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

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.

Goal-oriented a posteriori error estimation for the biharmonic problem based on an equilibrated moment tensor

“…(5.3) By the above Lemma 5.1 and following [6], we have the efficiency result: Lemma 5.2. Let 𝑢 IP be the discrete solution of (5.1) and 𝝈 eq ℎ be of (5.3).…”

“…We obtain an approximate solution by the 𝐶 0 interior penalty method (𝐶 0 IPDG); see [9,18,6]. Define the polynomial space for 𝐶 0 IPDG by…”

“…It is well known that for sufficiently large 𝜎 = 𝑂((𝑘 + 1) 2 ), there exists a positive constant 𝛽 such that the following coercivity result holds (see [9,18,6]): Construction of equilibrated moment tensor. We follow [6] for the construction of an equilibrated moment tensor. Define the symmetric piecewise polynomial tensor fields of order 𝑘 − 1 with the continuous normal-normal component…”

“…The construction of the potential reconstruction of Definition 3.1 can be found in [7,14,8,20], and the equilibrated moment tensor of Definition 3.2 can be found in [5,6]. We briefly describe their constructions in Section 5.…”

The Hellan–Herrmann–Johnson method for nonlinear shells

“…The HHJ method for fourth order Kirchhoff plates has been developed and analyzed in [22,23,27]. Later work has been done in the 80s [13,1], 90s [44] and recently after 20 years [11,25,7]. It overcomes the issue of C 1 -conformity by introducing the moment tensor as an additional tensor field leading to a mixed method.…”