<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" display="inline" overflow="scroll"><mml:mi>h</mml:mi></mml:math>-adaptive least-squares finite element methods for the 2D Stokes equations of any order with optimal convergence rates

Bringmann, Philipp; Carstensen, Carsten

doi:10.1016/j.camwa.2017.02.019

Cited by 11 publications

(3 citation statements)

References 22 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: 98%

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: 98%

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

Gantner,

Stevenson

2020

Preprint

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“…and define norms ⋅ ψ,t and ⋅ M,t in H ψ 0 (S, t) and H M 0 (S, t), respectively, as in (15). Finally, the (squared) re-scaled norm in the trial space is…”

Section: Robust Dpg Scheme For Large Domainsmentioning

confidence: 99%

“…Otherwise approximations suffer from long preasymptotic ranges of reduced convergence, a type of locking phenomenon. For a detailed analysis we refer to [27], and we note that a scaling of norms is required for least-squares methods as well, see [15,Section 3]. In order to not complicate the presentation, we restrain from providing all the details that are required to have a domain-robust approximation.…”

Section: Introductionmentioning

confidence: 99%

A DPG method for Reissner-Mindlin plates

Führer¹,

Heuer²,

Niemi³

2022

Preprint

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We present a discontinuous Petrov-Galerkin (DPG) method with optimal test functions for the Reissner-Mindlin plate bending model. Our method is based on a variational formulation that utilizes a Helmholtz decomposition of the shear force. It produces approximations of the primitive variables and the bending moments. For any canonical selection of boundary conditions the method converges quasi-optimally. In the case of hard-clamped convex plates, we prove that the lowest-order scheme is locking free. Several numerical experiments confirm our results.

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