Least-squares formulations for eigenvalue problems associated with linear elasticity

Bertrand, Fleurianne; Boffi, Daniele

doi:10.1016/j.camwa.2020.12.013

Cited by 11 publications

(19 citation statements)

References 11 publications

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“…In this paper we continue the discussion started in [4,5] about the spectrum of operators arising from least-squares finite element approximation of partial differential equations. In [4] several least-squares formulation associated with the Laplace problem and in [5] two formulations associated with the linear elasticity problem were considered. Here we continue the analysis of the two-field formulation presented in [5].…”

Section: Introductionmentioning

confidence: 77%

“…Applying the same strategy to (5) would lead to a non-linear problem. Following the original idea developed in [4] for the Laplace operator, in [5] the spectrum of the operators associated with the least-squares source formulation was considered.…”

Section: The Continuous Problemmentioning

confidence: 99%

“…This discrete eigenvalue problem has the same structure as in (9) (which is a typical mixed form structure) with the natural definition of matrices that are associated with the bilinear forms defined in (10). Looking at the problem from an algebraic point of view, the solution of this problem satisfies some properties that are discussed in [5,4]. In general, our discrete eigenvalue problem in (11) has the form of a generalized eigenvalue problem…”

Section: Galerkin Discretizationmentioning

confidence: 99%

“…In [4] several least-squares formulation associated with the Laplace problem and in [5] two formulations associated with the linear elasticity problem were considered. Here we continue the analysis of the two-field formulation presented in [5]. The two-field formulation was introduced in [7] for the approximation of the source problem and has the merit of providing a robust discretization also when the system approaches the incompressible limit.…”

Section: Introductionmentioning

confidence: 99%

“…The aim of this paper is two-fold. First, we complete the analysis presented in [5] where only a sketch of the main ideas were indicated. Actually, since the leastsquares formulation is not symmetric, the convergence analysis should take into account the dual problem for which we provide a careful description in this paper.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

On the spectrum of an operator associated with least-squares finite elements for linear elasticity

Alzaben¹,

Bertrand²,

Boffi³

2021

Preprint

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In this paper we provide some more details on the numerical analysis and we present some enlightening numerical results related to the spectrum of a finite element least-squares approximation of the linear elasticity formulation introduced recently. We show that, although the formulation is robust in the incompressible limit for the source problem, its spectrum is strongly dependent on the Lamé parameters and on the underlying mesh.

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Section: Introductionmentioning

confidence: 77%

Section: The Continuous Problemmentioning

confidence: 99%

Section: Galerkin Discretizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On the spectrum of an operator associated with least-squares finite elements for linear elasticity

Alzaben¹,

Bertrand²,

Boffi³

2021

Preprint

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A stabilized nonconforming Crouzeix–Raviart finite element method for the elasticity eigenvalue problem with the pure traction boundary condition

Zhang

Han

Yang

2023

Math Methods in App Sciences

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The linear elasticity problem has important applications in structural analysis and engineering design. It is shown that the classical Crouzeix–Raviart finite element method is unstable to solve the linear elasticity eigenvalue problem with the pure traction boundary condition. In this paper, first we propose a new scheme with the stabilization terms. Then we prove the a priori error estimates for approximate eigenpairs. Finally, we carry out numerical experiments using the stabilized scheme. Theoretical analysis and numerical results show that this scheme is locking free and efficient.

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Recent Advances in Least-Squares and Discontinuous Petrov–Galerkin Finite Element Methods

Bertrand¹,

Demkowicz²,

Gopalakrishnan³

2021

Computers & Mathematics with Applications

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Least-squares formulations for eigenvalue problems associated with linear elasticity

Cited by 11 publications

References 11 publications

On the spectrum of an operator associated with least-squares finite elements for linear elasticity

On the spectrum of an operator associated with least-squares finite elements for linear elasticity

A stabilized nonconforming Crouzeix–Raviart finite element method for the elasticity eigenvalue problem with the pure traction boundary condition

Recent Advances in Least-Squares and Discontinuous Petrov–Galerkin Finite Element Methods

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