Abstract
In this paper, we study the approximation of eigenvalues arising from the mixed Hellinger–Reissner elasticity problem by using a simple finite element introduced recently by one of the authors. We prove that the method converges when a residual type error estimator is considered and that the estimator decays optimally with respect to the number of degrees of freedom. A postprocessing technique originally proposed in a different context is discussed and tested numerically.
Funding source: Deutsche Forschungsgemeinschaft
Award Identifier / Grant number: BE 6511/1-1
Funding source: Alexander von Humboldt-Stiftung
Award Identifier / Grant number: STA 402/14-1
Funding statement: The first author gratefully acknowledges support by the Deutsche Forschungsgemeinschaft in the Priority Program SPP 1748 Simulation Techniques in Solid Mechanics, Development of Non-standard Discretization Methods, Mechanical and Mathematical Analysis, under the project number BE 6511/1-1. The second author is a member of the INdAM Research group GNCS, and his research is partially supported by IMATI/CNR and by PRIN/MIUR. The research of the third author was supported by the Alexander von Humboldt Foundation through the Humboldt Research Fellowship for Postdoctoral Researchers and in the project Approximation and reconstruction of stresses in the deformed configuration for hyperelastic material models (STA 402/14-1) by the DFG via the priority program 1748 Reliable Simulation Techniques in Solid Mechanics, Development of Non-standard Discretization Methods, Mechanical and Mathematical Analysis.
References
[1] D. Boffi, D. Gallistl, F. Gardini and L. Gastaldi, Optimal convergence of adaptive FEM for eigenvalue clusters in mixed form, Math. Comp. 86 (2017), no. 307, 2213–2237. 10.1090/mcom/3212Search in Google Scholar
[2] D. Boffi and L. Gastaldi, Adaptive finite element method for the Maxwell eigenvalue problem, SIAM J. Numer. Anal. 57 (2019), no. 1, 478–494. 10.1137/18M1179389Search in Google Scholar
[3] D. Boffi, L. Gastaldi, R. Rodríguez and I. Šebestová, A posteriori error estimates for Maxwell’s eigenvalue problem, J. Sci. Comput. 78 (2019), no. 2, 1250–1271. 10.1007/s10915-018-0808-5Search in Google Scholar
[4] C. Carstensen, M. Feischl, M. Page and D. Praetorius, Axioms of adaptivity, Comput. Math. Appl. 67 (2014), no. 6, 1195–1253. 10.1016/j.camwa.2013.12.003Search in Google Scholar PubMed PubMed Central
[5] C. Carstensen, D. Gallistl and J. Gedicke, Residual-based a posteriori error analysis for symmetric mixed Arnold–Winther FEM, Numer. Math. 142 (2019), no. 2, 205–234. 10.1007/s00211-019-01029-7Search in Google Scholar
[6]
C. Carstensen, D. Gallistl and M. Schedensack,
[7] C. Carstensen and J. Hu, An extended Argyris finite element method with optimal standard adaptive and multigrid V-cycle algorithms, preprint (2019). Search in Google Scholar
[8] C. Carstensen and H. Rabus, Axioms of adaptivity with separate marking for data resolution, SIAM J. Numer. Anal. 55 (2017), no. 6, 2644–2665. 10.1137/16M1068050Search in Google Scholar
[9] L. Chen, J. Hu and X. Huang, Fast auxiliary space preconditioners for linear elasticity in mixed form, Math. Comp. 87 (2018), no. 312, 1601–1633. 10.1090/mcom/3285Search in Google Scholar
[10] L. Chen, J. Hu, X. Huang and H. Man, Residual-based a posteriori error estimates for symmetric conforming mixed finite elements for linear elasticity problems, Sci. China Math. 61 (2018), no. 6, 973–992. 10.1007/s11425-017-9181-2Search in Google Scholar
[11] J. Douglas, Jr. and J. E. Roberts, Mixed finite element methods for second order elliptic problems, Mat. Apl. Comput. 1 (1982), no. 1, 91–103. Search in Google Scholar
[12] R. G. Durán, L. Gastaldi and C. Padra, A posteriori error estimators for mixed approximations of eigenvalue problems, Math. Models Methods Appl. Sci. 9 (1999), no. 8, 1165–1178. 10.1142/S021820259900052XSearch in Google Scholar
[13] D. Gallistl, An optimal adaptive FEM for eigenvalue clusters, Numer. Math. 130 (2015), no. 3, 467–496. 10.1007/s00211-014-0671-8Search in Google Scholar
[14] J. Gedicke and A. Khan, Arnold-Winther mixed finite elements for Stokes eigenvalue problems, SIAM J. Sci. Comput. 40 (2018), no. 5, A3449–A3469. 10.1137/17M1162032Search in Google Scholar
[15] B. Gong, J. Han, J. Sun and Z. Zhang, A shifted-inverse adaptive multigrid method for the elastic eigenvalue problem, Commun. Comput. Phys. 27 (2020), no. 1, 251–273. 10.4208/cicp.OA-2018-0293Search in Google Scholar
[16]
J. Hu,
Finite element approximations of symmetric tensors on simplicial grids in
[17]
J. Hu and R. Ma,
Partial relaxation of
[18] J. Hu and S. Zhang, A family of conforming mixed finite elements for linear elasticity on triangular grids, preprint (2015), https://arxiv.org/abs/1406.7457v2. 10.1007/s11425-014-4953-5Search in Google Scholar
© 2021 Walter de Gruyter GmbH, Berlin/Boston
