Partial relaxation of C⁰ vertex continuity of stresses of conforming mixed finite elements for the elasticity problem

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 T₁,..., T_N which are successively refined from an initial mesh T₀ through a newest vertex bisection strategy, admit a crucial hierarchical structure, namely, a newly added vertex x_e of the mesh T_ℓ is the midpoint of an edge e of the coarse mesh T_ℓ−1. Such a hierarchical structure is explored to partially relax the C⁰ vertex continuity of symmetric matrix-valued functions in the discrete stress space of the original element on T_ℓ and results in an extended discrete stress space: for such an internal vertex x_e located at the coarse edge e with the unit tangential vector t_e and the unit normal vector n_e = t⊥_e, the pure tangential component basis function φ_xe(x)t_et^T_e of the original discrete stress space associated to vertex x_e is split into two basis functions φ⁺_xe(x)t_et^T_e and φ⁻_xe(x)t_et^T_e along edge e, where φ_xe(x) is the nodal basis function of the scalar-valued Lagrange element of order k (k is equal to the polynomial degree of the discrete stress) on T_ℓ with φ⁺_xe(x) and φ⁻_xe(x) denoted its two restrictions on two sides of e, respectively. Since the remaining two basis functions φ_xe(x)n_en^T_e, φ_xe(x)(n_et^T_e + t_en^T_e) are the same as those associated to x_e of the original discrete stress space, the number of the global basis functions associated to x_e of the extended discrete stress space becomes four rather than three (for the original discrete stress space). As a result, though the extended discrete stress space on T_ℓ is still a H(div) subspace, the pure tangential component along the coarse edge e of discrete stresses in it is not necessarily continuous at such vertices like x_e. A feature of this extended discrete stress space is its nestedness in the sense that a space on a coarse mesh T is a subspace of a space on any refinement T of 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.