Adaptive Morley FEM for the von Kármán equations with optimal convergence rates

Carstensen, Carsten; Nataraj, Neela

doi:10.48550/arxiv.1908.08013

Cited by 2 publications

(3 citation statements)

References 29 publications

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“…This is critical in eigenvalue analysis or problems with low-order terms; for e.g. in [15,18]. The 2 orthogonality in Lemma 2.2.e also allows a direct proof of Theorem 2.3 that circumvents the a posteriori error analysis of the consistency term as part of the medius analysis [27].…”

Section: (A)mentioning

confidence: 99%

Lowest-order equivalent nonstandard finite element methods for biharmonic plates

Carstensen¹,

Nataraj²

2021

Preprint

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The popular (piecewise) quadratic schemes for the biharmonic equation based on triangles are the nonconforming Morley finite element, the discontinuous Galerkin, the 0 interior penalty, and the WOPSIP schemes. Those methods are modified in their right-hand side ∈ −2 (Ω) replaced by • ( M ) and then are quasi-optimal in their respective discrete norms. The smoother M is defined for a piecewise smooth input function by a (generalized) Morley interpolation M followed by a companion operator . An abstract framework for the error analysis in the energy, weaker and piecewise Sobolev norms for the schemes is outlined and applied to the biharmonic equation. Three errors are also equivalent in some particular discrete norm from [Carstensen, Gallistl, Nataraj: Comparison results of nonstandard 2 finite element methods for the biharmonic problem, ESAIM Math. Model. Numer. Anal. (2015)] without data oscillations. This paper extends the work [Veeser, Zanotti: Quasioptimal nonconforming methods for symmetric elliptic problems, SIAM J. Numer. Anal. 56 (2018)] to the discontinuous Galerkin scheme and adds error estimates in weaker and piecewise Sobolev norms.

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Section: (A)mentioning

confidence: 99%

Lowest-order equivalent nonstandard finite element methods for biharmonic plates

Carstensen¹,

Nataraj²

2021

Preprint

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“…This is critical e.g. in eigenvalue analysis or for problems with low-order terms [22,26]. The proof of the bestapproximation of Theorem 3.2 for the modified scheme (3.9), however, does not require the 2 orthogonality in (2.9).…”

Section: Companion Operatormentioning

confidence: 99%

“…On the other hand, the efficiency requires some other additional benefit of the companion operator, which actually motivated its first design in a posteriori error control. Rate-optimal adaptive nonconforming FEM are analyzed in [4,14,15,22,25,47] and the references therein. Amongst all second-order schemes, the Morley FEM appears to be the most simple and method of choice for fourth-order problems [21].…”

Section: Introductionmentioning

confidence: 99%

A priori and a posteriori error analysis of the Crouzeix-Raviart and Morley FEM with original and modified righthand sides

Carstensen¹,

Nataraj²

2021

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This article on nonconforming schemes for harmonic problems simultaneously treats the Crouzeix-Raviart ( = 1) and the Morley finite elements ( = 2) for the original and for modified right-hand side in the dual space * := − (Ω) to the energy space := 0 (Ω). The smoother : nc → in this paper is a companion operator, that is a linear and bounded right-inverse to the nonconforming interpolation operator nc : → nc , and modifies the discrete right-hand side ℎ := • ∈ * nc . The best-approximation property of the modified scheme from Veeser et al. ( 2018) is recovered and complemented with an analysis of the convergence rates in weaker Sobolev norms. Examples with oscillating data show that the original method may fail to enjoy the best-approximation property but can also be better than the modified scheme. The a posteriori analysis of this paper concerns data oscillations of various types in a class of right-hand sides ∈ * . The reliable error estimates involve explicit constants and can be recommended for explicit error control of the piecewise energy norm. The efficiency follows solely up to data oscillations and examples illustrate this can be problematic.

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Adaptive Morley FEM for the von Kármán equations with optimal convergence rates

Cited by 2 publications

References 29 publications

Lowest-order equivalent nonstandard finite element methods for biharmonic plates

Lowest-order equivalent nonstandard finite element methods for biharmonic plates

A priori and a posteriori error analysis of the Crouzeix-Raviart and Morley FEM with original and modified righthand sides

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