New results concerning the DWR method for some nonconforming FEM

An abstract property (H) is the key to a complete a priori error analysis in the (discrete) energy norm for several nonstandard finite element methods in the recent work [Lowest-order equivalent nonstandard finite element methods for biharmonic plates, Carstensen and Nataraj, M2AN, 2022]. This paper investigates the impact of (H) to the a posteriori error analysis and establishes known and novel explicit residualbased a posteriori error estimates. The abstract framework applies to Morley, two versions of discontinuous Galerkin, C 0 interior penalty, as well as weakly overpenalized symmetric interior penalty schemes for the biharmonic equation with a general source term in H −2(Ω).

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“…This subsection motivates the abstract a posteriori error analysis by a recollection [10,11,14,19,20,29,41] for = 1 and [2,13,35,36]…”

Section: Crouzeix-raviart and Morley Femmentioning

confidence: 99%

Unifying a posteriori error analysis of five piecewise quadratic discretisations for the biharmonic equation

Carstensen

Gräßle

Nataraj

2023

Journal of Numerical Mathematics

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A Priori and a Posteriori Error Analysis of the Crouzeix–Raviart and Morley FEM with Original and Modified Right-Hand Sides

Carstensen

Nataraj

2021

Computational Methods in Applied Mathematics

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This article on nonconforming schemes for m harmonic problems simultaneously treats the Crouzeix–Raviart ( m = 1 {m=1} ) and the Morley finite elements ( m = 2 {m=2} ) for the original and for modified right-hand side F in the dual space V * := H - m ⁢ ( Ω ) {V^{*}:=H^{-m}(\Omega)} to the energy space V := H 0 m ⁢ ( Ω ) {V:=H^{m}_{0}(\Omega)} . The smoother J : V nc → V {J:V_{\mathrm{nc}}\to V} in this paper is a companion operator, that is a linear and bounded right-inverse to the nonconforming interpolation operator I nc : V → V nc {I_{\mathrm{nc}}:V\to V_{\mathrm{nc}}} , and modifies the discrete right-hand side F h := F ∘ J ∈ V nc * {F_{h}:=F\circ J\in V_{\mathrm{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 F ∈ V * {F\in V^{*}} . 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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New results concerning the DWR method for some nonconforming FEM

Cited by 2 publications

References 12 publications

Unifying a posteriori error analysis of five piecewise quadratic discretisations for the biharmonic equation

Unifying a posteriori error analysis of five piecewise quadratic discretisations for the biharmonic equation

A Priori and a Posteriori Error Analysis of the Crouzeix–Raviart and Morley FEM with Original and Modified Right-Hand Sides

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