The popular (piecewise) quadratic schemes for the biharmonic equation based on triangles are the nonconforming Morley finite element, the discontinuous Galerkin, the $C^0$ interior penalty, and the WOPSIP schemes. Those methods are modified in their right-hand side $F\in H^{-2}(\Omega)$ replaced by $F\circ (JI_{\rm M}) $ and then are quasi-optimal in their respective discrete norms. The smoother $JI_{\rm M}$ is defined for a piecewise smooth input function by a (generalized) Morley interpolation $I_{\rm M}$ followed by a companion operator $J$. 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 $P_2$ finite element methods for the biharmonic problem, ESAIM Math. Model. Numer. Anal. (2015)] without data oscillations. This paper extends the work [Veeser, Zanotti: Quasi-optimal 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.

翻译：以三角形为基础的双声调方程式的流行(假的)二次方程式是不兼容的 Morley 限制元素、不连续的 Galerkin 、 $C$0 内罚和 WOPSIIP 方案。 这些方法的右侧修改 $F\ in H ⁇ -2} (\ Omega), 由$F\circ (Ji ⁇ rm M}) 取代, 然后在其各自的离散规范中是准最佳的。 平滑的 $Jürm M} 被一个( 通用的) Morley 间调 $I ⁇ m} 平滑输入函数定义, 由伴机操作操作操作操作 $J$J$。 这些方法的错误分析的抽象框架, 较弱和小的 Soborbolevle 规范被概述并适用于双声方方方方方程式。 在某种特定的离散规范中, 与[Carstenticsen, Gall, Gall, Nataryalalal: non rudeal ral ralal rude rude rual rual-deal-deal-de sal romax.