Lower order mixed elements for the linear elasticity problem in 2D and 3D

TL;DR

This paper introduces new lower order mixed finite elements for 2D and 3D linear elasticity problems, demonstrating stability and optimal convergence through macro-element techniques and confirming results with numerical experiments.

Contribution

The paper constructs novel lower order mixed elements for elasticity in 2D and 3D, enriching existing stress spaces with macro-element bubbles and establishing stability and convergence.

Findings

Discrete stability and optimal convergence proved for new elements.

Numerical experiments confirm theoretical convergence rates.

Extension of stability results to 3D mixed elements.

Abstract

In this paper, we construct two lower order mixed elements for the linear elasticity problem in the Hellinger-Reissner formulation, one for the 2D problem and one for the 3D problem, both on macro-element meshes. The discrete stress spaces enrich the analogous $P_k$ stress spaces in [J. Hu and S. Zhang, arxiv, 2014, J. Hu and S. Zhang, Sci. China Math., 2015] with simple macro-element bubble functions, and the discrete displacement spaces are discontinuous piecewise $P_{k-1}$ polynomial spaces, with $k=2,3$, respectively. Discrete stability and optimal convergence is proved by using the macro-element technique. As a byproduct, the discrete stability and optimal convergence of the $P_2-P_1$ mixed element in [L. Chen and X. Huang, SIAM J. Numer. Anal., 2022] in 3D is proved on another macro-element mesh. For the mixed element in 2D, an $H^2$-conforming composite element is constructed and an exact discrete elasticity sequence is presented. Numerical experiments confirm the theoretical results.