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

TL;DR

This paper introduces a new a posteriori error estimator for biharmonic equations using an equilibration approach, applicable to finite element methods, with a focus on local construction of equilibrated moment tensors.

Contribution

It presents a novel local equilibration-based error estimator for biharmonic equations, applicable to interior penalty discontinuous Galerkin and HHJ mixed formulations.

Findings

Provides a reliable error bound based on the two-energies principle.

Enables local computation of equilibrated moment tensors.

Applicable to multiple finite element formulations.

Abstract

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.