The Johnson-Krizek-Mercier elasticity element in any dimensions

TL;DR

This paper generalizes the Johnson-Krizek-Mercier elasticity element to higher dimensions, introduces reduced stress spaces in 3D, and provides optimal error estimates for numerical solutions.

Contribution

It extends mixed finite element methods for linear elasticity to any dimension and offers new reduced stress spaces with proven optimal error bounds.

Findings

Generalization of Johnson--Krizek-Mercier element to higher dimensions

Reduction of stress space to 24 degrees of freedom in 3D

Proofs of optimal and improved error estimates

Abstract

Mixed methods for linear elasticity with strongly symmetric stresses of lowest order are studied in this paper. On each simplex, the stress space has piecewise linear components with respect to its Alfeld split (which connects the vertices to barycenter), generalizing the Johnson--Mercier two-dimensional element to higher dimensions. Further reductions in the stress space in the three-dimensional case (to 24 degrees of freedom per tetrahedron) are possible when the displacement space is reduced to local rigid displacements. Proofs of optimal error estimates of numerical solutions and improved error estimates via postprocessing and the duality argument are presented.