An Equilibration Based A Posteriori Error Estimate for the Biharmonic Equation and Two Finite Element Methods
Provides a new error estimation technique for finite element methods solving the biharmonic equation, which is important for computational mechanics and numerical analysis.
The paper develops an a posteriori error estimator for the biharmonic equation using the two-energies principle, requiring a locally computed equilibrated moment tensor. The method is applied to both the Interior Penalty Discontinuous Galerkin and Hellan-Herrmann-Johnson formulations.
We develop an a posteriori error estimator 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 consists of symmetric tensor fields with continuous normal-normal components. It is known from the Hellan-Herrmann-Johnson (HHJ) mixed formulation. We propose a construction that is totally local. The procedure can also be applied to the original HHJ formulation, which directly provides an equilibrated moment tensor.