An analysis of the TDNNS method using natural norms

TL;DR

This paper provides a rigorous mathematical analysis of the TDNNS finite element method for mixed elasticity, establishing stability, trace operators, and optimal error estimates in natural function spaces.

Contribution

It introduces a comprehensive theoretical framework for TDNNS, including stability and error analysis using H(curl) and H(div div) spaces, addressing previous gaps in the method's analysis.

Findings

The TDNNS method is stable in H(curl) and H(div div) norms.

Optimal order a-priori error estimates are established.

Trace operators for normal-normal stress are defined and analyzed.

Abstract

The Tangential-Displacement Normal-Normal-Stress (TDNNS) method is a finite element method for mixed elasticity. As the name suggests, the tangential component of the displacement vector as well as the normal-normal component of the stress are the degrees of freedom of the finite elements. The TDNNS method was shown to converge of optimal order, and to be robust with respect to shear and volume locking. However, the method is slightly nonconforming, and an analysis with respect to the natural norms of the arising spaces was still missing. We present a sound mathematical theory of the infinite dimensional problem using the space H(curl) for the displacement. We define the space H(div div) for the stresses and provide trace operators for the normal-normal stress. Moreover, the finite element problem is shown to be stable with respect to the H(curl) and a discrete H(div div) norm. A-priori error estimates of optimal order with respect to these norms are obtained.