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 Pk + 1- Pk 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≥ n. For the lower order case (n- 2 ≤ 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.

Original languageEnglish (US)
Pages (from-to)569-604
Number of pages36
JournalNumerische Mathematik
Volume141
Issue number2
DOIs
StatePublished - Feb 13 2019

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Mixed Methods
Linear Elasticity
Elasticity
Grid
Local Approximation
Schur Complement
Poisson's Ratio
Lagrange multipliers
Mixed Finite Element Method
Poisson ratio
Order of Convergence
Stability and Convergence
Numerical Experiment
Finite element method
Approximation
Vertex of a graph
Experiments

All Science Journal Classification (ASJC) codes

  • Computational Mathematics
  • Applied Mathematics

Cite this

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New hybridized mixed methods for linear elasticity and optimal multilevel solvers. / Gong, Shihua; Wu, Shuonan; Xu, Jinchao.

In: Numerische Mathematik, Vol. 141, No. 2, 13.02.2019, p. 569-604.

Research output: Contribution to journalArticle

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