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<p>Lower and upper bounds for eigenvalues help estimate the location interval of eigenvalues, which is of practical meanings especially for those problems of which the eigenvalues cannot be exactly obtained. In this paper, we study the lower and upper bounds for linear elasticity eigenvalues by displacement-pressure mixed finite element schemes. By applying expansion identities for the error of eigenvalues, lower and upper numerically computable bounds for the eigenvalues are derived based on certain mathematical hypotheses. For the schemes studied here, roughly speaking, the accuracy loss of the local approximation of the discrete <b>displacement</b> may lead to lower bound and that of <b>pressure</b> to upper bound. By utilizing the min-max principle and perturbation theory for the solution operator, theoretical lower and upper bounds can be controlled by setting proper Lamé parameters.</p>

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Asymptotic Lower and Upper Bounds for Linear Elasticity Eigenvalues

  • 期刊论文

  • 发表日期：

    2026年2月15日

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  • Yifan Yue,
  • Hongtao Chen,
  • Shuo Zhang

摘要

Lower and upper bounds for eigenvalues help estimate the location interval of eigenvalues, which is of practical meanings especially for those problems of which the eigenvalues cannot be exactly obtained. In this paper, we study the lower and upper bounds for linear elasticity eigenvalues by displacement-pressure mixed finite element schemes. By applying expansion identities for the error of eigenvalues, lower and upper numerically computable bounds for the eigenvalues are derived based on certain mathematical hypotheses. For the schemes studied here, roughly speaking, the accuracy loss of the local approximation of the discrete displacement may lead to lower bound and that of pressure to upper bound. By utilizing the min-max principle and perturbation theory for the solution operator, theoretical lower and upper bounds can be controlled by setting proper Lamé parameters.