Research Article

A two-energies principle for the biharmonic equation and an a posteriori error estimator for an interior penalty discontinuous Galerkin approximation^⋆

¹ Faculty of Mathematics, Ruhr-University, 44780 Bochum, Germany
² Dept. of Math. Univ. of Houston Houston, TX 77204-3008, U.S.A
³ Inst. of Math., Univ. of Augsburg, 86159 Augsburg, Germany

^* Corresponding author: rohop@math.uh.edu; hoppe@math.uni-augsburg.de

Received: 30 August 2016
Accepted: 23 November 2016

Abstract

We consider an a posteriori error estimator for the Interior Penalty Discontinuous Galerkin (IPDG) approximation of the biharmonic equation based on the Hellan-Herrmann-Johnson (HHJ) mixed formulation. The error estimator is derived from a two-energies principle for the HHJ formulation and amounts to the construction of an equilibrated moment tensor which is done by local interpolation. The reliability estimate is a direct consequence of the two-energies principle and does not involve generic constants. The efficiency of the estimator follows by showing that it can be bounded from above by a residual-type estimator known to be efficient. A documentation of numerical results illustrates the performance of the estimator.