An Adaptive Finite Element Scheme for the Hellinger–Reissner Elasticity Mixed Eigenvalue Problem
-
Fleurianne Bertrand
,
Daniele Boffi
und
Rui Ma
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.
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© 2021 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Sino–German Computational and Applied Mathematics
- An Adaptive Finite Element Scheme for the Hellinger–Reissner Elasticity Mixed Eigenvalue Problem
- Adaptive Directional Compression of High-Frequency Helmholtz Boundary Element Matrices
- Hierarchical Argyris Finite Element Method for Adaptive and Multigrid Algorithms
- On the Threshold Condition for Dörfler Marking
- An Unfitted dG Scheme for Coupled Bulk-Surface PDEs on Complex Geometries
- Dimensionally Consistent Preconditioning for Saddle-Point Problems
- An Optimal Multilevel Method with One Smoothing Step for the Morley Element
- Kernel Embedding Based Variational Approach for Low-Dimensional Approximation of Dynamical Systems
- 𝑯(curl 2)-Conforming Spectral Element Method for Quad-Curl Problems
- Defects in Active Nematics – Algorithms for Identification and Tracking
- Dual-Weighted Residual A Posteriori Error Estimates for a Penalized Phase-Field Slit Discontinuity Problem
- Fast Tensor Product Schwarz Smoothers for High-Order Discontinuous Galerkin Methods
- Error Analysis of an Unconditionally Energy Stable Local Discontinuous Galerkin Scheme for the Cahn–Hilliard Equation with Concentration-Dependent Mobility
Artikel in diesem Heft
- Frontmatter
- Sino–German Computational and Applied Mathematics
- An Adaptive Finite Element Scheme for the Hellinger–Reissner Elasticity Mixed Eigenvalue Problem
- Adaptive Directional Compression of High-Frequency Helmholtz Boundary Element Matrices
- Hierarchical Argyris Finite Element Method for Adaptive and Multigrid Algorithms
- On the Threshold Condition for Dörfler Marking
- An Unfitted dG Scheme for Coupled Bulk-Surface PDEs on Complex Geometries
- Dimensionally Consistent Preconditioning for Saddle-Point Problems
- An Optimal Multilevel Method with One Smoothing Step for the Morley Element
- Kernel Embedding Based Variational Approach for Low-Dimensional Approximation of Dynamical Systems
- 𝑯(curl 2)-Conforming Spectral Element Method for Quad-Curl Problems
- Defects in Active Nematics – Algorithms for Identification and Tracking
- Dual-Weighted Residual A Posteriori Error Estimates for a Penalized Phase-Field Slit Discontinuity Problem
- Fast Tensor Product Schwarz Smoothers for High-Order Discontinuous Galerkin Methods
- Error Analysis of an Unconditionally Energy Stable Local Discontinuous Galerkin Scheme for the Cahn–Hilliard Equation with Concentration-Dependent Mobility
