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.

中文翻译：

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具有右侧和原始右侧的Crouzeix-Raviart和Morley有限元的先验和后验误差分析

本文针对m个调和问题的非协调方案同时针对原始和修改后的右手同时对待了Crouzeix-Raviart（m = 1 {m = 1}）和Morley有限元（m = 2 {m = 2}）双空间V中的F *：= H-m⁢（Ω）{V ^ {*}：= H ^ {-m}（\ Omega）}到能量空间V：= H 0 m⁢（Ω）{ V：= H ^ {m} _ {0}（\ Omega）}。本文中的平滑器J：V nc→V {J：V _ {\ mathrm {nc}} \ to V}是伴随算子，它是线性且有界，与不合格插值算子I nc：V→ V nc {I _ {\ mathrm {nc}}：V \至V _ {\ mathrm {nc}}}，并修改离散的右侧F h：= F∘J∈V nc * {F_ {h}： = F \ circ J \ in V _ {\ mathrm {nc}} ^ {**}}。Veeser等人的改进方案的最佳逼近性质。（2018）被恢复并补充了较弱的Sobolev规范中的收敛速度分析。带有振荡数据的示例表明，原始方法可能无法享受最佳逼近特性，但也可能比修改后的方案更好。本文的后验分析涉及一类右侧F∈V * {F \ in V ^ {*}}中各种类型的数据振荡。可靠的误差估计包括显式常数，可以建议将其用于分段能量范式的显式误差控制。效率仅取决于数据振荡，示例表明这可能是有问题的。本文的后验分析涉及一类右侧F∈V * {F \ in V ^ {*}}中各种类型的数据振荡。可靠的误差估计包括显式常数，可以建议将其用于分段能量范式的显式误差控制。效率仅取决于数据振荡，示例表明这可能是有问题的。本文的后验分析涉及一类右侧F∈V * {F \ in V ^ {*}}中各种类型的数据振荡。可靠的误差估计包括显式常数，可以建议将其用于分段能量范式的显式误差控制。效率仅取决于数据振荡，示例表明这可能是有问题的。