Corpus ID: 211677516

Least-squares for linear elasticity eigenvalue problem

@article{Bertrand2020LeastsquaresFL,
  title={Least-squares for linear elasticity eigenvalue problem},
  author={Fleurianne Bertrand and Daniele Boffi},
  journal={ArXiv},
  year={2020},
  volume={abs/2003.00449}
}
We study the approximation of the spectrum of least-squares operators arising from linear elasticity. We consider a two-field (stress/displacement) and a three-field (stress/displacement/vorticity) formulation; other formulations might be analyzed with similar techniques. We prove a priori estimates and we confirm the theoretical results with simple two-dimensional numerical experiments. 

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References

SHOWING 1-9 OF 9 REFERENCES
First order least-squares formulations for eigenvalue problems
In this paper we discuss spectral properties of operators associated with the least-squares finite-element approximation of elliptic partial differential equations. The convergence of the discreteExpand
Eigenvalues of Block Matrices Arising from Problems in Fluid Mechanics
Block matrices with a special structure arise from mixed finite element discretizations of incompressible flow problems. This paper is concerned with an analysis of the eigenvalue problem for suchExpand
Least-Squares Methods for Elasticity and Stokes Equations with Weakly Imposed Symmetry
TLDR
Two least-squares methods for the numerical solution of linear elasticity and Stokes equations in both two and three dimensions are developed and analyzes and an optimal L2{L^{2}} norm error estimates for the displacement/velocity are established. Expand
Arnold-Winther Mixed Finite Elements for Stokes Eigenvalue Problems
TLDR
With the help of the local post-processing, a reliable a posteriori error estimator is derived which is shown to be empirically efficient and confirms numerically the proven higher order convergence of the post-processed eigenvalues for convex domains with smooth eigenfunctions. Expand
Finite element approximation of eigenvalue problems
  • D. Boffi
  • Mathematics, Computer Science
  • Acta Numerica
  • 2010
TLDR
The final part tries to introduce the reader to the fascinating setting of differential forms and homological techniques with the description of the Hodge–Laplace eigenvalue problem and its mixed equivalent formulations. Expand
The L2 Norm Error Estimates for the Div Least-Squares Method
TLDR
This paper establishes optimal $L^2$ norm error estimates for the primal variable under the minimum regularity requirement through a refined duality argument. Expand
Quadrilateral H(div) Finite Elements
TLDR
This work considers the approximation properties of quadrilateral finite element spaces of vector fields defined by the Piola transform, and derives new estimates for approximation byquadrilateral Raviart--Thomas elements (requiring less regularity) and proposes a new quadrilaterally finite element space which provides optimal order approximation in H(div). Expand
SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems
The Scalable Library for Eigenvalue Problem Computations (SLEPc) is a software library for computing a few eigenvalues and associated eigenvectors of a large sparse matrix or matrix pencil. It hasExpand
Least-Squares Methods for Linear Elasticity
TLDR
It is shown that the homogeneous least-squares functional is elliptic and continuous in the H({\rm div};\,\Omega)^d \times H^1(\Omega]^d$ norm, which immediately implies optimal error estimates for finite element subspaces of the L2 norm. Expand