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We develop an a posteriori error estimator for the Interior PenaltyDiscontinuous Galerkin approximation of the biharmonic equation with continuousfinite elements. The error bound is based on the two-energies principle andrequires the computation of an equilibrated moment tensor. The natural spacefor the moment tensor consists of symmetric tensor fields with continuousnormal-normal components. It is known from the Hellan-Herrmann-Johnson (HHJ)mixed formulation. We propose a construction that is totally local. Theprocedure can also be applied to the original HHJ formulation, which directlyprovides an equilibrated moment tensor.

An Equilibration Based A Posteriori Error Estimate for the Biharmonic Equation and Two Finite Element Methods