An Adaptive Least-Squares FEM for Linear Elasticity with Optimal Convergence Rates

Bringmann, Philipp; Carstensen, Carsten; Starke, Gerhard

doi:10.1137/16m1083797

Cited by 16 publications

(6 citation statements)

References 18 publications

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“…(c) Optimal convergence of adaptive least-square finite element methods driven by an equivalent weighted error estimators has been already proved for the Poisson problem in [CP15,Car20], the linear elasticity problem [BCS18], and the Stokes problem [BC17]. However, to the best of our knowledge, convergence for adaptive algorithms driven by the natural estimator is only known for the Poisson problem if D örfler marking with a sufficiently large bulk parameter is used, see [CPB17], where Q-linear convergence has been demonstrated.…”

Section: Theorem 33 (Convergence For (Pure) Homogeneous Dirichlet) Th...mentioning

confidence: 99%

Further results on a space-time FOSLS formulation of parabolic PDEs

Gantner,

Stevenson

2020

Preprint

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In [2019, Space-time least-squares finite elements for parabolic equations, arXiv:1911.01942] by F ührer& Karkulik, well-posedness of a space-time First-Order System Least-Squares formulation of the heat equation was proven. In the present work, this result is generalized to general second order parabolic PDEs with possibly inhomogenoeus boundary conditions, and plain convergence of a standard adaptive finite element method driven by the least-squares estimator is demonstrated. The proof of the latter easily extends to a large class of least-squares formulations.

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Section: Theorem 33 (Convergence For (Pure) Homogeneous Dirichlet) Th...mentioning

confidence: 99%

Further results on a space-time FOSLS formulation of parabolic PDEs

Gantner,

Stevenson

2020

Preprint

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“…The commonly used method is the LSM. (25) In the evaluation of the flatness error Δ, we determine the direction of the reference plane Γ by continuously iterating the normal vector of the reference plane Γ. The normal vector direction A, B, C of the reference plane Γ is the most important parameter, and the specific position parameter D of the reference plane Γ has no effect on the evaluation of the flatness error.…”

Section: Mathematical Model Of Flatnessmentioning

confidence: 99%

An Intelligent Vision Mobile Measurement Approach for High-speed Inspection of Large-size Flatness

Qiu¹,

Xiao²,

Wan³

et al. 2022

Sensors and Materials

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In current science and engineering, the demand for large-size surface detection has increased considerably. However, large-size surface detection presents some challenges, such as very large detection target area, discontinuous detection surface, and low detection accuracy. The current detection methods are mainly based on in-position detection and cannot meet the requirements of detection speed and accuracy for large-size surface detection. In this paper, we propose a fast detection method for large-scale flatness based on an intelligent vision mobile platform (IVMP). Specifically, by establishing the path optimization model, beam adjustment model, and largescale flatness calculation model for the IVMP, the binocular vision acquisition of large-scale target information and fast large-scale shape detection are realized. The rigidly fixed position relation of binocular vision is considered, and the parameters of the main camera can be obtained through an error equation, then the parameters of the assistant camera can be acquired quickly to calculate the three-dimensional coordinates of space points. The particle swarm optimization algorithm is integrated into the differential evolution algorithm to improve the detection speed. The IVMP is applied to the flatness detection of a satellite antenna. The experimental results show that the detection precision and efficiency of the IVMP are clearly higher than those of the laser tracking and theodolite systems.

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“…In contrast to this, only very little is known about convergence of adaptive LSFEM; see [12,8,14,15]. To the best of our knowledge, plain convergence of adaptive LSFEM when using the built-in error estimator has so far only been addressed in [15].…”

Section: Introductionmentioning

confidence: 99%

“…To the best of our knowledge, plain convergence of adaptive LSFEM when using the built-in error estimator has so far only been addressed in [15]. The other works [12,8,14] deal with optimal convergence results, but rely on alternative error estimators.…”

Section: Introductionmentioning

confidence: 99%

A short note on plain convergence of adaptive least-squares finite element methods

Führer,

Praetorius

2020

Preprint

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We show that adaptive least-squares finite element methods driven by the canonical least-squares functional converge under weak conditions on PDE operator, meshrefinement, and marking strategy. Contrary to prior works, our plain convergence does neither rely on sufficiently fine initial meshes nor on severe restrictions on marking parameters. Finally, we prove that convergence is still valid if a contractive iterative solver is used to obtain the approximate solutions (e.g., the preconditioned conjugate gradient method with optimal preconditioner). The results apply within a fairly abstract framework which covers a variety of model problems.

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An Adaptive Least-Squares FEM for Linear Elasticity with Optimal Convergence Rates

Cited by 16 publications

References 18 publications

Further results on a space-time FOSLS formulation of parabolic PDEs

Further results on a space-time FOSLS formulation of parabolic PDEs

An Intelligent Vision Mobile Measurement Approach for High-speed Inspection of Large-size Flatness

A short note on plain convergence of adaptive least-squares finite element methods

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