Partial Relaxation of 𝐶0 Vertex Continuity of Stresses of Conforming Mixed Finite Elements for the Elasticity Problem

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 𝒯 1 , … , 𝒯 N {\mathcal{T}_{1},\ldots,\mathcal{T}_{N}} which are successively refined from an initial mesh 𝒯 0 {\mathcal{T}_{0}} through a newest vertex bisection strategy, admit a crucial hierarchical structure, namely, a newly added vertex 𝒙 e {\boldsymbol{x}_{e}} of the mesh 𝒯 ℓ {\mathcal{T}_{\ell}} is the midpoint of an edge e of the coarse mesh 𝒯 ℓ - 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 𝒯 ℓ {\mathcal{T}_{\ell}} and results in an extended discrete stress space: for such an internal vertex 𝒙 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 ⊥ {n_{e}=t_{e}^{\perp}} , the pure tangential component basis function φ 𝒙 e ⁢ ( 𝒙 ) ⁢ t e ⁢ t e T {\varphi_{\boldsymbol{x}_{e}}(\boldsymbol{x})t_{e}t_{e}^{T}} of the original discrete stress space associated to vertex 𝒙 e {\boldsymbol{x}_{e}} is split into two basis functions φ 𝒙 e + ⁢ ( 𝒙 ) ⁢ t e ⁢ t e T {\varphi_{\boldsymbol{x}_{e}}^{+}(\boldsymbol{x})t_{e}t_{e}^{T}} and φ 𝒙 e - ⁢ ( 𝒙 ) ⁢ t e ⁢ t e T {\varphi_{\boldsymbol{x}_{e}}^{-}(\boldsymbol{x})t_{e}t_{e}^{T}} along edge e, where φ 𝒙 e ⁢ ( 𝒙 ) {\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 𝒯 ℓ {\mathcal{T}_{\ell}} with φ 𝒙 e + ⁢ ( 𝒙 ) {\varphi_{\boldsymbol{x}_{e}}^{+}(\boldsymbol{x})} and φ 𝒙 e - ⁢ ( 𝒙 ) {\varphi_{\boldsymbol{x}_{e}}^{-}(\boldsymbol{x})} denoted its two restrictions on two sides of e, respectively. Since the remaining two basis functions φ 𝒙 e ⁢ ( 𝒙 ) ⁢ n e ⁢ n e T {\varphi_{\boldsymbol{x}_{e}}(\boldsymbol{x})n_{e}n_{e}^{T}} , φ 𝒙 e ⁢ ( 𝒙 ) ⁢ ( n e ⁢ t e T + t 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 𝒙 e {\boldsymbol{x}_{e}} of the original discrete stress space, the number of the global basis functions associated to 𝒙 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 𝒯 ℓ {\mathcal{T}_{\ell}} is still a H ⁢ ( 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 𝒙 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 𝒯 {\mathcal{T}} is a subspace of a space on any refinement 𝒯 ^ {\hat{\mathcal{T}}} of 𝒯 {\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.