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 residual-based a posteriori error estimates. The abstract framework applies to Morley, two versions of discontinuous Galerkin, $C^0$ interior penalty, as well as weakly over-penalized symmetric interior penalty schemes for the biharmonic equation with a general source term in $H^{-2}(\Omega)$.

翻译：抽象性质（H）是近期研究[Lowest-order equivalent nonstandard finite element methods for biharmonic plates, Carstensen and Nataraj, M2AN, 2022]中多种非标准有限元方法在（离散）能量范数下进行完整先验误差分析的关键。本文探究了性质（H）对后验误差分析的影响，并建立了已知和新型显式残差型后验误差估计。该抽象框架适用于Morley元、两种间断伽辽金格式、$C^0$内罚格式以及弱超罚对称内罚格式，求解带有一般源项$H^{-2}(\Omega)$的双调和方程。