New low-order mixed finite element methods for linear elasticity

TL;DR

This paper develops new low-order, $H(\operatorname{div})$-conforming finite elements for symmetric tensors in arbitrary dimensions by enriching $\mathbb{P}_2(K;\mathbb{S})$ with the $(d+1)$-order normal-normal face bubble, yielding a reduced space with $d(d+1)^2$ DoFs. It constructs two- and higher-dimensional elasticity complexes that couple stress and displacement through Bell-type discretizations and yields robust mixed finite element methods for linear elasticity with uniform $\lambda$-robust error and displacement superconvergence. A reduced variant is also formulated to further lower computational cost while preserving stability via exact discrete sequences and commuting projections. Numerical results on the unit square confirm the predicted convergence rates under various $\lambda$ and demonstrate the robustness of the methods, including the displacement superconvergence in the non-reduced scheme. Collectively, the work provides dimension-agnostic, provably stable and accurate low-order finite element tools for symmetric-tensor elasticity problems with practical implementation guidance.

Abstract

New low-order $H(\textrm{div})$-conforming finite elements for symmetric tensors are constructed in arbitrary dimension. The space of shape functions is defined by enriching the symmetric quadratic polynomial space with the $(d+1)$-order normal-normal face bubble space. The reduced counterpart has only $d(d+1)^2$ degrees of freedom. Basis functions are explicitly given in terms of barycentric coordinates. Low-order conforming finite element elasticity complexes starting from the Bell element, are developed in two dimensions. These finite elements for symmetric tensors are applied to devise robust mixed finite element methods for the linear elasticity problem, which possess the uniform error estimates with respect to the Lamé coefficient $λ$, and superconvergence for the displacement. Numerical results are provided to verify the theoretical convergence rates.

Figures (2)

• Figure 1: The finite element elasticity complex \ref{['disstresscomplex']} on a triangle. The DoFs $\nabla v(\texttt{v})$ and $\nabla^2 v(\texttt{v})$ at each vertex are shown by one circle.
• Figure 2: The reduced finite element elasticity complex \ref{['reduceddisstresscomplex']} on a triangle.

Theorems & Definitions (42)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof