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The popular (piecewise) quadratic schemes for the biharmonic equation basedon triangles are the nonconforming Morley finite element, the discontinuousGalerkin, the $C^0$ interior penalty, and the WOPSIP schemes. Those methods aremodified 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 andpiecewise Sobolev norms for the schemes is outlined and applied to thebiharmonic equation. Three errors are also equivalent in some particulardiscrete norm from [Carstensen, Gallistl, Nataraj: Comparison results ofnonstandard $P_2$ finite element methods for the biharmonic problem, ESAIMMath. Model. Numer. Anal. (2015)] without data oscillations. This paper extendsthe work [Veeser, Zanotti: Quasi-optimal nonconforming methods for symmetricelliptic problems, SIAM J. Numer. Anal. 56 (2018)] to the discontinuousGalerkin scheme and adds error estimates in weaker and piecewise Sobolev norms.

Lowest-order equivalent nonstandard finite element methods for biharmonic plates