Partial Relaxation of 𝐶0 Vertex Continuity of Stresses of Conforming Mixed Finite Elements for the Elasticity Problem
AbstractA 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},\ldots,\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}_{e}} 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: for such an internal vertex {\boldsymbol{x}_{e}} located at the coarse edge e with the unit tangential vector {t_{e}} and the unit normal vector {n_{e}=t_{e}^{\perp}}, the pure tangential component basis function {\varphi_{\boldsymbol{x}_{e}}(\boldsymbol{x})t_{e}t_{e}^{T}} of the original discrete stress space associated to vertex {\boldsymbol{x}_{e}} is split into two basis functions {\varphi_{\boldsymbol{x}_{e}}^{+}(\boldsymbol{x})t_{e}t_{e}^{T}} and {\varphi_{\boldsymbol{x}_{e}}^{-}(\boldsymbol{x})t_{e}t_{e}^{T}} along edge e, where {\varphi_{\boldsymbol{x}_{e}}(\boldsymbol{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 {\mathcal{T}_{\ell}} with {\varphi_{\boldsymbol{x}_{e}}^{+}(\boldsymbol{x})} and {\varphi_{\boldsymbol{x}_{e}}^{-}(\boldsymbol{x})} denoted its two restrictions on two sides of e, respectively. Since the remaining two basis functions {\varphi_{\boldsymbol{x}_{e}}(\boldsymbol{x})n_{e}n_{e}^{T}}, {\varphi_{\boldsymbol{x}_{e}}(\boldsymbol{x})(n_{e}t_{e}^{T}+t_{e}n_{e}^{T})} are the same as those associated to {\boldsymbol{x}_{e}} of the original discrete stress space, the number of the global basis functions associated to {\boldsymbol{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 {\mathcal{T}_{\ell}} is still a {H(\operatorname{div})} subspace, the pure tangential component along the coarse edge e of discrete stresses in it is not necessarily continuous at such vertices like {\boldsymbol{x}_{e}}. 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.