Spectral Analysis of a Mixed Method for Linear Elasticity

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Original languageEnglish
Pages (from-to)1885-1917
Journal / PublicationSIAM Journal on Numerical Analysis
Issue number4
Online published26 Jul 2023
Publication statusPublished - 2023



The purpose of this paper is to analyze a mixed method for the linear elasticity eigenvalue problem, which approximates numerically the stress, displacement, and rotation, by piecewise (k + 1), k, and (k + 1) th degree polynomial functions (k ≥ 1), respectively. The numerical eigenfunction of stress is symmetric. By the discrete H1-stability of numerical displacement, we prove an (hk+2) approximation to the L2-orthogonal projection of the eigenspace of exact displacement for the eigenvalue problem with a proper regularity assumption. Thus via postprocessing, we obtain a better approximation to the eigenspace of exact displacement for the eigenproblem than conventional methods. We also prove that numerical approximation to the eigenfunction of stress is locking free with respect to the Poisson ratio. We introduce a hybridization to reduce the mixed method to a condensed eigenproblem and prove an O (h2) initial approximation (independent of the inverse of the elasticity operator) of the eigenvalue for the nonlinear eigenproblem by using the discrete H1 -stability of numerical displacement, while only an (h) approximation can be obtained if we use the traditional inf-sup condition. Finally, we report some numerical experiments. © 2023 The Author(s).

Research Area(s)

  • linear elasticity, eigenvalue problem, error estimates, mixed methods

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