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

Abstract 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 residualbased a posteriori error estimates. The abstract framework applies to Morley, two versions of discontinuous Galerkin, C0 interior penalty, as well as weakly overpenalized symmetric interior penalty schemes for the biharmonic equation with a general source term in H−2(Ω).