Primer of Adaptive Finite Element Methods

Nochetto, Ricardo H.; Veeser, Andreas

doi:10.1007/978-3-642-24079-9_3

Cited by 65 publications

(68 citation statements)

References 50 publications

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“…Therefore, either the set of marked elements or the set of marked edges can be used as an input to the NVB-based mesh-refinement routine. Furthermore, NVB refinements lead to nested (Lagrange) finite element spaces (see [44, p.179])-an important ingredient in the proof of the contraction property for adaptive finite element approximations, see [44,Section 5] (note that nestedness is not guaranteed for other mesh-refinement techniques, such as red-green or red-green-blue refinements).…”

Section: Adaptive Finite Element Methods (Fem)mentioning

confidence: 99%

T-IFISS: a toolbox for adaptive FEM computation

Bespalov

Rocchi

Silvester

2021

Computers & Mathematics with Applications

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T-IFISS is a finite element software package for studying finite element solution algorithms for deterministic and parametric elliptic partial differential equations. The emphasis is on self-adaptive algorithms with rigorous error control using a variety of a posteriori error estimation techniques. The open-source MATLAB framework provides a computational laboratory for experimentation and exploration, enabling users to quickly develop new discretizations and test alternative algorithms. The package is also valuable as a teaching tool for students who want to learn about state-of-the-art finite element methodology. and is compatible with Windows, Linux and MacOS computers.3 We would almost certainly use an iterative solver preconditioned with an algebraic multigrid V-cycle if we were trying to solve the same PDE problem in three spatial dimensions. 4 The default uniform refinement in T-IFISS is by three bisections (see Figure 1(d)). However, there is an option to switch to the so-called red uniform refinement (i.e., the one obtained by connecting the edge midpoints of each triangle); this can be done by setting subdivPar = 1 within the function adiff adaptive main.m.

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Section: Adaptive Finite Element Methods (Fem)mentioning

confidence: 99%

T-IFISS: a toolbox for adaptive FEM computation

Bespalov

Rocchi

Silvester

2021

Computers & Mathematics with Applications

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“…We mention the recursive algorithms by Mitchell [24] for d = 2 and Kossaczky [21] for d = 3. We focus on the special case d = 2, and follow Binev, Dahmen, and DeVore [5] and Nochetto and Veeser [29], but the key Theorem 2 holds for any d ≥ 2 as shown by Stevenson [34]. We refer to Nochetto, Siebert, and Veeser [28] for a rather complete discussion for d ≥ 2.…”

Section: The Bisection Methodsmentioning

confidence: 99%

Why Adaptive Finite Element Methods Outperform Classical Ones

Nochetto

2011

Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)

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Adaptive finite element methods (AFEM) are a fundamental numerical tool in science and engineering. They are known to outperform classical FEM in practice and deliver optimal convergence rates when the latter cannot. This paper surveys recent progress in the theory of AFEM which explains their success and provides a solid mathematical framework for further developments.Mathematics Subject Classification (2010). Primary 65N30, 65N50, 65N15; Secondary 41A25.

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“…[14] proved the optimal cardinality of the AFEM using a decay between consecutive loops. A survey and an axiomatic presentation of the proof of optimal convergence rates for adaptive finite element methods can be found in [28,29,12]. For non-conforming elements, see [4].…”

Section: Introductionmentioning

confidence: 99%

Asymptotically Exact A Posteriori Error Analysis for the Mixed Laplace Eigenvalue Problem

Bertrand

Boffi

Stenberg

2019

Computational Methods in Applied Mathematics

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A posteriori error estimates for the mixed numerical approximation of the Laplace eigenvalue problem can be derived using a reconstruction in the standard H 0 1 ‐conforming space for the primal variable of the mixed problem. In the case of Raviart–Thomas finite elements of arbitrary polynomial degree the resulting error estimator constitutes a guaranteed upper bound for the error and is shown to be local efficient. This paper shows that the extension for the BDM‐element is not straightforward.

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Primer of Adaptive Finite Element Methods

Cited by 65 publications

References 50 publications

T-IFISS: a toolbox for adaptive FEM computation

T-IFISS: a toolbox for adaptive FEM computation

Why Adaptive Finite Element Methods Outperform Classical Ones

Asymptotically Exact A Posteriori Error Analysis for the Mixed Laplace Eigenvalue Problem

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