Abstract
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 đŻ 1 , ⊠, đŻ N {\mathcal{T}_{1},\ldots,\mathcal{T}_{N}} which are successively refined from an initial mesh đŻ 0 {\mathcal{T}_{0}} through a newest vertex bisection strategy, admit a crucial hierarchical structure, namely, a newly added vertex đ e {\boldsymbol{x}_{e}} of the mesh đŻ â {\mathcal{T}_{\ell}} is the midpoint of an edge e of the coarse mesh đŻ â - 1 {\mathcal{T}_{\ell-1}} . Such a hierarchical structure is explored to partially relax the C 0 {C^{0}} vertex continuity of symmetric matrix-valued functions in the discrete stress space of the original element on đŻ â {\mathcal{T}_{\ell}} and results in an extended discrete stress space: for such an internal vertex đ e {\boldsymbol{x}_{e}} located at the coarse edge e with the unit tangential vector t e {t_{e}} and the unit normal vector n e = t e â„ {n_{e}=t_{e}^{\perp}} , the pure tangential component basis function Ï đ e âą ( đ ) âą t e âą t e T {\varphi_{\boldsymbol{x}_{e}}(\boldsymbol{x})t_{e}t_{e}^{T}} of the original discrete stress space associated to vertex đ e {\boldsymbol{x}_{e}} is split into two basis functions Ï đ e + âą ( đ ) âą t e âą t e T {\varphi_{\boldsymbol{x}_{e}}^{+}(\boldsymbol{x})t_{e}t_{e}^{T}} and Ï đ e - âą ( đ ) âą t e âą t e T {\varphi_{\boldsymbol{x}_{e}}^{-}(\boldsymbol{x})t_{e}t_{e}^{T}} along edge e, where Ï đ e âą ( đ ) {\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 đŻ â {\mathcal{T}_{\ell}} with Ï đ e + âą ( đ ) {\varphi_{\boldsymbol{x}_{e}}^{+}(\boldsymbol{x})} and Ï đ e - âą ( đ ) {\varphi_{\boldsymbol{x}_{e}}^{-}(\boldsymbol{x})} denoted its two restrictions on two sides of e, respectively. Since the remaining two basis functions Ï đ e âą ( đ ) âą n e âą n e T {\varphi_{\boldsymbol{x}_{e}}(\boldsymbol{x})n_{e}n_{e}^{T}} , Ï đ e âą ( đ ) âą ( n e âą t e T + t e âą 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 đ e {\boldsymbol{x}_{e}} of the original discrete stress space, the number of the global basis functions associated to đ e {\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 đŻ â {\mathcal{T}_{\ell}} is still a H âą ( div ) {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 đ e {\boldsymbol{x}_{e}} . A feature of this extended discrete stress space is its nestedness in the sense that a space on a coarse mesh đŻ {\mathcal{T}} is a subspace of a space on any refinement đŻ ^ {\hat{\mathcal{T}}} of đŻ {\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.
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