Weakly symmetric stress equilibration and a posteriori error estimation for linear elasticity

TL;DR

This paper introduces a stress equilibration method for linear elasticity that provides reliable a posteriori error estimates, especially effective for nearly incompressible materials, using weakly symmetric stress reconstructions.

Contribution

It develops a novel weakly symmetric stress reconstruction technique based on finite element approximations, ensuring guaranteed error bounds and local efficiency for nearly incompressible elasticity problems.

Findings

Provides guaranteed upper bounds for error estimates.

Ensures local efficiency uniformly in the incompressible limit.

Constructs stress reconstructions using Raviart-Thomas elements with weak symmetry.

Abstract

A stress equilibration procedure for linear elasticity is proposed and analyzed in this paper with emphasis on the behavior for (nearly) incompressible materials. Based on the displacement-pressure approximation computed with a stable finite element pair, it constructs an $H (\text{div})$-conforming, weakly symmetric stress reconstruction. Our focus is on the Taylor-Hood combination of continuous finite element spaces of polynomial degrees $k+1$ and $k$ for the displacement and the pressure, respectively. Our construction leads then to reconstructed stresses by Raviart-Thomas elements of degree $k$ which are weakly symmetric in the sense that its anti-symmetric part is zero tested against continuous piecewise polynomial functions of degree $k$. The computation is performed locally on a set of vertex patches covering the computational domain in the spirit of equilibration \cite{BraSch:08}. Due to the weak symmetry constraint, the local problems need to satisfy consistency conditions associated with all rigid body modes, in contrast to the case of Poisson's equation where only the constant modes are involved. The resulting error estimator is shown to constitute a guaranteed upper bound for the error with a constant that depends only on the shape regularity of the triangulation. Local efficiency, uniformly in the incompressible limit, is deduced from the upper bound by the residual error estimator.