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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  (Ω).

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