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

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 $\mathbf{x}$ 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.