Spectral Analysis of a Mixed Method for Linear Elasticity
Abstract.
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 (\(k\geq 1\)), respectively. The numerical eigenfunction of stress is symmetric. By the discrete \(H^1\)-stability of numerical displacement, we prove an \(O(h^{k+2})\) approximation to the \(L^{2}\)-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 \(O(h^2)\) initial approximation (independent of the inverse of the elasticity operator) of the eigenvalue for the nonlinear eigenproblem by using the discrete \(H^1\)-stability of numerical displacement, while only an \(O(h)\) approximation can be obtained if we use the traditional inf-sup condition. Finally, we report some numerical experiments.
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© 2023 Society for Industrial and Applied Mathematics.
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Submitted: 23 March 2022
Accepted: 27 March 2023
Published online: 26 July 2023
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City University of Hong Kong (CityU): 11302219
Funding: The work of the authors was partially supported by the Research Grants Council of the Hong Kong Special Administrative Region, China project CityU 11302219.
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