Johnson-Mercier Elasticity Element Study in Linear Elasticity with Symmetric Stresses
Core Concepts
Optimal error estimates and improved convergence rates in numerical solutions for linear elasticity using the Johnson-Mercier element.
Abstract
Introduction
Study of mixed methods for linear elasticity with symmetric stresses.
Challenges in finding suitable stress and displacement spaces satisfying stability conditions.
Two-Dimensional Elastostatic Problem
Introduction of the Johnson-Mercier element on triangular meshes.
Displacement space composed of piecewise linear functions.
Higher Dimensional Versions
Investigation of higher dimensional versions on Alfeld splits.
Extension to three dimensions by Kˇr´ıˇzek in 1982 before Alfeld's work.
Unisolvency Proofs
Unisolvency proof of the 3D version of Johnson-Mercier stress element.
Finite Element Discretization
Canonical finite element discretization and unisolvency degrees of freedom for the space Σh(T).
Error Analysis
Error estimates for numerical solutions and superconvergent error estimates via post-processing.
Piecewise Constant Displacements
Replacement of displacement space Vh with Wh to exact satisfaction of equilibrium equation.
Robustness in Incompressible Limit
Convergence analysis for nearly incompressible isotropic materials satisfying weaker ellipticity inequality.
The Johnson-Mercier elasticity element in any dimensions
"Decades of research have illuminated the nontrivial difficulties in finding such a pair of spaces, Σh × Vh."
"Composite stress elements are interesting because they promise liberation from the necessity of vertex degrees of freedom when using polynomial elements."