Partial Relaxation of 𝐶⁰ 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 𝒯 1 , … , 𝒯 N which are successively refined from an initial mesh 𝒯 0 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 C 0 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 𝒙 e ⁢ ( 𝒙 ) ⁢ t e ⁢ t e T of the original discrete stress space associated to vertex 𝒙 e is split into two basis functions φ 𝒙 e + ⁢ ( 𝒙 ) ⁢ t e ⁢ t e T and φ 𝒙 e - ⁢ ( 𝒙 ) ⁢ t e ⁢ t e T along edge e, where φ 𝒙 e ⁢ ( 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 𝒙 e + ⁢ ( 𝒙 ) and φ 𝒙 e - ⁢ ( 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 , φ 𝒙 e ⁢ ( 𝒙 ) ⁢ ( n e ⁢ t e T + t e ⁢ n e T ) 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.