An equilibration-based a posteriori error bound for the biharmonic equation and two finite element methods

Abstract

Abstract We develop an a posteriori error bound for the interior penalty discontinuous Galerkin approximation of the biharmonic equation with continuous finite elements. The error bound is based on the two-energies principle and requires the computation of an equilibrated moment tensor. The natural space for the moment tensor is that of symmetric tensor fields with continuous normal-normal components, and is well-known from the Hellan-Herrmann-Johnson mixed formulation. We propose a construction that is totally local. The procedure can also be applied to the original Hellan–Herrmann–Johnson formulation, which directly provides an equilibrated moment tensor.