Partial relaxation of C^0 vertex continuity of stresses of conforming mixed finite elements for the elasticity problem

TL;DR

This paper introduces a partially relaxed $C^0$ vertex continuity in mixed finite elements for elasticity, enabling nested stress spaces and convergence proofs, with demonstrated effectiveness on adaptive meshes.

Contribution

It extends a conforming mixed finite element by relaxing vertex continuity, creating nested stress spaces that facilitate convergence analysis.

Findings

Nested stress spaces enable convergence proofs.

The method performs well on adaptive meshes.

Numerical experiments confirm effectiveness.

Abstract

A conforming triangular mixed element recently proposed by Hu and Zhang for linear elasticity is extended by rearranging the global degrees of freedom. More precisely, adaptive meshes $\mathcal{T}_1$, $\cdots$, $\mathcal{T}_N$ which are successively refined from an initial mesh $\mathcal{T}_0$ through a newest vertex bisection strategy, admit a crucial hierarchical structure, namely, a newly added vertex $\boldsymbol{x}$ of the mesh $\mathcal{T}_\ell$ is the midpoint of an edge $e$ of the coarse mesh $\mathcal{T}_{\ell-1}$. Such a hierarchical structure is explored to partially relax the $C^0$ vertex continuity of symmetric matrix-valued functions in the discrete stress space of the original element on $\mathcal{T}_\ell$ and results in an extended discrete stress space. A feature of this extended discrete stress space is its nestedness in the sense that a space on a coarse mesh $\mathcal{T}$ is a subspace of a space on any refinement $\hat{\mathcal{T}}$ of $\mathcal{T}$, which allows a proof of convergence of a standard adaptive algorithm. The idea is extended to impose a general traction boundary condition on the discrete level. Numerical experiments are provided to illustrate performance on both uniform and adaptive meshes.